Theorem ltrec 8871

Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
ltrec	|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )

Proof of Theorem ltrec

Step	Hyp	Ref	Expression
1		1red 8003	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 1 e. RR )
2		simprl 529	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. RR )
3		simpll 527	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. RR )
4		simplr 528	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < A )
5		ltmuldiv 8862	. . . 4 |- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( 1 x. A ) < B <-> 1 < ( B / A ) ) )
6	1, 2, 3, 4, 5	syl112anc 1253	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 x. A ) < B <-> 1 < ( B / A ) ) )
7	3	recnd 8017	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. CC )
8	7	mulid2d 8007	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 x. A ) = A )
9	8	breq1d 4028	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 x. A ) < B <-> A < B ) )
10	2	recnd 8017	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. CC )
11	3, 4	gt0ap0d 8617	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A # 0 )
12	10, 7, 11	divrecapd 8781	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( B / A ) = ( B x. ( 1 / A ) ) )
13	12	breq2d 4030	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 < ( B / A ) <-> 1 < ( B x. ( 1 / A ) ) ) )
14	6, 9, 13	3bitr3d 218	. 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> 1 < ( B x. ( 1 / A ) ) ) )
15	3, 11	rerecclapd 8822	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 / A ) e. RR )
16		simprr 531	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < B )
17		ltdivmul 8864	. . 3 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> 1 < ( B x. ( 1 / A ) ) ) )
18	1, 15, 2, 16, 17	syl112anc 1253	. 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> 1 < ( B x. ( 1 / A ) ) ) )
19	14, 18	bitr4d 191	1 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2160   class class class wbr 4018  (class class class)co 5897   RRcr 7841   0cc0 7842   1c1 7843   x. cmul 7847   < clt 8023   / cdiv 8660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661

This theorem is referenced by:  lerec  8872  ltdiv2  8875  ltrec1  8876  reclt1  8884  recgt1  8885  ltreci  8900  nnrecl  9205  ltrecd  9747