Theorem divcanap5 8857

Description: Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)

Assertion

Ref	Expression
divcanap5	|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) )

Proof of Theorem divcanap5

Step	Hyp	Ref	Expression
1		dividap 8844	. . . 4 |- ( ( C e. CC /\ C # 0 ) -> ( C / C ) = 1 )
2	1	oveq1d 6015	. . 3 |- ( ( C e. CC /\ C # 0 ) -> ( ( C / C ) x. ( A / B ) ) = ( 1 x. ( A / B ) ) )
3	2	3ad2ant3 1044	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( C / C ) x. ( A / B ) ) = ( 1 x. ( A / B ) ) )
4		simp3l 1049	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
5		simp1 1021	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
6		simp3 1023	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( C e. CC /\ C # 0 ) )
7		simp2 1022	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( B e. CC /\ B # 0 ) )
8		divmuldivap 8855	. . 3 |- ( ( ( C e. CC /\ A e. CC ) /\ ( ( C e. CC /\ C # 0 ) /\ ( B e. CC /\ B # 0 ) ) ) -> ( ( C / C ) x. ( A / B ) ) = ( ( C x. A ) / ( C x. B ) ) )
9	4, 5, 6, 7, 8	syl22anc 1272	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( C / C ) x. ( A / B ) ) = ( ( C x. A ) / ( C x. B ) ) )
10		divclap 8821	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC )
11	10	3expb 1228	. . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( A / B ) e. CC )
12	11	mulid2d 8161	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( 1 x. ( A / B ) ) = ( A / B ) )
13	12	3adant3 1041	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( 1 x. ( A / B ) ) = ( A / B ) )
14	3, 9, 13	3eqtr3d 2270	1 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   CCcc 7993   0cc0 7995   1c1 7996   x. cmul 8000   # cap 8724   / cdiv 8815

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816

This theorem is referenced by:  divcanap7  8864  divadddivap  8870  divcanap5d  8960  8th4div3  9326  flodddiv4  12442  pigt3  15512