Theorem divcanap5 8244

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 8231	. . . 4 |- ( ( C e. CC /\ C # 0 ) -> ( C / C ) = 1 )
2	1	oveq1d 5683	. . 3 |- ( ( C e. CC /\ C # 0 ) -> ( ( C / C ) x. ( A / B ) ) = ( 1 x. ( A / B ) ) )
3	2	3ad2ant3 967	. 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 972	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
5		simp1 944	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
6		simp3 946	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( C e. CC /\ C # 0 ) )
7		simp2 945	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( B e. CC /\ B # 0 ) )
8		divmuldivap 8242	. . 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 1176	. 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 8208	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC )
11	10	3expb 1145	. . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( A / B ) e. CC )
12	11	mulid2d 7569	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( 1 x. ( A / B ) ) = ( A / B ) )
13	12	3adant3 964	. 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 2129	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 103   /\ w3a 925   = wceq 1290   e. wcel 1439   class class class wbr 3853  (class class class)co 5668   CCcc 7411   0cc0 7413   1c1 7414   x. cmul 7418   # cap 8121   / cdiv 8202

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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-id 4131  df-po 4134  df-iso 4135  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203

This theorem is referenced by:  divcanap7  8251  divadddivap  8257  divcanap5d  8347  8th4div3  8698  flodddiv4  11275