Theorem mulcand 10660

Description: Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
mulcand.1	|- ( ph -> A e. CC )
mulcand.2	|- ( ph -> B e. CC )
mulcand.3	|- ( ph -> C e. CC )
mulcand.4	|- ( ph -> C =/= 0 )

Assertion

Ref	Expression
mulcand	|- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) )

Proof of Theorem mulcand

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcand.3	. . . 4 |- ( ph -> C e. CC )
2		mulcand.4	. . . 4 |- ( ph -> C =/= 0 )
3		recex 10659	. . . 4 |- ( ( C e. CC /\ C =/= 0 ) -> E. x e. CC ( C x. x ) = 1 )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ph -> E. x e. CC ( C x. x ) = 1 )
5		oveq2 6658	. . . 4 |- ( ( C x. A ) = ( C x. B ) -> ( x x. ( C x. A ) ) = ( x x. ( C x. B ) ) )
6		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> x e. CC )
7	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> C e. CC )
8	6, 7	mulcomd 10061	. . . . . . . 8 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. C ) = ( C x. x ) )
9		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( C x. x ) = 1 )
10	8, 9	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. C ) = 1 )
11	10	oveq1d 6665	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. A ) = ( 1 x. A ) )
12		mulcand.1	. . . . . . . 8 |- ( ph -> A e. CC )
13	12	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> A e. CC )
14	6, 7, 13	mulassd 10063	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. A ) = ( x x. ( C x. A ) ) )
15	13	mulid2d 10058	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. A ) = A )
16	11, 14, 15	3eqtr3d 2664	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. A ) ) = A )
17	10	oveq1d 6665	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. B ) = ( 1 x. B ) )
18		mulcand.2	. . . . . . . 8 |- ( ph -> B e. CC )
19	18	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> B e. CC )
20	6, 7, 19	mulassd 10063	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. B ) = ( x x. ( C x. B ) ) )
21	19	mulid2d 10058	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. B ) = B )
22	17, 20, 21	3eqtr3d 2664	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. B ) ) = B )
23	16, 22	eqeq12d 2637	. . . 4 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. ( C x. A ) ) = ( x x. ( C x. B ) ) <-> A = B ) )
24	5, 23	syl5ib 234	. . 3 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( C x. A ) = ( C x. B ) -> A = B ) )
25	4, 24	rexlimddv 3035	. 2 |- ( ph -> ( ( C x. A ) = ( C x. B ) -> A = B ) )
26		oveq2 6658	. 2 |- ( A = B -> ( C x. A ) = ( C x. B ) )
27	25, 26	impbid1 215	1 |- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  mulcan2d  10661  mulcanad  10662  mulcan  10664  div11  10713  eqneg  10745  qredeq  15371  cncongr1  15381  prmirredlem  19841  tanarg  24365  quad2  24566  atandm2  24604  lgseisenlem2  25101  frrusgrord0  27204