Theorem ltdiv23 10456

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 10038	. . . . . . 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 10284	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) e. RR )
5	4	3expb 1197	. . . . . 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 1016	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( A / B ) e. RR )
8		simp3 998	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> C e. RR )
9		simp2 997	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( B e. RR /\ 0 < B ) )
10		ltmul1 10413	. . . 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 1228	. . 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 1225	. 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 9599	. . . . . 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 9599	. . . . . 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 10342	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) x. B ) = A )
19	18	3adant3 1016	. . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) x. B ) = A )
20	19	breq1d 4466	. 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 9594	. . . . . . . 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 1014	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
25		ltdiv1 10427	. . . . 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 1275	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
27		recn 9599	. . . . . . . . 9 |- ( C e. RR -> C e. CC )
28	27	adantr 465	. . . . . . . 8 |- ( ( C e. RR /\ 0 < C ) -> C e. CC )
29		gt0ne0 10038	. . . . . . . 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 10252	. . . . . . . 8 |- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> ( ( C x. B ) / C ) = B )
32	31	3expb 1197	. . . . . . 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 1014	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
35	34	breq2d 4468	. . . 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 1223	. 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 973   = wceq 1395   e. wcel 1819   =/= wne 2652   class class class wbr 4456  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   x. cmul 9514   < clt 9645   / cdiv 10227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228

This theorem is referenced by:  ltdiv23i  10490  ltdiv23d  11337  divrcnv  13676  prmind2  14240  lebnumii  21592  bposlem2  23686  pntibndlem1  23900  stoweidlem7  31992