Theorem mulcanapd 8631

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 8623	. . . 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 5896	. . . 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 7992	. . . . . . . 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 2220	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. C ) = 1 )
11	10	oveq1d 5903	. . . . . 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 7994	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. A ) = ( x x. ( C x. A ) ) )
15	13	mulid2d 7989	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. A ) = A )
16	11, 14, 15	3eqtr3d 2228	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. A ) ) = A )
17	10	oveq1d 5903	. . . . . 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 7994	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. B ) = ( x x. ( C x. B ) ) )
21	19	mulid2d 7989	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. B ) = B )
22	17, 20, 21	3eqtr3d 2228	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. B ) ) = B )
23	16, 22	eqeq12d 2202	. . . 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 2609	. 2 |- ( ph -> ( ( C x. A ) = ( C x. B ) -> A = B ) )
26		oveq2 5896	. 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 1363   e. wcel 2158   E.wrex 2466   class class class wbr 4015  (class class class)co 5888   CCcc 7822   0cc0 7824   1c1 7825   x. cmul 7829   # cap 8551

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552

This theorem is referenced by:  mulcanap2d  8632  mulcanapad  8633  mulcanap  8635  div11ap  8670  eqneg  8702  dvdscmulr  11840  qredeq  12109  cncongr1  12116  lgseisenlem2  14691