Theorem mulcanapd 8707

Description: Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)

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
mulcanapd	|- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) )

Proof of Theorem mulcanapd

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		recexap 8699	. . . 4 |- ( ( C e. CC /\ C # 0 ) -> E. x e. CC ( C x. x ) = 1 )
4	1, 2, 3	syl2anc 411	. . 3 |- ( ph -> E. x e. CC ( C x. x ) = 1 )
5		oveq2 5933	. . . 4 |- ( ( C x. A ) = ( C x. B ) -> ( x x. ( C x. A ) ) = ( x x. ( C x. B ) ) )
6		simprl 529	. . . . . . . . 9 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> x e. CC )
7	1	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> C e. CC )
8	6, 7	mulcomd 8067	. . . . . . . 8 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. C ) = ( C x. x ) )
9		simprr 531	. . . . . . . 8 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( C x. x ) = 1 )
10	8, 9	eqtrd 2229	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. C ) = 1 )
11	10	oveq1d 5940	. . . . . 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 276	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> A e. CC )
14	6, 7, 13	mulassd 8069	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. A ) = ( x x. ( C x. A ) ) )
15	13	mulid2d 8064	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. A ) = A )
16	11, 14, 15	3eqtr3d 2237	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. A ) ) = A )
17	10	oveq1d 5940	. . . . . 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 276	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> B e. CC )
20	6, 7, 19	mulassd 8069	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. B ) = ( x x. ( C x. B ) ) )
21	19	mulid2d 8064	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. B ) = B )
22	17, 20, 21	3eqtr3d 2237	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. B ) ) = B )
23	16, 22	eqeq12d 2211	. . . 4 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. ( C x. A ) ) = ( x x. ( C x. B ) ) <-> A = B ) )
24	5, 23	imbitrid 154	. . 3 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( C x. A ) = ( C x. B ) -> A = B ) )
25	4, 24	rexlimddv 2619	. 2 |- ( ph -> ( ( C x. A ) = ( C x. B ) -> A = B ) )
26		oveq2 5933	. 2 |- ( A = B -> ( C x. A ) = ( C x. B ) )
27	25, 26	impbid1 142	1 |- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4034  (class class class)co 5925   CCcc 7896   0cc0 7898   1c1 7899   x. cmul 7903   # cap 8627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628

This theorem is referenced by:  mulcanap2d  8708  mulcanapad  8709  mulcanap  8711  div11ap  8746  eqneg  8778  dvdscmulr  12004  qredeq  12291  cncongr1  12298  lgseisenlem2  15420