Theorem dmdcanap 8907

Description: Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.)

Assertion

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

Proof of Theorem dmdcanap

Step	Hyp	Ref	Expression
1		simp1l 1047	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> A e. CC )
2		simp3 1025	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> C e. CC )
3		simp1r 1048	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> A # 0 )
4		divclap 8863	. . . 4 |- ( ( C e. CC /\ A e. CC /\ A # 0 ) -> ( C / A ) e. CC )
5	2, 1, 3, 4	syl3anc 1273	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> ( C / A ) e. CC )
6		simp2l 1049	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> B e. CC )
7		simp2r 1050	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> B # 0 )
8		div23ap 8876	. . 3 |- ( ( A e. CC /\ ( C / A ) e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( ( A x. ( C / A ) ) / B ) = ( ( A / B ) x. ( C / A ) ) )
9	1, 5, 6, 7, 8	syl112anc 1277	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> ( ( A x. ( C / A ) ) / B ) = ( ( A / B ) x. ( C / A ) ) )
10		divcanap2 8865	. . . 4 |- ( ( C e. CC /\ A e. CC /\ A # 0 ) -> ( A x. ( C / A ) ) = C )
11	2, 1, 3, 10	syl3anc 1273	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> ( A x. ( C / A ) ) = C )
12	11	oveq1d 6038	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> ( ( A x. ( C / A ) ) / B ) = ( C / B ) )
13	9, 12	eqtr3d 2265	1 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ C e. CC ) -> ( ( A / B ) x. ( C / A ) ) = ( C / B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2201   class class class wbr 4089  (class class class)co 6023   CCcc 8035   0cc0 8037   x. cmul 8042   # cap 8766   / cdiv 8857

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-id 4392  df-po 4395  df-iso 4396  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-iota 5288  df-fun 5330  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858

This theorem is referenced by:  dmdcanapd  9005