Theorem mulcanapd 8422

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 8414	. . . 4 |- ( ( C e. CC /\ C # 0 ) -> E. x e. CC ( C x. x ) = 1 )
4	1, 2, 3	syl2anc 408	. . 3 |- ( ph -> E. x e. CC ( C x. x ) = 1 )
5		oveq2 5782	. . . 4 |- ( ( C x. A ) = ( C x. B ) -> ( x x. ( C x. A ) ) = ( x x. ( C x. B ) ) )
6		simprl 520	. . . . . . . . 9 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> x e. CC )
7	1	adantr 274	. . . . . . . . 9 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> C e. CC )
8	6, 7	mulcomd 7787	. . . . . . . 8 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. C ) = ( C x. x ) )
9		simprr 521	. . . . . . . 8 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( C x. x ) = 1 )
10	8, 9	eqtrd 2172	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. C ) = 1 )
11	10	oveq1d 5789	. . . . . 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 274	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> A e. CC )
14	6, 7, 13	mulassd 7789	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. A ) = ( x x. ( C x. A ) ) )
15	13	mulid2d 7784	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. A ) = A )
16	11, 14, 15	3eqtr3d 2180	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. A ) ) = A )
17	10	oveq1d 5789	. . . . . 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 274	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> B e. CC )
20	6, 7, 19	mulassd 7789	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. B ) = ( x x. ( C x. B ) ) )
21	19	mulid2d 7784	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. B ) = B )
22	17, 20, 21	3eqtr3d 2180	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. B ) ) = B )
23	16, 22	eqeq12d 2154	. . . 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 153	. . 3 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( C x. A ) = ( C x. B ) -> A = B ) )
25	4, 24	rexlimddv 2554	. 2 |- ( ph -> ( ( C x. A ) = ( C x. B ) -> A = B ) )
26		oveq2 5782	. 2 |- ( A = B -> ( C x. A ) = ( C x. B ) )
27	25, 26	impbid1 141	1 |- ( ph -> ( ( C x. A ) = ( C x. B ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   CCcc 7618   0cc0 7620   1c1 7621   x. cmul 7625   # cap 8343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344

This theorem is referenced by:  mulcanap2d  8423  mulcanapad  8424  mulcanap  8426  div11ap  8460  eqneg  8492  dvdscmulr  11522  qredeq  11777  cncongr1  11784