Theorem lediv2t 5847

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

Assertion

Ref	Expression
lediv2t	|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> (A <_ B <-> (C / B) <_ (C / A)))

Proof of Theorem lediv2t

Step	Hyp	Ref	Expression
1		lemul2t 5797	. . 3 |- ((((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))))
2		gt0ne0t 5600	. . . . . 6 |- ((B e. RR /\ 0 < B) -> B =/= 0)
3		rerecclt 5767	. . . . . 6 |- ((B e. RR /\ B =/= 0) -> (1 / B) e. RR)
4	2, 3	syldan 467	. . . . 5 |- ((B e. RR /\ 0 < B) -> (1 / B) e. RR)
5	4	3ad2ant2 800	. . . 4 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> (1 / B) e. RR)
6		gt0ne0t 5600	. . . . . 6 |- ((A e. RR /\ 0 < A) -> A =/= 0)
7		rerecclt 5767	. . . . . 6 |- ((A e. RR /\ A =/= 0) -> (1 / A) e. RR)
8	6, 7	syldan 467	. . . . 5 |- ((A e. RR /\ 0 < A) -> (1 / A) e. RR)
9	8	3ad2ant1 799	. . . 4 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> (1 / A) e. RR)
10		pm3.26 319	. . . . 5 |- ((C e. RR /\ 0 < C) -> C e. RR)
11	10	3ad2ant3 801	. . . 4 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> C e. RR)
12	5, 9, 11	3jca 818	. . 3 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> ((1 / B) e. RR /\ (1 / A) e. RR /\ C e. RR))
13		pm3.27 323	. . . 4 |- ((C e. RR /\ 0 < C) -> 0 < C)
14	13	3ad2ant3 801	. . 3 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> 0 < C)
15	1, 12, 14	sylanc 471	. 2 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> ((1 / B) <_ (1 / A) <-> (C x. (1 / B)) <_ (C x. (1 / A))))
16		lerect 5841	. . 3 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> (A <_ B <-> (1 / B) <_ (1 / A)))
17	16	3adant3 798	. 2 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> (A <_ B <-> (1 / B) <_ (1 / A)))
18		divrect 5710	. . . . . . . 8 |- ((C e. CC /\ B e. CC /\ B =/= 0) -> (C / B) = (C x. (1 / B)))
19	18	3expb 833	. . . . . . 7 |- ((C e. CC /\ (B e. CC /\ B =/= 0)) -> (C / B) = (C x. (1 / B)))
20		recnt 5293	. . . . . . 7 |- (C e. RR -> C e. CC)
21		recnt 5293	. . . . . . . . 9 |- (B e. RR -> B e. CC)
22	21	adantr 389	. . . . . . . 8 |- ((B e. RR /\ 0 < B) -> B e. CC)
23	22, 2	jca 288	. . . . . . 7 |- ((B e. RR /\ 0 < B) -> (B e. CC /\ B =/= 0))
24	19, 20, 23	syl2an 454	. . . . . 6 |- ((C e. RR /\ (B e. RR /\ 0 < B)) -> (C / B) = (C x. (1 / B)))
25	24	3adant2 797	. . . . 5 |- ((C e. RR /\ (A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> (C / B) = (C x. (1 / B)))
26		divrect 5710	. . . . . . . 8 |- ((C e. CC /\ A e. CC /\ A =/= 0) -> (C / A) = (C x. (1 / A)))
27	26	3expb 833	. . . . . . 7 |- ((C e. CC /\ (A e. CC /\ A =/= 0)) -> (C / A) = (C x. (1 / A)))
28		recnt 5293	. . . . . . . . 9 |- (A e. RR -> A e. CC)
29	28	adantr 389	. . . . . . . 8 |- ((A e. RR /\ 0 < A) -> A e. CC)
30	29, 6	jca 288	. . . . . . 7 |- ((A e. RR /\ 0 < A) -> (A e. CC /\ A =/= 0))
31	27, 20, 30	syl2an 454	. . . . . 6 |- ((C e. RR /\ (A e. RR /\ 0 < A)) -> (C / A) = (C x. (1 / A)))
32	31	3adant3 798	. . . . 5 |- ((C e. RR /\ (A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> (C / A) = (C x. (1 / A)))
33	25, 32	breq12d 2626	. . . 4 |- ((C e. RR /\ (A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> ((C / B) <_ (C / A) <-> (C x. (1 / B)) <_ (C x. (1 / A))))
34	33	3coml 839	. . 3 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ C e. RR) -> ((C / B) <_ (C / A) <-> (C x. (1 / B)) <_ (C x. (1 / A))))
35	34	3adant3r 856	. 2 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> ((C / B) <_ (C / A) <-> (C x. (1 / B)) <_ (C x. (1 / A))))
36	15, 17, 35	3bitr4d 549	1 |- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> (A <_ B <-> (C / B) <_ (C / A)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956   =/= wne 1582   class class class wbr 2614  (class class class)co 3954  CCcc 5212  RRcr 5213  0cc0 5214  1c1 5215   x. cmul 5219   / cdiv 5274   <_ cle 5275   < clt 5466

This theorem is referenced by:  efaddlem22 7309  nndivlub 10358

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

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