Theorem apmul1 8891

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 8884	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( 1 / C ) e. CC )
5	1, 2, 4	mulassd 8126	. . . . 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 8886	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( C x. ( 1 / C ) ) = 1 )
7	6	oveq2d 5978	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. ( C x. ( 1 / C ) ) ) = ( A x. 1 ) )
8	1	mulridd 8119	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. 1 ) = A )
9	5, 7, 8	3eqtrd 2243	. . . 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 8126	. . . . 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 5978	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. ( C x. ( 1 / C ) ) ) = ( B x. 1 ) )
13	10	mulridd 8119	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. 1 ) = B )
14	11, 12, 13	3eqtrd 2243	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( B x. C ) x. ( 1 / C ) ) = B )
15	9, 14	breq12d 4067	. . 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 8123	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. C ) e. CC )
17	10, 2	mulcld 8123	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B x. C ) e. CC )
18		mulext1 8715	. . . 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 8715	. . 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 2177   class class class wbr 4054  (class class class)co 5962   CCcc 7953   0cc0 7955   1c1 7956   x. cmul 7960   # cap 8684   / cdiv 8775

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-id 4353  df-po 4356  df-iso 4357  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-iota 5246  df-fun 5287  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776

This theorem is referenced by:  apmul2  8892  divap1d  8904  apdivmuld  8916  qapne  9790  apcxp2  15496