Theorem divdivap1 8287

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

Assertion

Ref	Expression
divdivap1	|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) )

Proof of Theorem divdivap1

Step	Hyp	Ref	Expression
1		ax-1cn 7535	. . . . 5 |- 1 e. CC
2		1ap0 8164	. . . . 5 |- 1 # 0
3	1, 2	pm3.2i 267	. . . 4 |- ( 1 e. CC /\ 1 # 0 )
4		divdivdivap 8277	. . . 4 |- ( ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) /\ ( ( C e. CC /\ C # 0 ) /\ ( 1 e. CC /\ 1 # 0 ) ) ) -> ( ( A / B ) / ( C / 1 ) ) = ( ( A x. 1 ) / ( B x. C ) ) )
5	3, 4	mpanr2 430	. . 3 |- ( ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / ( C / 1 ) ) = ( ( A x. 1 ) / ( B x. C ) ) )
6	5	3impa 1141	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / ( C / 1 ) ) = ( ( A x. 1 ) / ( B x. C ) ) )
7		div1 8267	. . . . 5 |- ( C e. CC -> ( C / 1 ) = C )
8	7	oveq2d 5706	. . . 4 |- ( C e. CC -> ( ( A / B ) / ( C / 1 ) ) = ( ( A / B ) / C ) )
9	8	ad2antrl 475	. . 3 |- ( ( ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / ( C / 1 ) ) = ( ( A / B ) / C ) )
10	9	3adant1 964	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / ( C / 1 ) ) = ( ( A / B ) / C ) )
11		mulid1 7582	. . . 4 |- ( A e. CC -> ( A x. 1 ) = A )
12	11	oveq1d 5705	. . 3 |- ( A e. CC -> ( ( A x. 1 ) / ( B x. C ) ) = ( A / ( B x. C ) ) )
13	12	3ad2ant1 967	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A x. 1 ) / ( B x. C ) ) = ( A / ( B x. C ) ) )
14	6, 10, 13	3eqtr3d 2135	1 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 927   = wceq 1296   e. wcel 1445   class class class wbr 3867  (class class class)co 5690   CCcc 7445   0cc0 7447   1c1 7448   x. cmul 7452   # cap 8155   / cdiv 8236

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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237

This theorem is referenced by:  recdivap2  8289  divdivap1d  8386  sin01bnd  11213