Theorem divcan5t 5745

Description: Cancellation of common factor in a ratio.

Assertion

Ref	Expression
divcan5t	|- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> ((C x. A) / (C x. B)) = (A / B))

Proof of Theorem divcan5t

Step	Hyp	Ref	Expression
1		dividt 5730	. . . 4 |- ((C e. CC /\ C =/= 0) -> (C / C) = 1)
2	1	opreq1d 3966	. . 3 |- ((C e. CC /\ C =/= 0) -> ((C / C) x. (A / B)) = (1 x. (A / B)))
3	2	3ad2ant3 801	. 2 |- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> ((C / C) x. (A / B)) = (1 x. (A / B)))
4		divmuldivt 5744	. . 3 |- ((((C e. CC /\ C e. CC) /\ (A e. CC /\ B e. CC)) /\ (C =/= 0 /\ B =/= 0)) -> ((C / C) x. (A / B)) = ((C x. A) / (C x. B)))
5		pm3.26 319	. . . . . . 7 |- ((C e. CC /\ C =/= 0) -> C e. CC)
6	5, 5	jca 288	. . . . . 6 |- ((C e. CC /\ C =/= 0) -> (C e. CC /\ C e. CC))
7		pm3.26 319	. . . . . . 7 |- ((B e. CC /\ B =/= 0) -> B e. CC)
8	7	anim2i 335	. . . . . 6 |- ((A e. CC /\ (B e. CC /\ B =/= 0)) -> (A e. CC /\ B e. CC))
9	6, 8	anim12i 333	. . . . 5 |- (((C e. CC /\ C =/= 0) /\ (A e. CC /\ (B e. CC /\ B =/= 0))) -> ((C e. CC /\ C e. CC) /\ (A e. CC /\ B e. CC)))
10	9	3impb 828	. . . 4 |- (((C e. CC /\ C =/= 0) /\ A e. CC /\ (B e. CC /\ B =/= 0)) -> ((C e. CC /\ C e. CC) /\ (A e. CC /\ B e. CC)))
11	10	3coml 839	. . 3 |- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> ((C e. CC /\ C e. CC) /\ (A e. CC /\ B e. CC)))
12		pm3.27 323	. . . . . 6 |- ((C e. CC /\ C =/= 0) -> C =/= 0)
13		pm3.27 323	. . . . . 6 |- ((B e. CC /\ B =/= 0) -> B =/= 0)
14	12, 13	anim12i 333	. . . . 5 |- (((C e. CC /\ C =/= 0) /\ (B e. CC /\ B =/= 0)) -> (C =/= 0 /\ B =/= 0))
15	14	ancoms 436	. . . 4 |- (((B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> (C =/= 0 /\ B =/= 0))
16	15	3adant1 796	. . 3 |- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> (C =/= 0 /\ B =/= 0))
17	4, 11, 16	sylanc 471	. 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)))
18		divclt 5689	. . . . 5 |- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A / B) e. CC)
19	18	3expb 833	. . . 4 |- ((A e. CC /\ (B e. CC /\ B =/= 0)) -> (A / B) e. CC)
20		mulid2t 5397	. . . 4 |- ((A / B) e. CC -> (1 x. (A / B)) = (A / B))
21	19, 20	syl 10	. . 3 |- ((A e. CC /\ (B e. CC /\ B =/= 0)) -> (1 x. (A / B)) = (A / B))
22	21	3adant3 798	. 2 |- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> (1 x. (A / B)) = (A / B))
23	3, 17, 22	3eqtr3d 1512	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 3   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956   =/= wne 1582  (class class class)co 3954  CCcc 5212  0cc0 5214  1c1 5215   x. cmul 5219   / cdiv 5274

This theorem is referenced by:  8th4div3 5986  efaddlem22 7309  eirrlem2 7339  eirrlem3 7340  sin01bndlem3 7419

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680

Copyright terms: Public domain