Theorem ltdiv2t 5849

Description: Division of a positive number by both sides of 'less than'.

Assertion

Ref	Expression
ltdiv2t	|- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < A /\ 0 < B /\ 0 < C)) -> (A < B <-> (C / B) < (C / A)))

Proof of Theorem ltdiv2t

Step	Hyp	Ref	Expression
1		an6 901	. 2 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < A /\ 0 < B /\ 0 < C)) <-> ((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)))
2		ltrect 5846	. . . 4 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> (A < B <-> (1 / B) < (1 / A)))
3	2	3adant3 798	. . 3 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> (A < B <-> (1 / B) < (1 / A)))
4		ltmul2t 5801	. . . . . . . . . . . . 13 |- ((((1 / B) e. RR /\ (1 / A) e. RR /\ C e. RR) /\ 0 < C) -> ((1 / B) < (1 / A) <-> (C x. (1 / B)) < (C x. (1 / A))))
5		rerecclt 5773	. . . . . . . . . . . . 13 |- ((A e. RR /\ A =/= 0) -> (1 / A) e. RR)
6	4, 5	syl3anl2 873	. . . . . . . . . . . 12 |- ((((1 / B) e. RR /\ (A e. RR /\ A =/= 0) /\ C e. RR) /\ 0 < C) -> ((1 / B) < (1 / A) <-> (C x. (1 / B)) < (C x. (1 / A))))
7		rerecclt 5773	. . . . . . . . . . . 12 |- ((B e. RR /\ B =/= 0) -> (1 / B) e. RR)
8	6, 7	syl3anl1 872	. . . . . . . . . . 11 |- ((((B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0) /\ C e. RR) /\ 0 < C) -> ((1 / B) < (1 / A) <-> (C x. (1 / B)) < (C x. (1 / A))))
9		divrect 5716	. . . . . . . . . . . . . . . . . 18 |- ((C e. CC /\ B e. CC /\ B =/= 0) -> (C / B) = (C x. (1 / B)))
10		recnt 5300	. . . . . . . . . . . . . . . . . 18 |- (B e. RR -> B e. CC)
11	9, 10	syl3an2 859	. . . . . . . . . . . . . . . . 17 |- ((C e. CC /\ B e. RR /\ B =/= 0) -> (C / B) = (C x. (1 / B)))
12	11	3expb 833	. . . . . . . . . . . . . . . 16 |- ((C e. CC /\ (B e. RR /\ B =/= 0)) -> (C / B) = (C x. (1 / B)))
13	12	3adant3 798	. . . . . . . . . . . . . . 15 |- ((C e. CC /\ (B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0)) -> (C / B) = (C x. (1 / B)))
14		divrect 5716	. . . . . . . . . . . . . . . . . 18 |- ((C e. CC /\ A e. CC /\ A =/= 0) -> (C / A) = (C x. (1 / A)))
15		recnt 5300	. . . . . . . . . . . . . . . . . 18 |- (A e. RR -> A e. CC)
16	14, 15	syl3an2 859	. . . . . . . . . . . . . . . . 17 |- ((C e. CC /\ A e. RR /\ A =/= 0) -> (C / A) = (C x. (1 / A)))
17	16	3expb 833	. . . . . . . . . . . . . . . 16 |- ((C e. CC /\ (A e. RR /\ A =/= 0)) -> (C / A) = (C x. (1 / A)))
18	17	3adant2 797	. . . . . . . . . . . . . . 15 |- ((C e. CC /\ (B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0)) -> (C / A) = (C x. (1 / A)))
19	13, 18	breq12d 2628	. . . . . . . . . . . . . 14 |- ((C e. CC /\ (B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0)) -> ((C / B) < (C / A) <-> (C x. (1 / B)) < (C x. (1 / A))))
20	19	3coml 839	. . . . . . . . . . . . 13 |- (((B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0) /\ C e. CC) -> ((C / B) < (C / A) <-> (C x. (1 / B)) < (C x. (1 / A))))
21		recnt 5300	. . . . . . . . . . . . 13 |- (C e. RR -> C e. CC)
22	20, 21	syl3an3 860	. . . . . . . . . . . 12 |- (((B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0) /\ C e. RR) -> ((C / B) < (C / A) <-> (C x. (1 / B)) < (C x. (1 / A))))
23	22	adantr 389	. . . . . . . . . . 11 |- ((((B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0) /\ C e. RR) /\ 0 < C) -> ((C / B) < (C / A) <-> (C x. (1 / B)) < (C x. (1 / A))))
24	8, 23	bitr4d 530	. . . . . . . . . 10 |- ((((B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0) /\ C e. RR) /\ 0 < C) -> ((1 / B) < (1 / A) <-> (C / B) < (C / A)))
25	24	ex 373	. . . . . . . . 9 |- (((B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0) /\ C e. RR) -> (0 < C -> ((1 / B) < (1 / A) <-> (C / B) < (C / A))))
26	25	3com12 836	. . . . . . . 8 |- (((A e. RR /\ A =/= 0) /\ (B e. RR /\ B =/= 0) /\ C e. RR) -> (0 < C -> ((1 / B) < (1 / A) <-> (C / B) < (C / A))))
27	26	3exp 831	. . . . . . 7 |- ((A e. RR /\ A =/= 0) -> ((B e. RR /\ B =/= 0) -> (C e. RR -> (0 < C -> ((1 / B) < (1 / A) <-> (C / B) < (C / A))))))
28	27	imp4a 364	. . . . . 6 |- ((A e. RR /\ A =/= 0) -> ((B e. RR /\ B =/= 0) -> ((C e. RR /\ 0 < C) -> ((1 / B) < (1 / A) <-> (C / B) < (C / A)))))
29	28	3imp 826	. . . . 5 |- (((A e. RR /\ A =/= 0) /\ (B e. RR /\ B =/= 0) /\ (C e. RR /\ 0 < C)) -> ((1 / B) < (1 / A) <-> (C / B) < (C / A)))
30		pm3.26 319	. . . . . 6 |- ((B e. RR /\ 0 < B) -> B e. RR)
31		gt0ne0t 5606	. . . . . 6 |- ((B e. RR /\ 0 < B) -> B =/= 0)
32	30, 31	jca 288	. . . . 5 |- ((B e. RR /\ 0 < B) -> (B e. RR /\ B =/= 0))
33	29, 32	syl3an2 859	. . . 4 |- (((A e. RR /\ A =/= 0) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> ((1 / B) < (1 / A) <-> (C / B) < (C / A)))
34		pm3.26 319	. . . . 5 |- ((A e. RR /\ 0 < A) -> A e. RR)
35		gt0ne0t 5606	. . . . 5 |- ((A e. RR /\ 0 < A) -> A =/= 0)
36	34, 35	jca 288	. . . 4 |- ((A e. RR /\ 0 < A) -> (A e. RR /\ A =/= 0))
37	33, 36	syl3an1 858	. . 3 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> ((1 / B) < (1 / A) <-> (C / B) < (C / A)))
38	3, 37	bitrd 527	. 2 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> (A < B <-> (C / B) < (C / A)))
39	1, 38	sylbi 199	1 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < A /\ 0 < B /\ 0 < C)) -> (A < B <-> (C / B) < (C / A)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957   =/= wne 1584   class class class wbr 2616  (class class class)co 3960  CCcc 5219  RRcr 5220  0cc0 5221  1c1 5222   x. cmul 5226   / cdiv 5281   < clt 5473

This theorem is referenced by:  efcltlem1 7282  sin01gt0 7454  sincos6thpi 8692

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-nel 1587  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-en 4364  df-dom 4365  df-sdom 4366  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-mp 5076  df-ltp 5077  df-plpr 5151  df-mpr 5152  df-enr 5153  df-nr 5154  df-plr 5155  df-mr 5156  df-ltr 5157  df-0r 5158  df-1r 5159  df-m1r 5160  df-c 5227  df-0 5228  df-1 5229  df-i 5230  df-r 5231  df-plus 5232  df-mul 5233  df-lt 5234  df-sub 5343  df-neg 5345  df-pnf 5474  df-mnf 5475  df-xr 5476  df-ltxr 5477  df-le 5478  df-div 5686

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