MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  recreclt Structured version   Unicode version

Theorem recreclt 10234
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 10176 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
2 gt0ne0 9807 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
3 rereccl 10052 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
42, 3syldan 470 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
5 1re 9388 . . . . 5  |-  1  e.  RR
6 ltaddpos 9832 . . . . 5  |-  ( ( ( 1  /  A
)  e.  RR  /\  1  e.  RR )  ->  ( 0  <  (
1  /  A )  <->  1  <  ( 1  +  ( 1  /  A ) ) ) )
74, 5, 6sylancl 662 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  (
1  /  A )  <->  1  <  ( 1  +  ( 1  /  A ) ) ) )
81, 7mpbid 210 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
1  <  ( 1  +  ( 1  /  A ) ) )
9 readdcl 9368 . . . . 5  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 1  +  ( 1  /  A
) )  e.  RR )
105, 4, 9sylancr 663 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  +  ( 1  /  A ) )  e.  RR )
11 0lt1 9865 . . . . . 6  |-  0  <  1
12 0re 9389 . . . . . . . 8  |-  0  e.  RR
13 lttr 9454 . . . . . . . 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 1304 . . . . . . 7  |-  ( ( 1  +  ( 1  /  A ) )  e.  RR  ->  (
( 0  <  1  /\  1  <  ( 1  +  ( 1  /  A ) ) )  ->  0  <  (
1  +  ( 1  /  A ) ) ) )
1510, 14syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 0  <  1  /\  1  < 
( 1  +  ( 1  /  A ) ) )  ->  0  <  ( 1  +  ( 1  /  A ) ) ) )
1611, 15mpani 676 . . . . 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 10231 . . . 4  |-  ( ( ( 1  +  ( 1  /  A ) )  e.  RR  /\  0  <  ( 1  +  ( 1  /  A
) ) )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 ) )
1910, 17, 18syl2anc 661 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 ) )
208, 19mpbid 210 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 )
21 ltaddpos 9832 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  <  1  <->  ( 1  /  A )  <  (
( 1  /  A
)  +  1 ) ) )
225, 4, 21sylancr 663 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  1  <->  ( 1  /  A )  <  ( ( 1  /  A )  +  1 ) ) )
2311, 22mpbii 211 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  <  ( (
1  /  A )  +  1 ) )
244recnd 9415 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  CC )
25 ax-1cn 9343 . . . . 5  |-  1  e.  CC
26 addcom 9558 . . . . 5  |-  ( ( ( 1  /  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2724, 25, 26sylancl 662 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2823, 27breqtrd 4319 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  <  ( 1  +  ( 1  /  A ) ) )
29 simpl 457 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
30 simpr 461 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  A )
31 ltrec1 10222 . . . 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 1219 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  A ) )
3328, 32mpbid 210 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  A )
3420, 33jca 532 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 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2609   class class class wbr 4295  (class class class)co 6094   CCcc 9283   RRcr 9284   0cc0 9285   1c1 9286    + caddc 9288    < clt 9421    / cdiv 9996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-po 4644  df-so 4645  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator