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Theorem sqrlem12 6614
Description: Lemma for square root theorem.
Hypotheses
Ref Expression
sqrlem1.1 |- A e. RR
sqrlem1.2 |- 0 < A
sqrlem9.3 |- B e. RR
sqrlem9.4 |- C e. RR
sqrlem9.5 |- 0 < B
sqrlem9.6 |- A < (B x. B)
sqrlem9.7 |- C = ((B + (A / B)) / (1 + 1))
sqrlem12.8 |- S = {x e. RR | (0 <_ x /\ (x x. x) <_ A)}
Assertion
Ref Expression
sqrlem12 |- (D e. S -> D < C)
Distinct variable groups:   x,A   x,B   x,S   x,C   x,D
Proof of Theorem sqrlem12
StepHypRef Expression
1 sqrlem1.1 . . . . . 6 |- A e. RR
2 sqrlem1.2 . . . . . 6 |- 0 < A
3 sqrlem12.8 . . . . . 6 |- S = {x e. RR | (0 <_ x /\ (x x. x) <_ A)}
41, 2, 3sqrlem4 6606 . . . . 5 |- (D e. S <-> (D e. RR /\ (0 <_ D /\ (D x. D) <_ A)))
54pm3.27bi 326 . . . 4 |- (D e. S -> (0 <_ D /\ (D x. D) <_ A))
65pm3.26d 321 . . 3 |- (D e. S -> 0 <_ D)
74pm3.26bi 322 . . . 4 |- (D e. S -> D e. RR)
8 0re 5412 . . . . 5 |- 0 e. RR
9 leloet 5491 . . . . 5 |- ((0 e. RR /\ D e. RR) -> (0 <_ D <-> (0 < D \/ 0 = D)))
108, 9mpan 693 . . . 4 |- (D e. RR -> (0 <_ D <-> (0 < D \/ 0 = D)))
117, 10syl 10 . . 3 |- (D e. S -> (0 <_ D <-> (0 < D \/ 0 = D)))
126, 11mpbid 195 . 2 |- (D e. S -> (0 < D \/ 0 = D))
135pm3.27d 325 . . . . . . 7 |- (D e. S -> (D x. D) <_ A)
14 sqrlem9.3 . . . . . . . . 9 |- B e. RR
15 sqrlem9.4 . . . . . . . . 9 |- C e. RR
16 sqrlem9.5 . . . . . . . . 9 |- 0 < B
17 sqrlem9.6 . . . . . . . . 9 |- A < (B x. B)
18 sqrlem9.7 . . . . . . . . 9 |- C = ((B + (A / B)) / (1 + 1))
191, 2, 14, 15, 16, 17, 18sqrlem11 6613 . . . . . . . 8 |- A < (C x. C)
20 axmulrcl 5246 . . . . . . . . . 10 |- ((D e. RR /\ D e. RR) -> (D x. D) e. RR)
2120anidms 434 . . . . . . . . 9 |- (D e. RR -> (D x. D) e. RR)
2215, 15remulcl 5307 . . . . . . . . . 10 |- (C x. C) e. RR
23 lelttrt 5496 . . . . . . . . . 10 |- (((D x. D) e. RR /\ A e. RR /\ (C x. C) e. RR) -> (((D x. D) <_ A /\ A < (C x. C)) -> (D x. D) < (C x. C)))
241, 22, 23mp3an23 905 . . . . . . . . 9 |- ((D x. D) e. RR -> (((D x. D) <_ A /\ A < (C x. C)) -> (D x. D) < (C x. C)))
257, 21, 243syl 20 . . . . . . . 8 |- (D e. S -> (((D x. D) <_ A /\ A < (C x. C)) -> (D x. D) < (C x. C)))
2619, 25mpan2i 697 . . . . . . 7 |- (D e. S -> ((D x. D) <_ A -> (D x. D) < (C x. C)))
2713, 26mpd 26 . . . . . 6 |- (D e. S -> (D x. D) < (C x. C))
2827adantr 389 . . . . 5 |- ((D e. S /\ 0 < D) -> (D x. D) < (C x. C))
291, 2, 14, 15, 16, 17, 18sqrlem9 6611 . . . . . . 7 |- 0 < C
30 breq2 2613 . . . . . . . . . . 11 |- (D = if(D e. RR, D, 0) -> (0 < D <-> 0 < if(D e. RR, D, 0)))
3130anbi1d 615 . . . . . . . . . 10 |- (D = if(D e. RR, D, 0) -> ((0 < D /\ 0 < C) <-> (0 < if(D e. RR, D, 0) /\ 0 < C)))
32 breq1 2612 . . . . . . . . . . 11 |- (D = if(D e. RR, D, 0) -> (D < C <-> if(D e. RR, D, 0) < C))
33 opreq12 3955 . . . . . . . . . . . . 13 |- ((D = if(D e. RR, D, 0) /\ D = if(D e. RR, D, 0)) -> (D x. D) = (if(D e. RR, D, 0) x. if(D e. RR, D, 0)))
3433anidms 434 . . . . . . . . . . . 12 |- (D = if(D e. RR, D, 0) -> (D x. D) = (if(D e. RR, D, 0) x. if(D e. RR, D, 0)))
3534breq1d 2619 . . . . . . . . . . 11 |- (D = if(D e. RR, D, 0) -> ((D x. D) < (C x. C) <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C)))
3632, 35bibi12d 627 . . . . . . . . . 10 |- (D = if(D e. RR, D, 0) -> ((D < C <-> (D x. D) < (C x. C)) <-> (if(D e. RR, D, 0) < C <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C))))
3731, 36imbi12d 624 . . . . . . . . 9 |- (D = if(D e. RR, D, 0) -> (((0 < D /\ 0 < C) -> (D < C <-> (D x. D) < (C x. C))) <-> ((0 < if(D e. RR, D, 0) /\ 0 < C) -> (if(D e. RR, D, 0) < C <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C)))))
388elimel 2384 . . . . . . . . . . 11 |- if(D e. RR, D, 0) e. RR
3938, 15lt2msq 5829 . . . . . . . . . 10 |- ((0 <_ if(D e. RR, D, 0) /\ 0 <_ C) -> (if(D e. RR, D, 0) < C <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C)))
408, 38ltle 5553 . . . . . . . . . 10 |- (0 < if(D e. RR, D, 0) -> 0 <_ if(D e. RR, D, 0))
418, 15ltle 5553 . . . . . . . . . 10 |- (0 < C -> 0 <_ C)
4239, 40, 41syl2an 454 . . . . . . . . 9 |- ((0 < if(D e. RR, D, 0) /\ 0 < C) -> (if(D e. RR, D, 0) < C <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C)))
4337, 42dedth 2373 . . . . . . . 8 |- (D e. RR -> ((0 < D /\ 0 < C) -> (D < C <-> (D x. D) < (C x. C))))
447, 43syl 10 . . . . . . 7 |- (D e. S -> ((0 < D /\ 0 < C) -> (D < C <-> (D x. D) < (C x. C))))
4529, 44mpan2i 697 . . . . . 6 |- (D e. S -> (0 < D -> (D < C <-> (D x. D) < (C x. C))))
4645imp 350 . . . . 5 |- ((D e. S /\ 0 < D) -> (D < C <-> (D x. D) < (C x. C)))
4728, 46mpbird 196 . . . 4 |- ((D e. S /\ 0 < D) -> D < C)
4847ex 373 . . 3 |- (D e. S -> (0 < D -> D < C))
49 breq1 2612 . . . . 5 |- (0 = D -> (0 < C <-> D < C))
5029, 49mpbii 193 . . . 4 |- (0 = D -> D < C)
5150a1i 8 . . 3 |- (D e. S -> (0 = D -> D < C))
5248, 51jaod 424 . 2 |- (D e. S -> ((0 < D \/ 0 = D) -> D < C))
5312, 52mpd 26 1 |- (D e. S -> D < C)
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  {crab 1640  ifcif 2351   class class class wbr 2609  (class class class)co 3948  RRcr 5205  0cc0 5206  1c1 5207   + caddc 5209   x. cmul 5211   / cdiv 5266   <_ cle 5267   < clt 5458
This theorem is referenced by:  sqrlem13 6615
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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