Theorem divdivap2 9048

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 8266	. . . . 5 |- 1 e. CC
2		1ap0 8912	. . . . 5 |- 1 # 0
3	1, 2	pm3.2i 272	. . . 4 |- ( 1 e. CC /\ 1 # 0 )
4		divdivdivap 9037	. . . 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 439	. . 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 1230	. 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 9027	. . . 4 |- ( A e. CC -> ( A / 1 ) = A )
8	7	3ad2ant1 1049	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( A / 1 ) = A )
9	8	oveq1d 6094	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / 1 ) / ( B / C ) ) = ( A / ( B / C ) ) )
10		mullid 8318	. . . . 5 |- ( B e. CC -> ( 1 x. B ) = B )
11	10	ad2antrl 494	. . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( 1 x. B ) = B )
12	11	3adant3 1048	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( 1 x. B ) = B )
13	12	oveq2d 6095	. 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 2279	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 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4128  (class class class)co 6079   CCcc 8171   0cc0 8173   1c1 8174   x. cmul 8178   # cap 8903   / cdiv 8996

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997

This theorem is referenced by:  divdivap2d  9147