Theorem divassap 8311

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 8300	. . 3 |- ( ( C e. CC /\ C # 0 ) -> ( 1 / C ) e. CC )
2		mulass 7623	. . 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 1219	. 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 7619	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
5	4	3adant3 969	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A x. B ) e. CC )
6		simp3l 977	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
7		simp3r 978	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
8		divrecap 8309	. . 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 1184	. 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. B ) / C ) = ( ( A x. B ) x. ( 1 / C ) ) )
10		simp2 950	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
11		divrecap 8309	. . . 4 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
12	10, 6, 7, 11	syl3anc 1184	. . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
13	12	oveq2d 5722	. 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 2142	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 103   /\ w3a 930   = wceq 1299   e. wcel 1448   class class class wbr 3875  (class class class)co 5706   CCcc 7498   0cc0 7500   1c1 7501   x. cmul 7505   # cap 8209   / cdiv 8293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294

This theorem is referenced by:  div23ap  8312  div32ap  8313  divmulassap  8316  divmulasscomap  8317  divassapzi  8383  divassapi  8389  divassapd  8447  zdivmul  8993  efi4p  11222