Theorem divmulap 8592

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 8591	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
2	1	3expb 1199	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) = ( iota_ x e. CC ( C x. x ) = A ) )
3	2	3adant2 1011	. . 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 2179	. 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 993	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
6		receuap 8587	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> E! x e. CC ( C x. x ) = A )
7	6	3expb 1199	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
8	7	3adant2 1011	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> E! x e. CC ( C x. x ) = A )
9		oveq2 5861	. . . . 5 |- ( x = B -> ( C x. x ) = ( C x. B ) )
10	9	eqeq1d 2179	. . . 4 |- ( x = B -> ( ( C x. x ) = A <-> ( C x. B ) = A ) )
11	10	riota2 5831	. . 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 409	. 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 973   = wceq 1348   e. wcel 2141   E!wreu 2450   class class class wbr 3989   iota_crio 5808  (class class class)co 5853   CCcc 7772   0cc0 7774   x. cmul 7779   # cap 8500   / cdiv 8589

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590

This theorem is referenced by:  divmulap2  8593  divcanap2  8597  divrecap  8605  divcanap3  8615  div0ap  8619  div1  8620  recrecap  8626  rec11ap  8627  divdivdivap  8630  ddcanap  8643  rerecclap  8647  div2negap  8652  divmulapzi  8680  divmulapd  8729  caucvgrelemrec  10943  odd2np1  11832  sqgcd  11984  oddprmdvds  12306