Theorem divmulap 8845

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 8844	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
2	1	3expb 1228	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
3	2	3adant2 1040	. . 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 2238	. 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 1022	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
6		receuap 8839	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> E! x e. CC ( C x. x ) = A )
7	6	3expb 1228	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
8	7	3adant2 1040	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
9		oveq2 6021	. . . . 5 |- ( x = B -> ( C x. x ) = ( C x. B ) )
10	9	eqeq1d 2238	. . . 4 |- ( x = B -> ( ( C x. x ) = A <-> ( C x. B ) = A ) )
11	10	riota2 5990	. . 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 411	. 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 191	1 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) = B <-> ( C x. B ) = A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E!wreu 2510   class class class wbr 4086   iota_crio 5965  (class class class)co 6013   CCcc 8020   0cc0 8022   x. cmul 8027   # cap 8751   / cdiv 8842

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843

This theorem is referenced by:  divmulap2  8846  divcanap2  8850  divrecap  8858  divcanap3  8868  div0ap  8872  div1  8873  recrecap  8879  rec11ap  8880  divdivdivap  8883  ddcanap  8896  rerecclap  8900  div2negap  8905  divmulapzi  8933  divmulapd  8982  caucvgrelemrec  11530  odd2np1  12424  sqgcd  12590  oddprmdvds  12917