Theorem mulcanapd 8900

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 8892	. . . 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 6036	. . . 4 |- ( ( C x. A ) = ( C x. B ) -> ( x x. ( C x. A ) ) = ( x x. ( C x. B ) ) )
6		simprl 531	. . . . . . . . 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 8260	. . . . . . . 8 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. C ) = ( C x. x ) )
9		simprr 533	. . . . . . . 8 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( C x. x ) = 1 )
10	8, 9	eqtrd 2264	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. C ) = 1 )
11	10	oveq1d 6043	. . . . . 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 8262	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. A ) = ( x x. ( C x. A ) ) )
15	13	mullidd 8257	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. A ) = A )
16	11, 14, 15	3eqtr3d 2272	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. A ) ) = A )
17	10	oveq1d 6043	. . . . . 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 8262	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( ( x x. C ) x. B ) = ( x x. ( C x. B ) ) )
21	19	mullidd 8257	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( 1 x. B ) = B )
22	17, 20, 21	3eqtr3d 2272	. . . . 5 |- ( ( ph /\ ( x e. CC /\ ( C x. x ) = 1 ) ) -> ( x x. ( C x. B ) ) = B )
23	16, 22	eqeq12d 2246	. . . 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 2656	. 2 |- ( ph -> ( ( C x. A ) = ( C x. B ) -> A = B ) )
26		oveq2 6036	. 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 1398   e. wcel 2202   E.wrex 2512   class class class wbr 4093  (class class class)co 6028   CCcc 8090   0cc0 8092   1c1 8093   x. cmul 8097   # cap 8820

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821

This theorem is referenced by:  mulcanap2d  8901  mulcanapad  8902  mulcanap  8904  div11ap  8939  eqneg  8971  dvdscmulr  12461  qredeq  12748  cncongr1  12755  lgseisenlem2  15890