Theorem apmul1 8680

Description: Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)

Assertion

Ref	Expression
apmul1	|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )

Proof of Theorem apmul1

Step	Hyp	Ref	Expression
1		simp1 987	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
2		simp3l 1015	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
3		simp3r 1016	. . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
4	2, 3	recclapd 8673	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( 1 / C ) e. CC )
5	1, 2, 4	mulassd 7918	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. C ) x. ( 1 / C ) ) = ( A x. ( C x. ( 1 / C ) ) ) )
6	2, 3	recidapd 8675	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( C x. ( 1 / C ) ) = 1 )
7	6	oveq2d 5857	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. ( C x. ( 1 / C ) ) ) = ( A x. 1 ) )
8	1	mulid1d 7912	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. 1 ) = A )
9	5, 7, 8	3eqtrd 2202	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. C ) x. ( 1 / C ) ) = A )
10		simp2 988	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
11	10, 2, 4	mulassd 7918	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( B x. C ) x. ( 1 / C ) ) = ( B x. ( C x. ( 1 / C ) ) ) )
12	6	oveq2d 5857	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. ( C x. ( 1 / C ) ) ) = ( B x. 1 ) )
13	10	mulid1d 7912	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. 1 ) = B )
14	11, 12, 13	3eqtrd 2202	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( B x. C ) x. ( 1 / C ) ) = B )
15	9, 14	breq12d 3994	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( ( A x. C ) x. ( 1 / C ) ) # ( ( B x. C ) x. ( 1 / C ) ) <-> A # B ) )
16	1, 2	mulcld 7915	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. C ) e. CC )
17	10, 2	mulcld 7915	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. C ) e. CC )
18		mulext1 8506	. . . 4 |- ( ( ( A x. C ) e. CC /\ ( B x. C ) e. CC /\ ( 1 / C ) e. CC ) -> ( ( ( A x. C ) x. ( 1 / C ) ) # ( ( B x. C ) x. ( 1 / C ) ) -> ( A x. C ) # ( B x. C ) ) )
19	16, 17, 4, 18	syl3anc 1228	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( ( A x. C ) x. ( 1 / C ) ) # ( ( B x. C ) x. ( 1 / C ) ) -> ( A x. C ) # ( B x. C ) ) )
20	15, 19	sylbird 169	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A # B -> ( A x. C ) # ( B x. C ) ) )
21		mulext1 8506	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
22	21	3adant3r 1225	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
23	20, 22	impbid 128	1 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   e. wcel 2136   class class class wbr 3981  (class class class)co 5841   CCcc 7747   0cc0 7749   1c1 7750   x. cmul 7754   # cap 8475   / cdiv 8564

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-br 3982  df-opab 4043  df-id 4270  df-po 4273  df-iso 4274  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-iota 5152  df-fun 5189  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565

This theorem is referenced by:  apmul2  8681  divap1d  8693  apdivmuld  8705  qapne  9573  apcxp2  13458