Theorem apmul1 8967

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 1023	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
2		simp3l 1051	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
3		simp3r 1052	. . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
4	2, 3	recclapd 8960	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( 1 / C ) e. CC )
5	1, 2, 4	mulassd 8202	. . . . 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 8962	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( C x. ( 1 / C ) ) = 1 )
7	6	oveq2d 6033	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. ( C x. ( 1 / C ) ) ) = ( A x. 1 ) )
8	1	mulridd 8195	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. 1 ) = A )
9	5, 7, 8	3eqtrd 2268	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. C ) x. ( 1 / C ) ) = A )
10		simp2 1024	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
11	10, 2, 4	mulassd 8202	. . . . 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 6033	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. ( C x. ( 1 / C ) ) ) = ( B x. 1 ) )
13	10	mulridd 8195	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. 1 ) = B )
14	11, 12, 13	3eqtrd 2268	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( B x. C ) x. ( 1 / C ) ) = B )
15	9, 14	breq12d 4101	. . 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 8199	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. C ) e. CC )
17	10, 2	mulcld 8199	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. C ) e. CC )
18		mulext1 8791	. . . 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 1273	. . 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 8791	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
22	21	3adant3r 1261	. 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 1004   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   x. cmul 8036   # cap 8760   / cdiv 8851

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852

This theorem is referenced by:  apmul2  8968  divap1d  8980  apdivmuld  8992  qapne  9872  apcxp2  15662