Theorem divmulap 8058

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 8057	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
2	1	3expb 1142	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
3	2	3adant2 960	. . 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 2093	. 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 942	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
6		receuap 8054	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> E! x e. CC ( C x. x ) = A )
7	6	3expb 1142	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
8	7	3adant2 960	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
9		oveq2 5602	. . . . 5 |- ( x = B -> ( C x. x ) = ( C x. B ) )
10	9	eqeq1d 2093	. . . 4 |- ( x = B -> ( ( C x. x ) = A <-> ( C x. B ) = A ) )
11	10	riota2 5572	. . 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 922   = wceq 1287   e. wcel 1436   E!wreu 2357   class class class wbr 3814   iota_crio 5549  (class class class)co 5594   CCcc 7269   0cc0 7271   x. cmul 7276   # cap 7976   / cdiv 8055

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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-mulrcl 7365  ax-addcom 7366  ax-mulcom 7367  ax-addass 7368  ax-mulass 7369  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-1rid 7373  ax-0id 7374  ax-rnegex 7375  ax-precex 7376  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382  ax-pre-mulgt0 7383  ax-pre-mulext 7384

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-id 4087  df-po 4090  df-iso 4091  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-iota 4937  df-fun 4974  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-reap 7970  df-ap 7977  df-div 8056

This theorem is referenced by:  divmulap2  8059  divcanap2  8063  divrecap  8071  divcanap3  8081  div0ap  8085  div1  8086  recrecap  8092  rec11ap  8093  divdivdivap  8096  ddcanap  8109  rerecclap  8113  div2negap  8118  divmulapzi  8146  divmulapd  8194  caucvgrelemrec  10253  odd2np1  10667  sqgcd  10812