Theorem ltdiv23 10222

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 9803	. . . . . . 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 10049	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) e. RR )
5	4	3expb 1188	. . . . . 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 1008	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( A / B ) e. RR )
8		simp3 990	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> C e. RR )
9		simp2 989	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( B e. RR /\ 0 < B ) )
10		ltmul1 10178	. . . 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 1218	. . 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 1215	. 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 9371	. . . . . 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 9371	. . . . . 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 10107	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) x. B ) = A )
19	18	3adant3 1008	. . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) x. B ) = A )
20	19	breq1d 4301	. 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 9366	. . . . . . . 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 1006	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
25		ltdiv1 10192	. . . . 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 1265	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
27		recn 9371	. . . . . . . . 9 |- ( C e. RR -> C e. CC )
28	27	adantr 465	. . . . . . . 8 |- ( ( C e. RR /\ 0 < C ) -> C e. CC )
29		gt0ne0 9803	. . . . . . . 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 10017	. . . . . . . 8 |- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> ( ( C x. B ) / C ) = B )
32	31	3expb 1188	. . . . . . 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 1006	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
35	34	breq2d 4303	. . . 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 1213	. 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 965   = wceq 1369   e. wcel 1756   =/= wne 2605   class class class wbr 4291  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   x. cmul 9286   < clt 9417   / cdiv 9992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993

This theorem is referenced by:  ltdiv23i  10256  ltdiv23d  11082  divrcnv  13314  prmind2  13773  lebnumii  20537  bposlem2  22623  pntibndlem1  22837  stoweidlem7  29800