Theorem divcanap5 9010

Description: Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)

Assertion

Ref	Expression
divcanap5	|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) )

Proof of Theorem divcanap5

Step	Hyp	Ref	Expression
1		dividap 8997	. . . 4 |- ( ( C e. CC /\ C # 0 ) -> ( C / C ) = 1 )
2	1	oveq1d 6075	. . 3 |- ( ( C e. CC /\ C # 0 ) -> ( ( C / C ) x. ( A / B ) ) = ( 1 x. ( A / B ) ) )
3	2	3ad2ant3 1047	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( C / C ) x. ( A / B ) ) = ( 1 x. ( A / B ) ) )
4		simp3l 1052	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
5		simp1 1024	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
6		simp3 1026	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( C e. CC /\ C # 0 ) )
7		simp2 1025	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( B e. CC /\ B # 0 ) )
8		divmuldivap 9008	. . 3 |- ( ( ( C e. CC /\ A e. CC ) /\ ( ( C e. CC /\ C # 0 ) /\ ( B e. CC /\ B # 0 ) ) ) -> ( ( C / C ) x. ( A / B ) ) = ( ( C x. A ) / ( C x. B ) ) )
9	4, 5, 6, 7, 8	syl22anc 1275	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( C / C ) x. ( A / B ) ) = ( ( C x. A ) / ( C x. B ) ) )
10		divclap 8974	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC )
11	10	3expb 1231	. . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( A / B ) e. CC )
12	11	mullidd 8310	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( 1 x. ( A / B ) ) = ( A / B ) )
13	12	3adant3 1044	. 2 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( 1 x. ( A / B ) ) = ( A / B ) )
14	3, 9, 13	3eqtr3d 2275	1 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( C e. CC /\ C # 0 ) ) -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4115  (class class class)co 6060   CCcc 8143   0cc0 8145   1c1 8146   x. cmul 8150   # cap 8875   / cdiv 8968

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-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-mulrcl 8244  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-precex 8255  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261  ax-pre-mulgt0 8262  ax-pre-mulext 8263

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 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-iota 5319  df-fun 5361  df-fv 5367  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-reap 8869  df-ap 8876  df-div 8969

This theorem is referenced by:  divcanap7  9017  divadddivap  9023  divcanap5d  9113  8th4div3  9479  flodddiv4  12653  pigt3  15840