Theorem divmulap 7763

Description: Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)

Assertion

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

Proof of Theorem divmulap

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divvalap 7762	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
2	1	3expb 1139	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
3	2	3adant2 957	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
4	3	eqeq1d 2089	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) = B <-> ( iota_ x e. CC ( C x. x ) = A ) = B ) )
5		simp2 939	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
6		receuap 7759	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> E! x e. CC ( C x. x ) = A )
7	6	3expb 1139	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
8	7	3adant2 957	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
9		oveq2 5540	. . . . 5 |- ( x = B -> ( C x. x ) = ( C x. B ) )
10	9	eqeq1d 2089	. . . 4 |- ( x = B -> ( ( C x. x ) = A <-> ( C x. B ) = A ) )
11	10	riota2 5510	. . 3 |- ( ( B e. CC /\ E! x e. CC ( C x. x ) = A ) -> ( ( C x. B ) = A <-> ( iota_ x e. CC ( C x. x ) = A ) = B ) )
12	5, 8, 11	syl2anc 403	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( C x. B ) = A <-> ( iota_ x e. CC ( C x. x ) = A ) = B ) )
13	4, 12	bitr4d 189	1 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) = B <-> ( C x. B ) = A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   E!wreu 2350   class class class wbr 3785   iota_crio 5487  (class class class)co 5532   CCcc 6979   0cc0 6981   x. cmul 6986   # cap 7681   / cdiv 7760

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761

This theorem is referenced by:  divmulap2  7764  divcanap2  7768  divrecap  7776  divcanap3  7786  div0ap  7790  div1  7791  recrecap  7797  rec11ap  7798  divdivdivap  7801  ddcanap  7814  rerecclap  7818  div2negap  7823  divmulapzi  7851  divmulapd  7899  caucvgrelemrec  9865  odd2np1  10272  sqgcd  10418