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Theorem recreclt 10506
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 10450 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
2 gt0ne0 10080 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
3 rereccl 10326 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
42, 3syldan 472 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
5 1re 9643 . . . . 5  |-  1  e.  RR
6 ltaddpos 10105 . . . . 5  |-  ( ( ( 1  /  A
)  e.  RR  /\  1  e.  RR )  ->  ( 0  <  (
1  /  A )  <->  1  <  ( 1  +  ( 1  /  A ) ) ) )
74, 5, 6sylancl 666 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  (
1  /  A )  <->  1  <  ( 1  +  ( 1  /  A ) ) ) )
81, 7mpbid 213 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
1  <  ( 1  +  ( 1  /  A ) ) )
9 readdcl 9623 . . . . 5  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 1  +  ( 1  /  A
) )  e.  RR )
105, 4, 9sylancr 667 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  +  ( 1  /  A ) )  e.  RR )
11 0lt1 10137 . . . . . 6  |-  0  <  1
12 0re 9644 . . . . . . . 8  |-  0  e.  RR
13 lttr 9711 . . . . . . . 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 1350 . . . . . . 7  |-  ( ( 1  +  ( 1  /  A ) )  e.  RR  ->  (
( 0  <  1  /\  1  <  ( 1  +  ( 1  /  A ) ) )  ->  0  <  (
1  +  ( 1  /  A ) ) ) )
1510, 14syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 0  <  1  /\  1  < 
( 1  +  ( 1  /  A ) ) )  ->  0  <  ( 1  +  ( 1  /  A ) ) ) )
1611, 15mpani 680 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  ->  0  <  (
1  +  ( 1  /  A ) ) ) )
178, 16mpd 15 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  +  ( 1  /  A ) ) )
18 recgt1 10503 . . . 4  |-  ( ( ( 1  +  ( 1  /  A ) )  e.  RR  /\  0  <  ( 1  +  ( 1  /  A
) ) )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 ) )
1910, 17, 18syl2anc 665 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 ) )
208, 19mpbid 213 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 )
21 ltaddpos 10105 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  <  1  <->  ( 1  /  A )  <  (
( 1  /  A
)  +  1 ) ) )
225, 4, 21sylancr 667 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  1  <->  ( 1  /  A )  <  ( ( 1  /  A )  +  1 ) ) )
2311, 22mpbii 214 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  <  ( (
1  /  A )  +  1 ) )
244recnd 9670 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  CC )
25 ax-1cn 9598 . . . . 5  |-  1  e.  CC
26 addcom 9820 . . . . 5  |-  ( ( ( 1  /  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2724, 25, 26sylancl 666 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2823, 27breqtrd 4445 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  <  ( 1  +  ( 1  /  A ) ) )
29 simpl 458 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
30 simpr 462 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  A )
31 ltrec1 10494 . . . 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 1265 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  A ) )
3328, 32mpbid 213 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  A )
3420, 33jca 534 1  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  / 
( 1  +  ( 1  /  A ) ) )  <  1  /\  ( 1  /  (
1  +  ( 1  /  A ) ) )  <  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   class class class wbr 4420  (class class class)co 6302   CCcc 9538   RRcr 9539   0cc0 9540   1c1 9541    + caddc 9543    < clt 9676    / cdiv 10270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271
This theorem is referenced by: (None)
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