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Theorem recreclt 8754
Description: Given a positive number A, construct a new positive number less than both A and 1. (Contributed by NM, 28-Dec-2005.)
Assertion
Ref Expression
recreclt |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / ( 1 + ( 1 / A ) ) ) < 1 /\ ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )
Proof of Theorem recreclt
StepHypRef Expression
1 recgt0 8704 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
2 simpl 108 . . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A e. RR )
3 gt0ap0 8484 . . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
42, 3rerecclapd 8689 . . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
5 1re 7860 . . . . 5 |- 1 e. RR
6 ltaddpos 8310 . . . . 5 |- ( ( ( 1 / A ) e. RR /\ 1 e. RR ) -> ( 0 < ( 1 / A ) <-> 1 < ( 1 + ( 1 / A ) ) ) )
74, 5, 6sylancl 410 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 0 < ( 1 / A ) <-> 1 < ( 1 + ( 1 / A ) ) ) )
81, 7mpbid 146 . . 3 |- ( ( A e. RR /\ 0 < A ) -> 1 < ( 1 + ( 1 / A ) ) )
9 readdcl 7841 . . . . 5 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 1 + ( 1 / A ) ) e. RR )
105, 4, 9sylancr 411 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 + ( 1 / A ) ) e. RR )
11 0lt1 7985 . . . . . 6 |- 0 < 1
12 0re 7861 . . . . . . . 8 |- 0 e. RR
13 lttr 7934 . . . . . . . 8 |- ( ( 0 e. RR /\ 1 e. RR /\ ( 1 + ( 1 / A ) ) e. RR ) -> ( ( 0 < 1 /\ 1 < ( 1 + ( 1 / A ) ) ) -> 0 < ( 1 + ( 1 / A ) ) ) )
1412, 5, 13mp3an12 1309 . . . . . . 7 |- ( ( 1 + ( 1 / A ) ) e. RR -> ( ( 0 < 1 /\ 1 < ( 1 + ( 1 / A ) ) ) -> 0 < ( 1 + ( 1 / A ) ) ) )
1510, 14syl 14 . . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> ( ( 0 < 1 /\ 1 < ( 1 + ( 1 / A ) ) ) -> 0 < ( 1 + ( 1 / A ) ) ) )
1611, 15mpani 427 . . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 < ( 1 + ( 1 / A ) ) -> 0 < ( 1 + ( 1 / A ) ) ) )
178, 16mpd 13 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 + ( 1 / A ) ) )
18 recgt1 8751 . . . 4 |- ( ( ( 1 + ( 1 / A ) ) e. RR /\ 0 < ( 1 + ( 1 / A ) ) ) -> ( 1 < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 ) )
1910, 17, 18syl2anc 409 . . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 ) )
208, 19mpbid 146 . 2 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 )
21 ltaddpos 8310 . . . . . 6 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 0 < 1 <-> ( 1 / A ) < ( ( 1 / A ) + 1 ) ) )
225, 4, 21sylancr 411 . . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 0 < 1 <-> ( 1 / A ) < ( ( 1 / A ) + 1 ) ) )
2311, 22mpbii 147 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( ( 1 / A ) + 1 ) )
244recnd 7889 . . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. CC )
25 ax-1cn 7808 . . . . 5 |- 1 e. CC
26 addcom 7995 . . . . 5 |- ( ( ( 1 / A ) e. CC /\ 1 e. CC ) -> ( ( 1 / A ) + 1 ) = ( 1 + ( 1 / A ) ) )
2724, 25, 26sylancl 410 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) + 1 ) = ( 1 + ( 1 / A ) ) )
2823, 27breqtrd 3990 . . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( 1 + ( 1 / A ) ) )
29 simpr 109 . . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < A )
30 ltrec1 8742 . . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( ( 1 + ( 1 / A ) ) e. RR /\ 0 < ( 1 + ( 1 / A ) ) ) ) -> ( ( 1 / A ) < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )
312, 29, 10, 17, 30syl22anc 1221 . . 3 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )
3228, 31mpbid 146 . 2 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 + ( 1 / A ) ) ) < A )
3320, 32jca 304 1 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / ( 1 + ( 1 / A ) ) ) < 1 /\ ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   class class class wbr 3965  (class class class)co 5818   CCcc 7713   RRcr 7714   0cc0 7715   1c1 7716   + caddc 7718   < clt 7895   / cdiv 8528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4252  df-po 4255  df-iso 4256  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-iota 5132  df-fun 5169  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529
This theorem is referenced by: (None)
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