Theorem divmulap 8435

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 8434	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
2	1	3expb 1182	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
3	2	3adant2 1000	. . 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 2148	. 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 982	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
6		receuap 8430	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> E! x e. CC ( C x. x ) = A )
7	6	3expb 1182	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
8	7	3adant2 1000	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
9		oveq2 5782	. . . . 5 |- ( x = B -> ( C x. x ) = ( C x. B ) )
10	9	eqeq1d 2148	. . . 4 |- ( x = B -> ( ( C x. x ) = A <-> ( C x. B ) = A ) )
11	10	riota2 5752	. . 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 408	. 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 190	1 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) = B <-> ( C x. B ) = A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   E!wreu 2418   class class class wbr 3929   iota_crio 5729  (class class class)co 5774   CCcc 7618   0cc0 7620   x. cmul 7625   # cap 8343   / cdiv 8432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433

This theorem is referenced by:  divmulap2  8436  divcanap2  8440  divrecap  8448  divcanap3  8458  div0ap  8462  div1  8463  recrecap  8469  rec11ap  8470  divdivdivap  8473  ddcanap  8486  rerecclap  8490  div2negap  8495  divmulapzi  8523  divmulapd  8572  caucvgrelemrec  10751  odd2np1  11570  sqgcd  11717