Theorem divdivdiv 6961

Description: Division of two ratios. Theorem I.15 of [Apostol] p. 18.

Assertion

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

Proof of Theorem divdivdiv

Step	Hyp	Ref	Expression
1		simprrl 458	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> D e. CC)
2		simprll 456	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> C e. CC)
3		simprlr 457	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> C =/= 0)
4		divcl 6901	. . . . . . 7 |- ((D e. CC /\ C e. CC /\ C =/= 0) -> (D / C) e. CC)
5	1, 2, 3, 4	syl111anc 1100	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (D / C) e. CC)
6		simpll 448	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> A e. CC)
7		simplrl 454	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> B e. CC)
8		simplrr 455	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> B =/= 0)
9		divcl 6901	. . . . . . 7 |- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A / B) e. CC)
10	6, 7, 8, 9	syl111anc 1100	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (A / B) e. CC)
11		mulcom 6459	. . . . . 6 |- (((D / C) e. CC /\ (A / B) e. CC) -> ((D / C) x. (A / B)) = ((A / B) x. (D / C)))
12	5, 10, 11	syl11anc 524	. . . . 5 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((D / C) x. (A / B)) = ((A / B) x. (D / C)))
13		simplr 449	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (B e. CC /\ B =/= 0))
14		simprl 450	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (C e. CC /\ C =/= 0))
15		divmuldiv 6956	. . . . . 6 |- (((A e. CC /\ D e. CC) /\ ((B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0))) -> ((A / B) x. (D / C)) = ((A x. D) / (B x. C)))
16	6, 1, 13, 14, 15	syl22anc 1101	. . . . 5 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((A / B) x. (D / C)) = ((A x. D) / (B x. C)))
17	12, 16	eqtrd 1925	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((D / C) x. (A / B)) = ((A x. D) / (B x. C)))
18	17	opreq2d 4898	. . 3 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((C / D) x. ((D / C) x. (A / B))) = ((C / D) x. ((A x. D) / (B x. C))))
19		simprr 451	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (D e. CC /\ D =/= 0))
20		divmuldiv 6956	. . . . . . 7 |- (((C e. CC /\ D e. CC) /\ ((D e. CC /\ D =/= 0) /\ (C e. CC /\ C =/= 0))) -> ((C / D) x. (D / C)) = ((C x. D) / (D x. C)))
21	2, 1, 19, 14, 20	syl22anc 1101	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((C / D) x. (D / C)) = ((C x. D) / (D x. C)))
22		mulcom 6459	. . . . . . . 8 |- ((C e. CC /\ D e. CC) -> (C x. D) = (D x. C))
23	2, 1, 22	syl11anc 524	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (C x. D) = (D x. C))
24	23	opreq1d 4897	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((C x. D) / (D x. C)) = ((D x. C) / (D x. C)))
25		mulcl 6456	. . . . . . . 8 |- ((D e. CC /\ C e. CC) -> (D x. C) e. CC)
26	1, 2, 25	syl11anc 524	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (D x. C) e. CC)
27		simprrr 459	. . . . . . . 8 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> D =/= 0)
28		mulne0 6887	. . . . . . . 8 |- (((D e. CC /\ D =/= 0) /\ (C e. CC /\ C =/= 0)) -> (D x. C) =/= 0)
29	1, 27, 2, 3, 28	syl22anc 1101	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (D x. C) =/= 0)
30		divid 6942	. . . . . . 7 |- (((D x. C) e. CC /\ (D x. C) =/= 0) -> ((D x. C) / (D x. C)) = 1)
31	26, 29, 30	syl11anc 524	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((D x. C) / (D x. C)) = 1)
32	21, 24, 31	3eqtrd 1929	. . . . 5 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((C / D) x. (D / C)) = 1)
33	32	opreq1d 4897	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (((C / D) x. (D / C)) x. (A / B)) = (1 x. (A / B)))
34		divcl 6901	. . . . . 6 |- ((C e. CC /\ D e. CC /\ D =/= 0) -> (C / D) e. CC)
35	2, 1, 27, 34	syl111anc 1100	. . . . 5 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (C / D) e. CC)
36		mulass 6461	. . . . 5 |- (((C / D) e. CC /\ (D / C) e. CC /\ (A / B) e. CC) -> (((C / D) x. (D / C)) x. (A / B)) = ((C / D) x. ((D / C) x. (A / B))))
37	35, 5, 10, 36	syl111anc 1100	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (((C / D) x. (D / C)) x. (A / B)) = ((C / D) x. ((D / C) x. (A / B))))
38		mulid2 6578	. . . . 5 |- ((A / B) e. CC -> (1 x. (A / B)) = (A / B))
39	10, 38	syl 12	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (1 x. (A / B)) = (A / B))
40	33, 37, 39	3eqtr3d 1934	. . 3 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((C / D) x. ((D / C) x. (A / B))) = (A / B))
41	18, 40	eqtr3d 1927	. 2 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((C / D) x. ((A x. D) / (B x. C))) = (A / B))
42		mulcl 6456	. . . . 5 |- ((A e. CC /\ D e. CC) -> (A x. D) e. CC)
43	6, 1, 42	syl11anc 524	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (A x. D) e. CC)
44		mulcl 6456	. . . . 5 |- ((B e. CC /\ C e. CC) -> (B x. C) e. CC)
45	7, 2, 44	syl11anc 524	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (B x. C) e. CC)
46		mulne0 6887	. . . . 5 |- (((B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> (B x. C) =/= 0)
47	46	ad2ant2lr 446	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (B x. C) =/= 0)
48		divcl 6901	. . . 4 |- (((A x. D) e. CC /\ (B x. C) e. CC /\ (B x. C) =/= 0) -> ((A x. D) / (B x. C)) e. CC)
49	43, 45, 47, 48	syl111anc 1100	. . 3 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((A x. D) / (B x. C)) e. CC)
50		divne0 6912	. . . 4 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> (C / D) =/= 0)
51	50	adantl 424	. . 3 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (C / D) =/= 0)
52		divmul 6896	. . 3 |- (((A / B) e. CC /\ ((A x. D) / (B x. C)) e. CC /\ ((C / D) e. CC /\ (C / D) =/= 0)) -> (((A / B) / (C / D)) = ((A x. D) / (B x. C)) <-> ((C / D) x. ((A x. D) / (B x. C))) = (A / B)))
53	10, 49, 35, 51, 52	syl112anc 1104	. 2 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (((A / B) / (C / D)) = ((A x. D) / (B x. C)) <-> ((C / D) x. ((A x. D) / (B x. C))) = (A / B)))
54	41, 53	mpbird 213	1 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((A / B) / (C / D)) = ((A x. D) / (B x. C)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   x. cmul 6391   / cdiv 6447

This theorem is referenced by:  divdivdivi 6966  recdiv 6967  divdiv1 6972  divdiv2 6973  qreccl 7453  georeclim 8502

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892

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