Theorem ltdiv23 10427

Description: Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)

Assertion

Ref	Expression
ltdiv23	|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) < C <-> ( A / C ) < B ) )

Proof of Theorem ltdiv23

Step	Hyp	Ref	Expression
1		simpl 457	. . . . . . 7 |- ( ( B e. RR /\ 0 < B ) -> B e. RR )
2		gt0ne0 10008	. . . . . . 7 |- ( ( B e. RR /\ 0 < B ) -> B =/= 0 )
3	1, 2	jca 532	. . . . . 6 |- ( ( B e. RR /\ 0 < B ) -> ( B e. RR /\ B =/= 0 ) )
4		redivcl 10254	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) e. RR )
5	4	3expb 1192	. . . . . 6 |- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) ) -> ( A / B ) e. RR )
6	3, 5	sylan2 474	. . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( A / B ) e. RR )
7	6	3adant3 1011	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( A / B ) e. RR )
8		simp3 993	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> C e. RR )
9		simp2 992	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( B e. RR /\ 0 < B ) )
10		ltmul1 10383	. . . 4 |- ( ( ( A / B ) e. RR /\ C e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) < C <-> ( ( A / B ) x. B ) < ( C x. B ) ) )
11	7, 8, 9, 10	syl3anc 1223	. . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( ( A / B ) < C <-> ( ( A / B ) x. B ) < ( C x. B ) ) )
12	11	3adant3r 1220	. 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) < C <-> ( ( A / B ) x. B ) < ( C x. B ) ) )
13		recn 9573	. . . . . 6 |- ( A e. RR -> A e. CC )
14	13	adantr 465	. . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> A e. CC )
15		recn 9573	. . . . . 6 |- ( B e. RR -> B e. CC )
16	15	ad2antrl 727	. . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> B e. CC )
17	2	adantl 466	. . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> B =/= 0 )
18	14, 16, 17	divcan1d 10312	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) x. B ) = A )
19	18	3adant3 1011	. . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) x. B ) = A )
20	19	breq1d 4452	. 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( ( A / B ) x. B ) < ( C x. B ) <-> A < ( C x. B ) ) )
21		remulcl 9568	. . . . . . . 8 |- ( ( C e. RR /\ B e. RR ) -> ( C x. B ) e. RR )
22	21	ancoms 453	. . . . . . 7 |- ( ( B e. RR /\ C e. RR ) -> ( C x. B ) e. RR )
23	22	adantrr 716	. . . . . 6 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
24	23	3adant1 1009	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
25		ltdiv1 10397	. . . . 5 |- ( ( A e. RR /\ ( C x. B ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
26	24, 25	syld3an2 1270	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
27		recn 9573	. . . . . . . . 9 |- ( C e. RR -> C e. CC )
28	27	adantr 465	. . . . . . . 8 |- ( ( C e. RR /\ 0 < C ) -> C e. CC )
29		gt0ne0 10008	. . . . . . . 8 |- ( ( C e. RR /\ 0 < C ) -> C =/= 0 )
30	28, 29	jca 532	. . . . . . 7 |- ( ( C e. RR /\ 0 < C ) -> ( C e. CC /\ C =/= 0 ) )
31		divcan3 10222	. . . . . . . 8 |- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> ( ( C x. B ) / C ) = B )
32	31	3expb 1192	. . . . . . 7 |- ( ( B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( C x. B ) / C ) = B )
33	15, 30, 32	syl2an 477	. . . . . 6 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
34	33	3adant1 1009	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
35	34	breq2d 4454	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) < ( ( C x. B ) / C ) <-> ( A / C ) < B ) )
36	26, 35	bitrd 253	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < B ) )
37	36	3adant2r 1218	. 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < B ) )
38	12, 20, 37	3bitrd 279	1 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) < C <-> ( A / C ) < B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   =/= wne 2657   class class class wbr 4442  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   x. cmul 9488   < clt 9619   / cdiv 10197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198

This theorem is referenced by:  ltdiv23i  10461  ltdiv23d  11303  divrcnv  13618  prmind2  14078  lebnumii  21196  bposlem2  23283  pntibndlem1  23497  stoweidlem7  31264