Theorem divassap 8677

Description: An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)

Assertion

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

Proof of Theorem divassap

Step	Hyp	Ref	Expression
1		recclap 8666	. . 3 |- ( ( C e. CC /\ C # 0 ) -> ( 1 / C ) e. CC )
2		mulass 7972	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( 1 / C ) e. CC ) -> ( ( A x. B ) x. ( 1 / C ) ) = ( A x. ( B x. ( 1 / C ) ) ) )
3	1, 2	syl3an3 1284	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. B ) x. ( 1 / C ) ) = ( A x. ( B x. ( 1 / C ) ) ) )
4		mulcl 7968	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
5	4	3adant3 1019	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. B ) e. CC )
6		simp3l 1027	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
7		simp3r 1028	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
8		divrecap 8675	. . 3 |- ( ( ( A x. B ) e. CC /\ C e. CC /\ C # 0 ) -> ( ( A x. B ) / C ) = ( ( A x. B ) x. ( 1 / C ) ) )
9	5, 6, 7, 8	syl3anc 1249	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. B ) / C ) = ( ( A x. B ) x. ( 1 / C ) ) )
10		simp2 1000	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
11		divrecap 8675	. . . 4 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
12	10, 6, 7, 11	syl3anc 1249	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
13	12	oveq2d 5912	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. ( B / C ) ) = ( A x. ( B x. ( 1 / C ) ) ) )
14	3, 9, 13	3eqtr4d 2232	1 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. B ) / C ) = ( A x. ( B / C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   class class class wbr 4018  (class class class)co 5896   CCcc 7839   0cc0 7841   1c1 7842   x. cmul 7846   # cap 8568   / cdiv 8659

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660

This theorem is referenced by:  div23ap  8678  div32ap  8679  divmulassap  8682  divmulasscomap  8683  divassapzi  8749  divassapi  8755  divassapd  8813  zdivmul  9373  efi4p  11757