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Theorem lt2mul2divt 5822
Description: 'Less than' relationship between division and multiplication.
Assertion
Ref Expression
lt2mul2divt |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> ((A x. B) < (C x. D) <-> (A / D) < (C / B)))
Proof of Theorem lt2mul2divt
StepHypRef Expression
1 axmulcom 5248 . . . . . . . . 9 |- ((C e. CC /\ D e. CC) -> (C x. D) = (D x. C))
2 recnt 5285 . . . . . . . . 9 |- (C e. RR -> C e. CC)
3 recnt 5285 . . . . . . . . 9 |- (D e. RR -> D e. CC)
41, 2, 3syl2an 454 . . . . . . . 8 |- ((C e. RR /\ D e. RR) -> (C x. D) = (D x. C))
54opreq1d 3960 . . . . . . 7 |- ((C e. RR /\ D e. RR) -> ((C x. D) / B) = ((D x. C) / B))
653ad2ant2 799 . . . . . 6 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ 0 < B) -> ((C x. D) / B) = ((D x. C) / B))
7 divasst 5704 . . . . . . 7 |- (((D e. CC /\ C e. CC /\ B e. CC) /\ B =/= 0) -> ((D x. C) / B) = (D x. (C / B)))
83ad2antlr 405 . . . . . . . . 9 |- (((C e. RR /\ D e. RR) /\ 0 < B) -> D e. CC)
983adant1 795 . . . . . . . 8 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ 0 < B) -> D e. CC)
102ad2antrl 406 . . . . . . . . 9 |- ((B e. RR /\ (C e. RR /\ D e. RR)) -> C e. CC)
11103adant3 797 . . . . . . . 8 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ 0 < B) -> C e. CC)
12 recnt 5285 . . . . . . . . 9 |- (B e. RR -> B e. CC)
13123ad2ant1 798 . . . . . . . 8 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ 0 < B) -> B e. CC)
149, 11, 133jca 817 . . . . . . 7 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ 0 < B) -> (D e. CC /\ C e. CC /\ B e. CC))
15 gt0ne0t 5592 . . . . . . . 8 |- ((B e. RR /\ 0 < B) -> B =/= 0)
16153adant2 796 . . . . . . 7 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ 0 < B) -> B =/= 0)
177, 14, 16sylanc 471 . . . . . 6 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ 0 < B) -> ((D x. C) / B) = (D x. (C / B)))
186, 17eqtrd 1499 . . . . 5 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ 0 < B) -> ((C x. D) / B) = (D x. (C / B)))
19183adant3r 855 . . . 4 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> ((C x. D) / B) = (D x. (C / B)))
20193adant1l 850 . . 3 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> ((C x. D) / B) = (D x. (C / B)))
2120breq2d 2620 . 2 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> (A < ((C x. D) / B) <-> A < (D x. (C / B))))
22 ltmuldivt 5817 . . 3 |- (((A e. RR /\ B e. RR /\ (C x. D) e. RR) /\ 0 < B) -> ((A x. B) < (C x. D) <-> A < ((C x. D) / B)))
23 pm3.26 319 . . . . 5 |- ((A e. RR /\ B e. RR) -> A e. RR)
24233ad2ant1 798 . . . 4 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> A e. RR)
25 pm3.27 323 . . . . 5 |- ((A e. RR /\ B e. RR) -> B e. RR)
26253ad2ant1 798 . . . 4 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> B e. RR)
27 axmulrcl 5246 . . . . 5 |- ((C e. RR /\ D e. RR) -> (C x. D) e. RR)
28273ad2ant2 799 . . . 4 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> (C x. D) e. RR)
2924, 26, 283jca 817 . . 3 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> (A e. RR /\ B e. RR /\ (C x. D) e. RR))
30 pm3.26 319 . . . 4 |- ((0 < B /\ 0 < D) -> 0 < B)
31303ad2ant3 800 . . 3 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> 0 < B)
3222, 29, 31sylanc 471 . 2 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> ((A x. B) < (C x. D) <-> A < ((C x. D) / B)))
33 ltdivmult 5819 . . 3 |- (((A e. RR /\ D e. RR /\ (C / B) e. RR) /\ 0 < D) -> ((A / D) < (C / B) <-> A < (D x. (C / B))))
34 pm3.27 323 . . . . 5 |- ((C e. RR /\ D e. RR) -> D e. RR)
35343ad2ant2 799 . . . 4 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> D e. RR)
36 redivclt 5756 . . . . . . . . 9 |- ((C e. RR /\ B e. RR /\ B =/= 0) -> (C / B) e. RR)
37153adant1 795 . . . . . . . . 9 |- ((C e. RR /\ B e. RR /\ 0 < B) -> B =/= 0)
3836, 37syl3dan3 868 . . . . . . . 8 |- ((C e. RR /\ B e. RR /\ 0 < B) -> (C / B) e. RR)
39383com12 835 . . . . . . 7 |- ((B e. RR /\ C e. RR /\ 0 < B) -> (C / B) e. RR)
40393adant3r 855 . . . . . 6 |- ((B e. RR /\ C e. RR /\ (0 < B /\ 0 < D)) -> (C / B) e. RR)
41403adant2r 853 . . . . 5 |- ((B e. RR /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> (C / B) e. RR)
42413adant1l 850 . . . 4 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> (C / B) e. RR)
4324, 35, 423jca 817 . . 3 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> (A e. RR /\ D e. RR /\ (C / B) e. RR))
44 pm3.27 323 . . . 4 |- ((0 < B /\ 0 < D) -> 0 < D)
45443ad2ant3 800 . . 3 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> 0 < D)
4633, 43, 45sylanc 471 . 2 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> ((A / D) < (C / B) <-> A < (D x. (C / B))))
4721, 32, 463bitr4d 548 1 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> ((A x. B) < (C x. D) <-> (A / D) < (C / B)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577   class class class wbr 2609  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206   x. cmul 5211   / cdiv 5266   < clt 5458
This theorem is referenced by:  efaddlem22 7301
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672
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