Theorem divdivap2 8252

Description: Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)

Assertion

Ref	Expression
divdivap2	|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( A / ( B / C ) ) = ( ( A x. C ) / B ) )

Proof of Theorem divdivap2

Step	Hyp	Ref	Expression
1		ax-1cn 7499	. . . . 5 |- 1 e. CC
2		1ap0 8128	. . . . 5 |- 1 # 0
3	1, 2	pm3.2i 267	. . . 4 |- ( 1 e. CC /\ 1 # 0 )
4		divdivdivap 8241	. . . 4 |- ( ( ( A e. CC /\ ( 1 e. CC /\ 1 # 0 ) ) /\ ( ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) ) -> ( ( A / 1 ) / ( B / C ) ) = ( ( A x. C ) / ( 1 x. B ) ) )
5	3, 4	mpanl2 427	. . 3 |- ( ( A e. CC /\ ( ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) ) -> ( ( A / 1 ) / ( B / C ) ) = ( ( A x. C ) / ( 1 x. B ) ) )
6	5	3impb 1140	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / 1 ) / ( B / C ) ) = ( ( A x. C ) / ( 1 x. B ) ) )
7		div1 8231	. . . 4 |- ( A e. CC -> ( A / 1 ) = A )
8	7	3ad2ant1 965	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( A / 1 ) = A )
9	8	oveq1d 5681	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / 1 ) / ( B / C ) ) = ( A / ( B / C ) ) )
10		mulid2 7547	. . . . 5 |- ( B e. CC -> ( 1 x. B ) = B )
11	10	ad2antrl 475	. . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( 1 x. B ) = B )
12	11	3adant3 964	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( 1 x. B ) = B )
13	12	oveq2d 5682	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. C ) / ( 1 x. B ) ) = ( ( A x. C ) / B ) )
14	6, 9, 13	3eqtr3d 2129	1 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( A / ( B / C ) ) = ( ( A x. C ) / B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 925   = wceq 1290   e. wcel 1439   class class class wbr 3851  (class class class)co 5666   CCcc 7409   0cc0 7411   1c1 7412   x. cmul 7416   # cap 8119   / cdiv 8200

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 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524

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 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-po 4132  df-iso 4133  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201

This theorem is referenced by:  divdivap2d  8351