Theorem divdivap1 9018

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 8237	. . . . 5 |- 1 e. CC
2		1ap0 8883	. . . . 5 |- 1 # 0
3	1, 2	pm3.2i 272	. . . 4 |- ( 1 e. CC /\ 1 # 0 )
4		divdivdivap 9008	. . . 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 1221	. 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 8998	. . . . 5 |- ( C e. CC -> ( C / 1 ) = C )
8	7	oveq2d 6075	. . . 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 1042	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / B ) / ( C / 1 ) ) = ( ( A / B ) / C ) )
11		mulrid 8288	. . . 4 |- ( A e. CC -> ( A x. 1 ) = A )
12	11	oveq1d 6074	. . 3 |- ( A e. CC -> ( ( A x. 1 ) / ( B x. C ) ) = ( A / ( B x. C ) ) )
13	12	3ad2ant1 1045	. 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 2275	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 1005   = wceq 1398   e. wcel 2205   class class class wbr 4115  (class class class)co 6059   CCcc 8142   0cc0 8144   1c1 8145   x. cmul 8149   # cap 8874   / cdiv 8967

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-mulrcl 8243  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-precex 8254  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-apti 8259  ax-pre-ltadd 8260  ax-pre-mulgt0 8261  ax-pre-mulext 8262

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-id 4420  df-po 4423  df-iso 4424  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-iota 5318  df-fun 5360  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-reap 8868  df-ap 8875  df-div 8968

This theorem is referenced by:  recdivap2  9020  divdivap1d  9117  sin01bnd  12473