HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version
Theorem ltdiv23t 5840
Description: Swap denominator with other side of 'less than'.
Assertion
Ref Expression
ltdiv23t |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < B /\ 0 < C)) -> ((A / B) < C <-> (A / C) < B))
Proof of Theorem ltdiv23t
StepHypRef Expression
1 ltmul1t 5786 . . . 4 |- ((((A / B) e. RR /\ C e. RR /\ B e. RR) /\ 0 < B) -> ((A / B) < C <-> ((A / B) x. B) < (C x. B)))
2 gt0ne0t 5592 . . . . . . . 8 |- ((B e. RR /\ 0 < B) -> B =/= 0)
32adantll 392 . . . . . . 7 |- (((A e. RR /\ B e. RR) /\ 0 < B) -> B =/= 0)
4 redivclt 5756 . . . . . . . 8 |- ((A e. RR /\ B e. RR /\ B =/= 0) -> (A / B) e. RR)
543expa 831 . . . . . . 7 |- (((A e. RR /\ B e. RR) /\ B =/= 0) -> (A / B) e. RR)
63, 5syldan 467 . . . . . 6 |- (((A e. RR /\ B e. RR) /\ 0 < B) -> (A / B) e. RR)
763adantl3 803 . . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> (A / B) e. RR)
8 3simp3 788 . . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> C e. RR)
98adantr 389 . . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> C e. RR)
10 3simp2 787 . . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> B e. RR)
1110adantr 389 . . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> B e. RR)
127, 9, 113jca 817 . . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> ((A / B) e. RR /\ C e. RR /\ B e. RR))
13 pm3.27 323 . . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> 0 < B)
141, 12, 13sylanc 471 . . 3 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> ((A / B) < C <-> ((A / B) x. B) < (C x. B)))
1514adantrr 395 . 2 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < B /\ 0 < C)) -> ((A / B) < C <-> ((A / B) x. B) < (C x. B)))
16 divcan1t 5689 . . . . . . . . 9 |- ((B e. CC /\ A e. CC /\ B =/= 0) -> ((A / B) x. B) = A)
17163com12 835 . . . . . . . 8 |- ((A e. CC /\ B e. CC /\ B =/= 0) -> ((A / B) x. B) = A)
18173expa 831 . . . . . . 7 |- (((A e. CC /\ B e. CC) /\ B =/= 0) -> ((A / B) x. B) = A)
19 recnt 5285 . . . . . . . 8 |- (A e. RR -> A e. CC)
20 recnt 5285 . . . . . . . 8 |- (B e. RR -> B e. CC)
2119, 20anim12i 333 . . . . . . 7 |- ((A e. RR /\ B e. RR) -> (A e. CC /\ B e. CC))
2218, 21sylan 448 . . . . . 6 |- (((A e. RR /\ B e. RR) /\ B =/= 0) -> ((A / B) x. B) = A)
233, 22syldan 467 . . . . 5 |- (((A e. RR /\ B e. RR) /\ 0 < B) -> ((A / B) x. B) = A)
24233adantl3 803 . . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> ((A / B) x. B) = A)
2524adantrr 395 . . 3 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < B /\ 0 < C)) -> ((A / B) x. B) = A)
2625breq1d 2619 . 2 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < B /\ 0 < C)) -> (((A / B) x. B) < (C x. B) <-> A < (C x. B)))
27 ltdiv1t 5805 . . . . 5 |- (((A e. RR /\ (C x. B) e. RR /\ C e. RR) /\ 0 < C) -> (A < (C x. B) <-> (A / C) < ((C x. B) / C)))
28 3simp1 786 . . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> A e. RR)
29 axmulrcl 5246 . . . . . . . 8 |- ((C e. RR /\ B e. RR) -> (C x. B) e. RR)
3029ancoms 436 . . . . . . 7 |- ((B e. RR /\ C e. RR) -> (C x. B) e. RR)
31303adant1 795 . . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (C x. B) e. RR)
3228, 31, 83jca 817 . . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (A e. RR /\ (C x. B) e. RR /\ C e. RR))
3327, 32sylan 448 . . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A < (C x. B) <-> (A / C) < ((C x. B) / C)))
34 gt0ne0t 5592 . . . . . . . 8 |- ((C e. RR /\ 0 < C) -> C =/= 0)
3534adantll 392 . . . . . . 7 |- (((B e. RR /\ C e. RR) /\ 0 < C) -> C =/= 0)
36 divcan3t 5718 . . . . . . . . . . 11 |- ((C e. CC /\ B e. CC /\ C =/= 0) -> ((C x. B) / C) = B)
3736, 20syl3an2 858 . . . . . . . . . 10 |- ((C e. CC /\ B e. RR /\ C =/= 0) -> ((C x. B) / C) = B)
38 recnt 5285 . . . . . . . . . 10 |- (C e. RR -> C e. CC)
3937, 38syl3an1 857 . . . . . . . . 9 |- ((C e. RR /\ B e. RR /\ C =/= 0) -> ((C x. B) / C) = B)
40393com12 835 . . . . . . . 8 |- ((B e. RR /\ C e. RR /\ C =/= 0) -> ((C x. B) / C) = B)
41403expa 831 . . . . . . 7 |- (((B e. RR /\ C e. RR) /\ C =/= 0) -> ((C x. B) / C) = B)
4235, 41syldan 467 . . . . . 6 |- (((B e. RR /\ C e. RR) /\ 0 < C) -> ((C x. B) / C) = B)
43423adantl1 801 . . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> ((C x. B) / C) = B)
4443breq2d 2620 . . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> ((A / C) < ((C x. B) / C) <-> (A / C) < B))
4533, 44bitrd 526 . . 3 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A < (C x. B) <-> (A / C) < B))
4645adantrl 394 . 2 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < B /\ 0 < C)) -> (A < (C x. B) <-> (A / C) < B))
4715, 26, 463bitrd 542 1 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < B /\ 0 < C)) -> ((A / B) < C <-> (A / C) < B))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577   class class class wbr 2609  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206   x. cmul 5211   / cdiv 5266   < clt 5458
This theorem is referenced by:  ltdiv23 5842  efaddlem23 7302
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672
Copyright terms: Public domain