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Theorem recp1lt1 10483
Description: Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
Assertion
Ref Expression
recp1lt1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  /  (
1  +  A ) )  <  1 )

Proof of Theorem recp1lt1
StepHypRef Expression
1 ltp1 10421 . . . . 5  |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
2 recn 9612 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
3 ax-1cn 9580 . . . . . 6  |-  1  e.  CC
4 addcom 9800 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
52, 3, 4sylancl 660 . . . . 5  |-  ( A  e.  RR  ->  ( A  +  1 )  =  ( 1  +  A ) )
61, 5breqtrd 4419 . . . 4  |-  ( A  e.  RR  ->  A  <  ( 1  +  A
) )
76adantr 463 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  <  ( 1  +  A ) )
82adantr 463 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  CC )
9 1re 9625 . . . . . . 7  |-  1  e.  RR
10 readdcl 9605 . . . . . . 7  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  +  A
)  e.  RR )
119, 10mpan 668 . . . . . 6  |-  ( A  e.  RR  ->  (
1  +  A )  e.  RR )
1211adantr 463 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 1  +  A
)  e.  RR )
1312recnd 9652 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 1  +  A
)  e.  CC )
14 0lt1 10115 . . . . . . 7  |-  0  <  1
15 addgtge0 10081 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  A  e.  RR )  /\  ( 0  <  1  /\  0  <_  A ) )  -> 
0  <  ( 1  +  A ) )
1614, 15mpanr1 681 . . . . . 6  |-  ( ( ( 1  e.  RR  /\  A  e.  RR )  /\  0  <_  A
)  ->  0  <  ( 1  +  A ) )
179, 16mpanl1 678 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <  ( 1  +  A ) )
1817gt0ne0d 10157 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 1  +  A
)  =/=  0 )
198, 13, 18divcan1d 10362 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( A  / 
( 1  +  A
) )  x.  (
1  +  A ) )  =  A )
2011recnd 9652 . . . . 5  |-  ( A  e.  RR  ->  (
1  +  A )  e.  CC )
2120mulid2d 9644 . . . 4  |-  ( A  e.  RR  ->  (
1  x.  ( 1  +  A ) )  =  ( 1  +  A ) )
2221adantr 463 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 1  x.  (
1  +  A ) )  =  ( 1  +  A ) )
237, 19, 223brtr4d 4425 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( A  / 
( 1  +  A
) )  x.  (
1  +  A ) )  <  ( 1  x.  ( 1  +  A ) ) )
24 simpl 455 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  RR )
2524, 12, 18redivcld 10413 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  /  (
1  +  A ) )  e.  RR )
26 ltmul1 10433 . . . 4  |-  ( ( ( A  /  (
1  +  A ) )  e.  RR  /\  1  e.  RR  /\  (
( 1  +  A
)  e.  RR  /\  0  <  ( 1  +  A ) ) )  ->  ( ( A  /  ( 1  +  A ) )  <  1  <->  ( ( A  /  ( 1  +  A ) )  x.  ( 1  +  A
) )  <  (
1  x.  ( 1  +  A ) ) ) )
279, 26mp3an2 1314 . . 3  |-  ( ( ( A  /  (
1  +  A ) )  e.  RR  /\  ( ( 1  +  A )  e.  RR  /\  0  <  ( 1  +  A ) ) )  ->  ( ( A  /  ( 1  +  A ) )  <  1  <->  ( ( A  /  ( 1  +  A ) )  x.  ( 1  +  A
) )  <  (
1  x.  ( 1  +  A ) ) ) )
2825, 12, 17, 27syl12anc 1228 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( A  / 
( 1  +  A
) )  <  1  <->  ( ( A  /  (
1  +  A ) )  x.  ( 1  +  A ) )  <  ( 1  x.  ( 1  +  A
) ) ) )
2923, 28mpbird 232 1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  /  (
1  +  A ) )  <  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   class class class wbr 4395  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    < clt 9658    <_ cle 9659    / cdiv 10247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248
This theorem is referenced by: (None)
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