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Theorem pntibndlem1 20738
Description: Lemma for pntibnd 20742. (Contributed by Mario Carneiro, 10-Apr-2016.)
Hypotheses
Ref Expression
pntibnd.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
pntibndlem1.1  |-  ( ph  ->  A  e.  RR+ )
pntibndlem1.l  |-  L  =  ( ( 1  / 
4 )  /  ( A  +  3 ) )
Assertion
Ref Expression
pntibndlem1  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )

Proof of Theorem pntibndlem1
StepHypRef Expression
1 pntibndlem1.l . . . 4  |-  L  =  ( ( 1  / 
4 )  /  ( A  +  3 ) )
2 4nn 9879 . . . . . 6  |-  4  e.  NN
3 nnrp 10363 . . . . . 6  |-  ( 4  e.  NN  ->  4  e.  RR+ )
4 rpreccl 10377 . . . . . 6  |-  ( 4  e.  RR+  ->  ( 1  /  4 )  e.  RR+ )
52, 3, 4mp2b 9 . . . . 5  |-  ( 1  /  4 )  e.  RR+
6 pntibndlem1.1 . . . . . 6  |-  ( ph  ->  A  e.  RR+ )
7 3nn 9878 . . . . . . 7  |-  3  e.  NN
8 nnrp 10363 . . . . . . 7  |-  ( 3  e.  NN  ->  3  e.  RR+ )
97, 8ax-mp 8 . . . . . 6  |-  3  e.  RR+
10 rpaddcl 10374 . . . . . 6  |-  ( ( A  e.  RR+  /\  3  e.  RR+ )  ->  ( A  +  3 )  e.  RR+ )
116, 9, 10sylancl 643 . . . . 5  |-  ( ph  ->  ( A  +  3 )  e.  RR+ )
12 rpdivcl 10376 . . . . 5  |-  ( ( ( 1  /  4
)  e.  RR+  /\  ( A  +  3 )  e.  RR+ )  ->  (
( 1  /  4
)  /  ( A  +  3 ) )  e.  RR+ )
135, 11, 12sylancr 644 . . . 4  |-  ( ph  ->  ( ( 1  / 
4 )  /  ( A  +  3 ) )  e.  RR+ )
141, 13syl5eqel 2367 . . 3  |-  ( ph  ->  L  e.  RR+ )
1514rpred 10390 . 2  |-  ( ph  ->  L  e.  RR )
1614rpgt0d 10393 . 2  |-  ( ph  ->  0  <  L )
17 rpcn 10362 . . . . . . 7  |-  ( ( 1  /  4 )  e.  RR+  ->  ( 1  /  4 )  e.  CC )
185, 17ax-mp 8 . . . . . 6  |-  ( 1  /  4 )  e.  CC
1918div1i 9488 . . . . 5  |-  ( ( 1  /  4 )  /  1 )  =  ( 1  /  4
)
20 rpre 10360 . . . . . . 7  |-  ( ( 1  /  4 )  e.  RR+  ->  ( 1  /  4 )  e.  RR )
215, 20mp1i 11 . . . . . 6  |-  ( ph  ->  ( 1  /  4
)  e.  RR )
22 3re 9817 . . . . . . 7  |-  3  e.  RR
2322a1i 10 . . . . . 6  |-  ( ph  ->  3  e.  RR )
2411rpred 10390 . . . . . 6  |-  ( ph  ->  ( A  +  3 )  e.  RR )
25 1lt4 9891 . . . . . . . . 9  |-  1  <  4
26 4re 9819 . . . . . . . . . 10  |-  4  e.  RR
27 4pos 9832 . . . . . . . . . 10  |-  0  <  4
28 recgt1 9652 . . . . . . . . . 10  |-  ( ( 4  e.  RR  /\  0  <  4 )  -> 
( 1  <  4  <->  ( 1  /  4 )  <  1 ) )
2926, 27, 28mp2an 653 . . . . . . . . 9  |-  ( 1  <  4  <->  ( 1  /  4 )  <  1 )
3025, 29mpbi 199 . . . . . . . 8  |-  ( 1  /  4 )  <  1
31 1lt3 9888 . . . . . . . 8  |-  1  <  3
325, 20ax-mp 8 . . . . . . . . 9  |-  ( 1  /  4 )  e.  RR
33 1re 8837 . . . . . . . . 9  |-  1  e.  RR
3432, 33, 22lttri 8945 . . . . . . . 8  |-  ( ( ( 1  /  4
)  <  1  /\  1  <  3 )  -> 
( 1  /  4
)  <  3 )
3530, 31, 34mp2an 653 . . . . . . 7  |-  ( 1  /  4 )  <  3
3635a1i 10 . . . . . 6  |-  ( ph  ->  ( 1  /  4
)  <  3 )
37 ltaddrp 10386 . . . . . . . 8  |-  ( ( 3  e.  RR  /\  A  e.  RR+ )  -> 
3  <  ( 3  +  A ) )
3822, 6, 37sylancr 644 . . . . . . 7  |-  ( ph  ->  3  <  ( 3  +  A ) )
39 3cn 9818 . . . . . . . 8  |-  3  e.  CC
406rpcnd 10392 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
41 addcom 8998 . . . . . . . 8  |-  ( ( 3  e.  CC  /\  A  e.  CC )  ->  ( 3  +  A
)  =  ( A  +  3 ) )
4239, 40, 41sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 3  +  A
)  =  ( A  +  3 ) )
4338, 42breqtrd 4047 . . . . . 6  |-  ( ph  ->  3  <  ( A  +  3 ) )
4421, 23, 24, 36, 43lttrd 8977 . . . . 5  |-  ( ph  ->  ( 1  /  4
)  <  ( A  +  3 ) )
4519, 44syl5eqbr 4056 . . . 4  |-  ( ph  ->  ( ( 1  / 
4 )  /  1
)  <  ( A  +  3 ) )
4633a1i 10 . . . . 5  |-  ( ph  ->  1  e.  RR )
47 0lt1 9296 . . . . . 6  |-  0  <  1
4847a1i 10 . . . . 5  |-  ( ph  ->  0  <  1 )
4911rpregt0d 10396 . . . . 5  |-  ( ph  ->  ( ( A  + 
3 )  e.  RR  /\  0  <  ( A  +  3 ) ) )
50 ltdiv23 9647 . . . . 5  |-  ( ( ( 1  /  4
)  e.  RR  /\  ( 1  e.  RR  /\  0  <  1 )  /\  ( ( A  +  3 )  e.  RR  /\  0  < 
( A  +  3 ) ) )  -> 
( ( ( 1  /  4 )  / 
1 )  <  ( A  +  3 )  <-> 
( ( 1  / 
4 )  /  ( A  +  3 ) )  <  1 ) )
5121, 46, 48, 49, 50syl121anc 1187 . . . 4  |-  ( ph  ->  ( ( ( 1  /  4 )  / 
1 )  <  ( A  +  3 )  <-> 
( ( 1  / 
4 )  /  ( A  +  3 ) )  <  1 ) )
5245, 51mpbid 201 . . 3  |-  ( ph  ->  ( ( 1  / 
4 )  /  ( A  +  3 ) )  <  1 )
531, 52syl5eqbr 4056 . 2  |-  ( ph  ->  L  <  1 )
54 0xr 8878 . . 3  |-  0  e.  RR*
55 rexr 8877 . . . 4  |-  ( 1  e.  RR  ->  1  e.  RR* )
5633, 55ax-mp 8 . . 3  |-  1  e.  RR*
57 elioo2 10697 . . 3  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  ( L  e.  ( 0 (,) 1 )  <->  ( L  e.  RR  /\  0  < 
L  /\  L  <  1 ) ) )
5854, 56, 57mp2an 653 . 2  |-  ( L  e.  ( 0 (,) 1 )  <->  ( L  e.  RR  /\  0  < 
L  /\  L  <  1 ) )
5915, 16, 53, 58syl3anbrc 1136 1  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    - cmin 9037    / cdiv 9423   NNcn 9746   3c3 9796   4c4 9797   RR+crp 10354   (,)cioo 10656  ψcchp 20330
This theorem is referenced by:  pntibndlem2a  20739  pntibndlem2  20740  pntibnd  20742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-rp 10355  df-ioo 10660
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