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Theorem recreclt 8333
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 8283 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
2 simpl 107 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
3 gt0ap0 8078 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A #  0 )
42, 3rerecclapd 8272 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
5 1re 7466 . . . . 5  |-  1  e.  RR
6 ltaddpos 7909 . . . . 5  |-  ( ( ( 1  /  A
)  e.  RR  /\  1  e.  RR )  ->  ( 0  <  (
1  /  A )  <->  1  <  ( 1  +  ( 1  /  A ) ) ) )
74, 5, 6sylancl 404 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  (
1  /  A )  <->  1  <  ( 1  +  ( 1  /  A ) ) ) )
81, 7mpbid 145 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
1  <  ( 1  +  ( 1  /  A ) ) )
9 readdcl 7447 . . . . 5  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 1  +  ( 1  /  A
) )  e.  RR )
105, 4, 9sylancr 405 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  +  ( 1  /  A ) )  e.  RR )
11 0lt1 7589 . . . . . 6  |-  0  <  1
12 0re 7467 . . . . . . . 8  |-  0  e.  RR
13 lttr 7538 . . . . . . . 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 1263 . . . . . . 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 421 . . . . 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 8330 . . . 4  |-  ( ( ( 1  +  ( 1  /  A ) )  e.  RR  /\  0  <  ( 1  +  ( 1  /  A
) ) )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 ) )
1910, 17, 18syl2anc 403 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 ) )
208, 19mpbid 145 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 )
21 ltaddpos 7909 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  <  1  <->  ( 1  /  A )  <  (
( 1  /  A
)  +  1 ) ) )
225, 4, 21sylancr 405 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  1  <->  ( 1  /  A )  <  ( ( 1  /  A )  +  1 ) ) )
2311, 22mpbii 146 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  <  ( (
1  /  A )  +  1 ) )
244recnd 7495 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  CC )
25 ax-1cn 7417 . . . . 5  |-  1  e.  CC
26 addcom 7598 . . . . 5  |-  ( ( ( 1  /  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2724, 25, 26sylancl 404 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2823, 27breqtrd 3861 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  <  ( 1  +  ( 1  /  A ) ) )
29 simpr 108 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  A )
30 ltrec1 8321 . . . 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 1175 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  A ) )
3228, 31mpbid 145 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  A )
3320, 32jca 300 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327   RRcr 7328   0cc0 7329   1c1 7330    + caddc 7332    < clt 7501    / cdiv 8113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114
This theorem is referenced by: (None)
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