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Theorem mulcan 5659
Description: Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18.
Hypotheses
Ref Expression
mulcan.1 |- A e. CC
mulcan.2 |- B e. CC
mulcan.3 |- C e. CC
mulcan.4 |- A =/= 0
Assertion
Ref Expression
mulcan |- ((A x. B) = (A x. C) <-> B = C)
Proof of Theorem mulcan
StepHypRef Expression
1 mulcan.1 . . . 4 |- A e. CC
2 mulcan.4 . . . 4 |- A =/= 0
31, 2recex 5658 . . 3 |- E.x e. CC (A x. x) = 1
4 mulcan.2 . . . . . . . . . 10 |- B e. CC
5 axmulass 5250 . . . . . . . . . 10 |- ((x e. CC /\ A e. CC /\ B e. CC) -> ((x x. A) x. B) = (x x. (A x. B)))
64, 5mp3an3 902 . . . . . . . . 9 |- ((x e. CC /\ A e. CC) -> ((x x. A) x. B) = (x x. (A x. B)))
7 mulcan.3 . . . . . . . . . 10 |- C e. CC
8 axmulass 5250 . . . . . . . . . 10 |- ((x e. CC /\ A e. CC /\ C e. CC) -> ((x x. A) x. C) = (x x. (A x. C)))
97, 8mp3an3 902 . . . . . . . . 9 |- ((x e. CC /\ A e. CC) -> ((x x. A) x. C) = (x x. (A x. C)))
106, 9eqeq12d 1481 . . . . . . . 8 |- ((x e. CC /\ A e. CC) -> (((x x. A) x. B) = ((x x. A) x. C) <-> (x x. (A x. B)) = (x x. (A x. C))))
111, 10mpan2 694 . . . . . . 7 |- (x e. CC -> (((x x. A) x. B) = ((x x. A) x. C) <-> (x x. (A x. B)) = (x x. (A x. C))))
12 opreq2 3954 . . . . . . 7 |- ((A x. B) = (A x. C) -> (x x. (A x. B)) = (x x. (A x. C)))
1311, 12syl5bir 210 . . . . . 6 |- (x e. CC -> ((A x. B) = (A x. C) -> ((x x. A) x. B) = ((x x. A) x. C)))
1413adantr 389 . . . . 5 |- ((x e. CC /\ (A x. x) = 1) -> ((A x. B) = (A x. C) -> ((x x. A) x. B) = ((x x. A) x. C)))
15 axmulcom 5248 . . . . . . . . 9 |- ((A e. CC /\ x e. CC) -> (A x. x) = (x x. A))
161, 15mpan 693 . . . . . . . 8 |- (x e. CC -> (A x. x) = (x x. A))
1716eqeq1d 1475 . . . . . . 7 |- (x e. CC -> ((A x. x) = 1 <-> (x x. A) = 1))
18 opreq1 3953 . . . . . . . . 9 |- ((x x. A) = 1 -> ((x x. A) x. B) = (1 x. B))
194mulid2 5305 . . . . . . . . 9 |- (1 x. B) = B
2018, 19syl6eq 1515 . . . . . . . 8 |- ((x x. A) = 1 -> ((x x. A) x. B) = B)
21 opreq1 3953 . . . . . . . . 9 |- ((x x. A) = 1 -> ((x x. A) x. C) = (1 x. C))
227mulid2 5305 . . . . . . . . 9 |- (1 x. C) = C
2321, 22syl6eq 1515 . . . . . . . 8 |- ((x x. A) = 1 -> ((x x. A) x. C) = C)
2420, 23eqeq12d 1481 . . . . . . 7 |- ((x x. A) = 1 -> (((x x. A) x. B) = ((x x. A) x. C) <-> B = C))
2517, 24syl6bi 214 . . . . . 6 |- (x e. CC -> ((A x. x) = 1 -> (((x x. A) x. B) = ((x x. A) x. C) <-> B = C)))
2625imp 350 . . . . 5 |- ((x e. CC /\ (A x. x) = 1) -> (((x x. A) x. B) = ((x x. A) x. C) <-> B = C))
2714, 26sylibd 202 . . . 4 |- ((x e. CC /\ (A x. x) = 1) -> ((A x. B) = (A x. C) -> B = C))
2827r19.23aiva 1736 . . 3 |- (E.x e. CC (A x. x) = 1 -> ((A x. B) = (A x. C) -> B = C))
293, 28ax-mp 7 . 2 |- ((A x. B) = (A x. C) -> B = C)
30 opreq2 3954 . 2 |- (B = C -> (A x. B) = (A x. C))
3129, 30impbi 157 1 |- ((A x. B) = (A x. C) <-> B = C)
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  E.wrex 1638  (class class class)co 3948  CCcc 5204  0cc0 5206  1c1 5207   x. cmul 5211
This theorem is referenced by:  mulcant2 5660  div11 5720  sqr2irrlem1 6654  cjreb 6716  0.999... 7181  ipasslem10 8430
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
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