Theorem divdivdiv 6757

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		mulcom 6255	. . . . . 6 |- (((D / C) e. CC /\ (A / B) e. CC) -> ((D / C) x. (A / B)) = ((A / B) x. (D / C)))
2		divcl 6697	. . . . . . . . . 10 |- ((D e. CC /\ C e. CC /\ C =/= 0) -> (D / C) e. CC)
3	2	3expb 947	. . . . . . . . 9 |- ((D e. CC /\ (C e. CC /\ C =/= 0)) -> (D / C) e. CC)
4	3	ancoms 482	. . . . . . . 8 |- (((C e. CC /\ C =/= 0) /\ D e. CC) -> (D / C) e. CC)
5	4	adantrr 429	. . . . . . 7 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> (D / C) e. CC)
6	5	adantl 422	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (D / C) e. CC)
7		divcl 6697	. . . . . . . 8 |- ((A e. CC /\ B e. CC /\ B =/= 0) -> (A / B) e. CC)
8	7	3expb 947	. . . . . . 7 |- ((A e. CC /\ (B e. CC /\ B =/= 0)) -> (A / B) e. CC)
9	8	adantr 423	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (A / B) e. CC)
10	1, 6, 9	sylanc 521	. . . . 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)))
11		divmuldiv 6752	. . . . . 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)))
12		pm3.26 344	. . . . . . 7 |- ((A e. CC /\ (B e. CC /\ B =/= 0)) -> A e. CC)
13		simprl 448	. . . . . . 7 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> D e. CC)
14	12, 13	anim12i 358	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (A e. CC /\ D e. CC))
15		simplr 447	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (B e. CC /\ B =/= 0))
16		simprl 448	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (C e. CC /\ C =/= 0))
17	15, 16	jca 308	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)))
18	11, 14, 17	sylanc 521	. . . . 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)))
19	10, 18	eqtrd 1762	. . . 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)))
20	19	opreq2d 4709	. . 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))))
21		divmuldiv 6752	. . . . . . . 8 |- (((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)))
22		simpll 446	. . . . . . . . 9 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> C e. CC)
23	22, 13	jca 308	. . . . . . . 8 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> (C e. CC /\ D e. CC))
24		pm3.22 484	. . . . . . . 8 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> ((D e. CC /\ D =/= 0) /\ (C e. CC /\ C =/= 0)))
25	21, 23, 24	sylanc 521	. . . . . . 7 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> ((C / D) x. (D / C)) = ((C x. D) / (D x. C)))
26		mulcom 6255	. . . . . . . . 9 |- ((C e. CC /\ D e. CC) -> (C x. D) = (D x. C))
27	26	ad2ant2r 443	. . . . . . . 8 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> (C x. D) = (D x. C))
28	27	opreq1d 4708	. . . . . . 7 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> ((C x. D) / (D x. C)) = ((D x. C) / (D x. C)))
29		divid 6738	. . . . . . . 8 |- (((D x. C) e. CC /\ (D x. C) =/= 0) -> ((D x. C) / (D x. C)) = 1)
30		axmulcl 6222	. . . . . . . . . 10 |- ((D e. CC /\ C e. CC) -> (D x. C) e. CC)
31	30	ancoms 482	. . . . . . . . 9 |- ((C e. CC /\ D e. CC) -> (D x. C) e. CC)
32	31	ad2ant2r 443	. . . . . . . 8 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> (D x. C) e. CC)
33		mulne0 6683	. . . . . . . . 9 |- (((D e. CC /\ D =/= 0) /\ (C e. CC /\ C =/= 0)) -> (D x. C) =/= 0)
34	33	ancoms 482	. . . . . . . 8 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> (D x. C) =/= 0)
35	29, 32, 34	sylanc 521	. . . . . . 7 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> ((D x. C) / (D x. C)) = 1)
36	25, 28, 35	3eqtrd 1766	. . . . . 6 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> ((C / D) x. (D / C)) = 1)
37	36	adantl 422	. . . . 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)
38	37	opreq1d 4708	. . . 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)))
39		mulass 6257	. . . . 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))))
40		divcl 6697	. . . . . . . 8 |- ((C e. CC /\ D e. CC /\ D =/= 0) -> (C / D) e. CC)
41	40	3expb 947	. . . . . . 7 |- ((C e. CC /\ (D e. CC /\ D =/= 0)) -> (C / D) e. CC)
42	41	adantlr 427	. . . . . 6 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> (C / D) e. CC)
43	42	adantl 422	. . . . 5 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (C / D) e. CC)
44	39, 43, 6, 9	syl3anc 975	. . . 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))))
45		mulid2 6374	. . . . 5 |- ((A / B) e. CC -> (1 x. (A / B)) = (A / B))
46	9, 45	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))
47	38, 44, 46	3eqtr3d 1771	. . 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))
48	20, 47	eqtr3d 1764	. 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))
49		divmul 6692	. . 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)))
50		divcl 6697	. . . 4 |- (((A x. D) e. CC /\ (B x. C) e. CC /\ (B x. C) =/= 0) -> ((A x. D) / (B x. C)) e. CC)
51		axmulcl 6222	. . . . . 6 |- ((A e. CC /\ D e. CC) -> (A x. D) e. CC)
52	51	ad2ant2r 443	. . . . 5 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ (D e. CC /\ D =/= 0)) -> (A x. D) e. CC)
53	52	adantrl 428	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (A x. D) e. CC)
54		axmulcl 6222	. . . . . 6 |- ((B e. CC /\ C e. CC) -> (B x. C) e. CC)
55	54	ad2ant2r 443	. . . . 5 |- (((B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> (B x. C) e. CC)
56	55	ad2ant2lr 444	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (B x. C) e. CC)
57		mulne0 6683	. . . . 5 |- (((B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> (B x. C) =/= 0)
58	57	ad2ant2lr 444	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (B x. C) =/= 0)
59	50, 53, 56, 58	syl3anc 975	. . 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)
60		divne0 6708	. . . . 5 |- (((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0)) -> (C / D) =/= 0)
61	60	adantl 422	. . . 4 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> (C / D) =/= 0)
62	43, 61	jca 308	. . 3 |- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((C / D) e. CC /\ (C / D) =/= 0))
63	49, 9, 59, 62	syl3anc 975	. 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)))
64	48, 63	mpbird 212	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 162   /\ wa 239   = wceq 1136   e. wcel 1138   =/= wne 1854  (class class class)co 4695  CCcc 6180  0cc0 6182  1c1 6183   x. cmul 6187   / cdiv 6243

This theorem is referenced by:  divdivdivi 6761  recdiv 6762  divdiv1 6767  divdiv2 6768  qreccl 7248  georeclim 8297

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-13 1149  ax-14 1150  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702  ax-rep 3243  ax-sep 3253  ax-nul 3260  ax-pow 3296  ax-pr 3339  ax-un 3601  ax-inf2 5540

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-3or 856  df-3an 857  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614  df-clab 1709  df-cleq 1714  df-clel 1717  df-ne 1856  df-nel 1857  df-ral 1943  df-rex 1944  df-reu 1945  df-rab 1946  df-v 2127  df-sbc 2287  df-csb 2374  df-dif 2430  df-un 2433  df-in 2436  df-ss 2438  df-pss 2440  df-nul 2702  df-if 2807  df-pw 2859  df-sn 2873  df-pr 2874  df-tp 2876  df-op 2877  df-uni 3000  df-int 3037  df-iun 3079  df-br 3159  df-opab 3214  df-tr 3230  df-eprel 3398  df-id 3401  df-po 3406  df-so 3419  df-fr 3440  df-we 3459  df-ord 3475  df-on 3476  df-lim 3477  df-suc 3478  df-om 3761  df-xp 3811  df-rel 3812  df-cnv 3813  df-co 3814  df-dm 3815  df-rn 3816  df-res 3817  df-ima 3818  df-fun 3819  df-fn 3820  df-f 3821  df-f1 3822  df-fo 3823  df-f1o 3824  df-fv 3825  df-opr 4697  df-oprab 4698  df-mpt 4817  df-1st 4831  df-2nd 4832  df-iota 4900  df-rdg 4951  df-1o 4988  df-oadd 4990  df-omul 4991  df-er 5129  df-ec 5131  df-qs 5134  df-en 5238  df-dom 5239  df-sdom 5240  df-undef 5367  df-riota 5371  df-ni 5948  df-pli 5949  df-mi 5950  df-lti 5951  df-plpq 5983  df-mpq 5984  df-enq 5985  df-nq 5986  df-plq 5987  df-mq 5988  df-rq 5989  df-ltq 5990  df-1q 5991  df-np 6034  df-1p 6035  df-plp 6036  df-mp 6037  df-ltp 6038  df-plpr 6112  df-mpr 6113  df-enr 6114  df-nr 6115  df-plr 6116  df-mr 6117  df-ltr 6118  df-0r 6119  df-1r 6120  df-m1r 6121  df-c 6188  df-0 6189  df-1 6190  df-i 6191  df-r 6192  df-plus 6193  df-mul 6194  df-lt 6195  df-sub 6307  df-neg 6309  df-pnf 6450  df-mnf 6451  df-xr 6452  df-ltxr 6453  df-le 6454  df-div 6688

Copyright terms: Public domain