Theorem divdivap1 8715

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 7939	. . . . 5 |- 1 e. CC
2		1ap0 8582	. . . . 5 |- 1 # 0
3	1, 2	pm3.2i 272	. . . 4 |- ( 1 e. CC /\ 1 # 0 )
4		divdivdivap 8705	. . . 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 438	. . 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 1196	. 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 8695	. . . . 5 |- ( C e. CC -> ( C / 1 ) = C )
8	7	oveq2d 5916	. . . 4 |- ( C e. CC -> ( ( A / B ) / ( C / 1 ) ) = ( ( A / B ) / C ) )
9	8	ad2antrl 490	. . 3 |- ( ( ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / ( C / 1 ) ) = ( ( A / B ) / C ) )
10	9	3adant1 1017	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / ( C / 1 ) ) = ( ( A / B ) / C ) )
11		mulrid 7989	. . . 4 |- ( A e. CC -> ( A x. 1 ) = A )
12	11	oveq1d 5915	. . 3 |- ( A e. CC -> ( ( A x. 1 ) / ( B x. C ) ) = ( A / ( B x. C ) ) )
13	12	3ad2ant1 1020	. 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 2230	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 104   /\ w3a 980   = wceq 1364   e. wcel 2160   class class class wbr 4021  (class class class)co 5900   CCcc 7844   0cc0 7846   1c1 7847   x. cmul 7851   # cap 8573   / cdiv 8664

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-po 4317  df-iso 4318  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665

This theorem is referenced by:  recdivap2  8717  divdivap1d  8814  sin01bnd  11806