Theorem ltdiv2t 5887

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 902	. 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 5884	. . . 4 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> (A < B <-> (1 / B) < (1 / A)))
3	2	3adant3 799	. . 3 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> (A < B <-> (1 / B) < (1 / A)))
4		ltmul2t 5831	. . . . . . . . . . . . 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 5803	. . . . . . . . . . . . 13 |- ((A e. RR /\ A =/= 0) -> (1 / A) e. RR)
6	4, 5	syl3anl2 874	. . . . . . . . . . . 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 5803	. . . . . . . . . . . 12 |- ((B e. RR /\ B =/= 0) -> (1 / B) e. RR)
8	6, 7	syl3anl1 873	. . . . . . . . . . 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 5739	. . . . . . . . . . . . . . . . . 18 |- ((C e. CC /\ B e. CC /\ B =/= 0) -> (C / B) = (C x. (1 / B)))
10		recnt 5313	. . . . . . . . . . . . . . . . . 18 |- (B e. RR -> B e. CC)
11	9, 10	syl3an2 860	. . . . . . . . . . . . . . . . 17 |- ((C e. CC /\ B e. RR /\ B =/= 0) -> (C / B) = (C x. (1 / B)))
12	11	3expb 834	. . . . . . . . . . . . . . . 16 |- ((C e. CC /\ (B e. RR /\ B =/= 0)) -> (C / B) = (C x. (1 / B)))
13	12	3adant3 799	. . . . . . . . . . . . . . 15 |- ((C e. CC /\ (B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0)) -> (C / B) = (C x. (1 / B)))
14		divrect 5739	. . . . . . . . . . . . . . . . . 18 |- ((C e. CC /\ A e. CC /\ A =/= 0) -> (C / A) = (C x. (1 / A)))
15		recnt 5313	. . . . . . . . . . . . . . . . . 18 |- (A e. RR -> A e. CC)
16	14, 15	syl3an2 860	. . . . . . . . . . . . . . . . 17 |- ((C e. CC /\ A e. RR /\ A =/= 0) -> (C / A) = (C x. (1 / A)))
17	16	3expb 834	. . . . . . . . . . . . . . . 16 |- ((C e. CC /\ (A e. RR /\ A =/= 0)) -> (C / A) = (C x. (1 / A)))
18	17	3adant2 798	. . . . . . . . . . . . . . 15 |- ((C e. CC /\ (B e. RR /\ B =/= 0) /\ (A e. RR /\ A =/= 0)) -> (C / A) = (C x. (1 / A)))
19	13, 18	breq12d 2631	. . . . . . . . . . . . . 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 840	. . . . . . . . . . . . 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 5313	. . . . . . . . . . . . 13 |- (C e. RR -> C e. CC)
22	20, 21	syl3an3 861	. . . . . . . . . . . 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 531	. . . . . . . . . 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 837	. . . . . . . 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 832	. . . . . . 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 827	. . . . 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 5618	. . . . . 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 860	. . . 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 5618	. . . . 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 859	. . 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 528	. 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 775   = wceq 956   e. wcel 958   =/= wne 1585   class class class wbr 2619  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234  1c1 5235   x. cmul 5239   / cdiv 5294   < clt 5486

This theorem is referenced by:  efcltlem1 7304  sin01gt0 7476  sincos6thpi 8711

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703

Copyright terms: Public domain