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Theorem recreclt 10219
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 10161 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
2 gt0ne0 9792 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
3 rereccl 10037 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
42, 3syldan 467 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
5 1re 9373 . . . . 5  |-  1  e.  RR
6 ltaddpos 9817 . . . . 5  |-  ( ( ( 1  /  A
)  e.  RR  /\  1  e.  RR )  ->  ( 0  <  (
1  /  A )  <->  1  <  ( 1  +  ( 1  /  A ) ) ) )
74, 5, 6sylancl 655 . . . 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 9353 . . . . 5  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 1  +  ( 1  /  A
) )  e.  RR )
105, 4, 9sylancr 656 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  +  ( 1  /  A ) )  e.  RR )
11 0lt1 9850 . . . . . 6  |-  0  <  1
12 0re 9374 . . . . . . . 8  |-  0  e.  RR
13 lttr 9439 . . . . . . . 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 1297 . . . . . . 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 669 . . . . 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 10216 . . . 4  |-  ( ( ( 1  +  ( 1  /  A ) )  e.  RR  /\  0  <  ( 1  +  ( 1  /  A
) ) )  -> 
( 1  <  (
1  +  ( 1  /  A ) )  <-> 
( 1  /  (
1  +  ( 1  /  A ) ) )  <  1 ) )
1910, 17, 18syl2anc 654 . . 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 9817 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  <  1  <->  ( 1  /  A )  <  (
( 1  /  A
)  +  1 ) ) )
225, 4, 21sylancr 656 . . . . 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 9400 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  CC )
25 ax-1cn 9328 . . . . 5  |-  1  e.  CC
26 addcom 9543 . . . . 5  |-  ( ( ( 1  /  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2724, 25, 26sylancl 655 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  +  1 )  =  ( 1  +  ( 1  /  A ) ) )
2823, 27breqtrd 4304 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  <  ( 1  +  ( 1  /  A ) ) )
29 simpl 454 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
30 simpr 458 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  A )
31 ltrec1 10207 . . . 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 1212 . . 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 529 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 1362    e. wcel 1755    =/= wne 2596   class class class wbr 4280  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    < clt 9406    / cdiv 9981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982
This theorem is referenced by: (None)
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