Theorem ltrec 9062

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 8193	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 1 e. RR )
2		simprl 531	. . . 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 529	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < A )
5		ltmuldiv 9053	. . . 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 1277	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 x. A ) < B <-> 1 < ( B / A ) ) )
7	3	recnd 8207	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. CC )
8	7	mulid2d 8197	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 x. A ) = A )
9	8	breq1d 4098	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 x. A ) < B <-> A < B ) )
10	2	recnd 8207	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. CC )
11	3, 4	gt0ap0d 8808	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A # 0 )
12	10, 7, 11	divrecapd 8972	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( B / A ) = ( B x. ( 1 / A ) ) )
13	12	breq2d 4100	. . 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 9013	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 / A ) e. RR )
16		simprr 533	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < B )
17		ltdivmul 9055	. . 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 1277	. 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 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   0cc0 8031   1c1 8032   x. cmul 8036   < clt 8213   / cdiv 8851

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852

This theorem is referenced by:  lerec  9063  ltdiv2  9066  ltrec1  9067  reclt1  9075  recgt1  9076  ltreci  9091  nnrecl  9399  ltrecd  9949