Theorem mulcant 5661

Description: Cancellation law for multiplication (full theorem form). Theorem I.7 of [Apostol] p. 18. Illustrates use of dedth 2373 and elimne0 5288.

Assertion

Ref	Expression
mulcant	|- (((A e. CC /\ B e. CC /\ C e. CC) /\ A =/= 0) -> ((A x. B) = (A x. C) <-> B = C))

Proof of Theorem mulcant

Step	Hyp	Ref	Expression
1		eleq1 1526	. . . . 5 |- (A = if(A =/= 0, A, 1) -> (A e. CC <-> if(A =/= 0, A, 1) e. CC))
2	1	3anbi1d 894	. . . 4 |- (A = if(A =/= 0, A, 1) -> ((A e. CC /\ B e. CC /\ C e. CC) <-> (if(A =/= 0, A, 1) e. CC /\ B e. CC /\ C e. CC)))
3		opreq1 3953	. . . . . 6 |- (A = if(A =/= 0, A, 1) -> (A x. B) = (if(A =/= 0, A, 1) x. B))
4		opreq1 3953	. . . . . 6 |- (A = if(A =/= 0, A, 1) -> (A x. C) = (if(A =/= 0, A, 1) x. C))
5	3, 4	eqeq12d 1481	. . . . 5 |- (A = if(A =/= 0, A, 1) -> ((A x. B) = (A x. C) <-> (if(A =/= 0, A, 1) x. B) = (if(A =/= 0, A, 1) x. C)))
6	5	bibi1d 617	. . . 4 |- (A = if(A =/= 0, A, 1) -> (((A x. B) = (A x. C) <-> B = C) <-> ((if(A =/= 0, A, 1) x. B) = (if(A =/= 0, A, 1) x. C) <-> B = C)))
7	2, 6	imbi12d 624	. . 3 |- (A = if(A =/= 0, A, 1) -> (((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) = (A x. C) <-> B = C)) <-> ((if(A =/= 0, A, 1) e. CC /\ B e. CC /\ C e. CC) -> ((if(A =/= 0, A, 1) x. B) = (if(A =/= 0, A, 1) x. C) <-> B = C))))
8		elimne0 5288	. . . 4 |- if(A =/= 0, A, 1) =/= 0
9	8	mulcant2 5660	. . 3 |- ((if(A =/= 0, A, 1) e. CC /\ B e. CC /\ C e. CC) -> ((if(A =/= 0, A, 1) x. B) = (if(A =/= 0, A, 1) x. C) <-> B = C))
10	7, 9	dedth 2373	. 2 |- (A =/= 0 -> ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) = (A x. C) <-> B = C)))
11	10	impcom 351	1 |- (((A e. CC /\ B e. CC /\ C e. CC) /\ A =/= 0) -> ((A x. B) = (A x. C) <-> B = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577  ifcif 2351  (class class class)co 3948  CCcc 5204  0cc0 5206  1c1 5207   x. cmul 5211

This theorem is referenced by:  mulcan2t 5662  mul0or 5663  conjmult 5753  sq01t 6582  ipasslem4 8424  eff1i 8665

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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