Theorem apmul1 8861

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 1000	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
2		simp3l 1028	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
3		simp3r 1029	. . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
4	2, 3	recclapd 8854	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( 1 / C ) e. CC )
5	1, 2, 4	mulassd 8096	. . . . 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 8856	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( C x. ( 1 / C ) ) = 1 )
7	6	oveq2d 5960	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. ( C x. ( 1 / C ) ) ) = ( A x. 1 ) )
8	1	mulridd 8089	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. 1 ) = A )
9	5, 7, 8	3eqtrd 2242	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. C ) x. ( 1 / C ) ) = A )
10		simp2 1001	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
11	10, 2, 4	mulassd 8096	. . . . 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 5960	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. ( C x. ( 1 / C ) ) ) = ( B x. 1 ) )
13	10	mulridd 8089	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. 1 ) = B )
14	11, 12, 13	3eqtrd 2242	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( B x. C ) x. ( 1 / C ) ) = B )
15	9, 14	breq12d 4057	. . 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 8093	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. C ) e. CC )
17	10, 2	mulcld 8093	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. C ) e. CC )
18		mulext1 8685	. . . 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 1250	. . 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 170	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A # B -> ( A x. C ) # ( B x. C ) ) )
21		mulext1 8685	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
22	21	3adant3r 1238	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
23	20, 22	impbid 129	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 104   <-> wb 105   /\ w3a 981   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   CCcc 7923   0cc0 7925   1c1 7926   x. cmul 7930   # cap 8654   / cdiv 8745

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746

This theorem is referenced by:  apmul2  8862  divap1d  8874  apdivmuld  8886  qapne  9760  apcxp2  15411