Theorem ltrec1 9161

Description: Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)

Assertion

Ref	Expression
ltrec1	|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / A ) < B <-> ( 1 / B ) < A ) )

Proof of Theorem ltrec1

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. RR )
2		simplr 529	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < A )
3	1, 2	gt0ap0d 8902	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A # 0 )
4	1, 3	rerecclapd 9107	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 / A ) e. RR )
5		recgt0 9123	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
6	5	adantr 276	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( 1 / A ) )
7		simpr 110	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( B e. RR /\ 0 < B ) )
8		ltrec 9156	. . 3 |- ( ( ( ( 1 / A ) e. RR /\ 0 < ( 1 / A ) ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / A ) < B <-> ( 1 / B ) < ( 1 / ( 1 / A ) ) ) )
9	4, 6, 7, 8	syl21anc 1273	. 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / A ) < B <-> ( 1 / B ) < ( 1 / ( 1 / A ) ) ) )
10	1	recnd 8301	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. CC )
11	10, 3	recrecapd 9058	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 / ( 1 / A ) ) = A )
12	11	breq2d 4120	. 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / B ) < ( 1 / ( 1 / A ) ) <-> ( 1 / B ) < A ) )
13	9, 12	bitrd 188	1 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / A ) < B <-> ( 1 / B ) < A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2203   class class class wbr 4108  (class class class)co 6049   RRcr 8125   0cc0 8126   1c1 8127   < clt 8307   / cdiv 8945

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946

This theorem is referenced by:  recreclt  9173  ltrec1d  10049  expnlbnd  11025