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Theorem recreclt 9671
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 9616 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
2 gt0ne0 9255 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
3 rereccl 9494 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
42, 3syldan 456 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
5 1re 8853 . . . . 5  |-  1  e.  RR
6 ltaddpos 9280 . . . . 5  |-  ( ( ( 1  /  A
)  e.  RR  /\  1  e.  RR )  ->  ( 0  <  (
1  /  A )  <->  1  <  ( 1  +  ( 1  /  A ) ) ) )
74, 5, 6sylancl 643 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  (
1  /  A )  <->  1  <  ( 1  +  ( 1  /  A ) ) ) )
81, 7mpbid 201 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
1  <  ( 1  +  ( 1  /  A ) ) )
9 readdcl 8836 . . . . 5  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 1  +  ( 1  /  A
) )  e.  RR )
105, 4, 9sylancr 644 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  +  ( 1  /  A ) )  e.  RR )
11 0lt1 9312 . . . . . 6  |-  0  <  1
12 0re 8854 . . . . . . . 8  |-  0  e.  RR
13 lttr 8915 . . . . . . . 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 1267 . . . . . . 7  |-  ( ( 1  +  ( 1  /  A ) )  e.  RR  ->  (
( 0  <  1  /\  1  <  ( 1  +  ( 1  /  A ) ) )  ->  0  <  (
1  +  ( 1  /  A ) ) ) )
1510, 14syl 15 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 0  <  1  /\  1  < 
( 1  +  ( 1  /  A ) ) )  ->  0  <  ( 1  +  ( 1  /  A ) ) ) )
1611, 15mpani 657 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  ->  0  <  (
1  +  ( 1  /  A ) ) ) )
178, 16mpd 14 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  +  ( 1  /  A ) ) )
18 recgt1 9668 . . . 4  |-  ( ( ( 1  +  ( 1  /  A ) )  e.  RR  /\  0  <  ( 1  +  ( 1  /  A
) ) )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 ) )
1910, 17, 18syl2anc 642 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 ) )
208, 19mpbid 201 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 )
21 ltaddpos 9280 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  <  1  <->  ( 1  /  A )  <  (
( 1  /  A
)  +  1 ) ) )
225, 4, 21sylancr 644 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  1  <->  ( 1  /  A )  <  ( ( 1  /  A )  +  1 ) ) )
2311, 22mpbii 202 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  <  ( (
1  /  A )  +  1 ) )
244recnd 8877 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  CC )
25 ax-1cn 8811 . . . . 5  |-  1  e.  CC
26 addcom 9014 . . . . 5  |-  ( ( ( 1  /  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2724, 25, 26sylancl 643 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2823, 27breqtrd 4063 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  <  ( 1  +  ( 1  /  A ) ) )
29 simpl 443 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
30 simpr 447 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  A )
31 ltrec1 9659 . . . 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 ) )
3229, 30, 10, 17, 31syl22anc 1183 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  A ) )
3328, 32mpbid 201 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  A )
3420, 33jca 518 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    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    / cdiv 9439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440
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