Theorem apmul1 9108

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 1028	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
2		simp3l 1056	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
3		simp3r 1057	. . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
4	2, 3	recclapd 9101	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( 1 / C ) e. CC )
5	1, 2, 4	mulassd 8339	. . . . 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 9103	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( C x. ( 1 / C ) ) = 1 )
7	6	oveq2d 6091	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. ( C x. ( 1 / C ) ) ) = ( A x. 1 ) )
8	1	mulridd 8333	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. 1 ) = A )
9	5, 7, 8	3eqtrd 2275	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. C ) x. ( 1 / C ) ) = A )
10		simp2 1029	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
11	10, 2, 4	mulassd 8339	. . . . 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 6091	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. ( C x. ( 1 / C ) ) ) = ( B x. 1 ) )
13	10	mulridd 8333	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. 1 ) = B )
14	11, 12, 13	3eqtrd 2275	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( B x. C ) x. ( 1 / C ) ) = B )
15	9, 14	breq12d 4138	. . 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 8336	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. C ) e. CC )
17	10, 2	mulcld 8336	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. C ) e. CC )
18		mulext1 8930	. . . 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 1278	. . 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 8930	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
22	21	3adant3r 1266	. 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 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   x. cmul 8174   # cap 8899   / cdiv 8992

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993

This theorem is referenced by:  apmul2  9109  divap1d  9121  apdivmuld  9133  qapne  10018  apcxp2  15964