Theorem ltdiv23 10504

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 459	. . . . . . 7 |- ( ( B e. RR /\ 0 < B ) -> B e. RR )
2		gt0ne0 10086	. . . . . . 7 |- ( ( B e. RR /\ 0 < B ) -> B =/= 0 )
3	1, 2	jca 535	. . . . . 6 |- ( ( B e. RR /\ 0 < B ) -> ( B e. RR /\ B =/= 0 ) )
4		redivcl 10333	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) e. RR )
5	4	3expb 1210	. . . . . 6 |- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) ) -> ( A / B ) e. RR )
6	3, 5	sylan2 477	. . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( A / B ) e. RR )
7	6	3adant3 1029	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( A / B ) e. RR )
8		simp3 1011	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> C e. RR )
9		simp2 1010	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( B e. RR /\ 0 < B ) )
10		ltmul1 10462	. . . 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 1269	. . 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 1266	. 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 9634	. . . . . 6 |- ( A e. RR -> A e. CC )
14	13	adantr 467	. . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> A e. CC )
15		recn 9634	. . . . . 6 |- ( B e. RR -> B e. CC )
16	15	ad2antrl 735	. . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> B e. CC )
17	2	adantl 468	. . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> B =/= 0 )
18	14, 16, 17	divcan1d 10391	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) x. B ) = A )
19	18	3adant3 1029	. . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) x. B ) = A )
20	19	breq1d 4415	. 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 9629	. . . . . . . 8 |- ( ( C e. RR /\ B e. RR ) -> ( C x. B ) e. RR )
22	21	ancoms 455	. . . . . . 7 |- ( ( B e. RR /\ C e. RR ) -> ( C x. B ) e. RR )
23	22	adantrr 724	. . . . . 6 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
24	23	3adant1 1027	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
25		ltdiv1 10476	. . . . 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 1316	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
27		recn 9634	. . . . . . . . 9 |- ( C e. RR -> C e. CC )
28	27	adantr 467	. . . . . . . 8 |- ( ( C e. RR /\ 0 < C ) -> C e. CC )
29		gt0ne0 10086	. . . . . . . 8 |- ( ( C e. RR /\ 0 < C ) -> C =/= 0 )
30	28, 29	jca 535	. . . . . . 7 |- ( ( C e. RR /\ 0 < C ) -> ( C e. CC /\ C =/= 0 ) )
31		divcan3 10301	. . . . . . . 8 |- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> ( ( C x. B ) / C ) = B )
32	31	3expb 1210	. . . . . . 7 |- ( ( B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( C x. B ) / C ) = B )
33	15, 30, 32	syl2an 480	. . . . . 6 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
34	33	3adant1 1027	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
35	34	breq2d 4417	. . . 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 257	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < B ) )
37	36	3adant2r 1264	. 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 283	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 188   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   =/= wne 2624   class class class wbr 4405  (class class class)co 6295   CCcc 9542   RRcr 9543   0cc0 9544   x. cmul 9549   < clt 9680   / cdiv 10276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277

This theorem is referenced by:  ltdiv23i  10538  ltdiv23d  11410  divrcnv  13922  prmind2  14647  lebnumii  22009  bposlem2  24225  pntibndlem1  24439  stoweidlem7  37877  divlt1lt  40414