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Theorem pntlemd 20743
Description: Lemma for pnt 20763. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  A is C^*,  B is c1,  L is λ,  D is c2, and  F is c3. (Contributed by Mario Carneiro, 13-Apr-2016.)
Hypotheses
Ref Expression
pntlem1.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
pntlem1.a  |-  ( ph  ->  A  e.  RR+ )
pntlem1.b  |-  ( ph  ->  B  e.  RR+ )
pntlem1.l  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
pntlem1.d  |-  D  =  ( A  +  1 )
pntlem1.f  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
Assertion
Ref Expression
pntlemd  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )

Proof of Theorem pntlemd
StepHypRef Expression
1 ioossre 10712 . . . 4  |-  ( 0 (,) 1 )  C_  RR
2 pntlem1.l . . . 4  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
31, 2sseldi 3178 . . 3  |-  ( ph  ->  L  e.  RR )
4 eliooord 10710 . . . . 5  |-  ( L  e.  ( 0 (,) 1 )  ->  (
0  <  L  /\  L  <  1 ) )
52, 4syl 15 . . . 4  |-  ( ph  ->  ( 0  <  L  /\  L  <  1
) )
65simpld 445 . . 3  |-  ( ph  ->  0  <  L )
73, 6elrpd 10388 . 2  |-  ( ph  ->  L  e.  RR+ )
8 pntlem1.d . . 3  |-  D  =  ( A  +  1 )
9 pntlem1.a . . . 4  |-  ( ph  ->  A  e.  RR+ )
10 1rp 10358 . . . 4  |-  1  e.  RR+
11 rpaddcl 10374 . . . 4  |-  ( ( A  e.  RR+  /\  1  e.  RR+ )  ->  ( A  +  1 )  e.  RR+ )
129, 10, 11sylancl 643 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  RR+ )
138, 12syl5eqel 2367 . 2  |-  ( ph  ->  D  e.  RR+ )
14 pntlem1.f . . 3  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
15 1re 8837 . . . . . . . 8  |-  1  e.  RR
16 ltaddrp 10386 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
1715, 9, 16sylancr 644 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
189rpcnd 10392 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
19 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
20 addcom 8998 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
2118, 19, 20sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
228, 21syl5eq 2327 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
2317, 22breqtrrd 4049 . . . . . 6  |-  ( ph  ->  1  <  D )
2413recgt1d 10404 . . . . . 6  |-  ( ph  ->  ( 1  <  D  <->  ( 1  /  D )  <  1 ) )
2523, 24mpbid 201 . . . . 5  |-  ( ph  ->  ( 1  /  D
)  <  1 )
2613rprecred 10401 . . . . . 6  |-  ( ph  ->  ( 1  /  D
)  e.  RR )
27 difrp 10387 . . . . . 6  |-  ( ( ( 1  /  D
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  /  D )  <  1  <->  ( 1  -  ( 1  /  D ) )  e.  RR+ ) )
2826, 15, 27sylancl 643 . . . . 5  |-  ( ph  ->  ( ( 1  /  D )  <  1  <->  ( 1  -  ( 1  /  D ) )  e.  RR+ ) )
2925, 28mpbid 201 . . . 4  |-  ( ph  ->  ( 1  -  (
1  /  D ) )  e.  RR+ )
30 3nn0 9983 . . . . . . . . 9  |-  3  e.  NN0
31 2nn 9877 . . . . . . . . 9  |-  2  e.  NN
3230, 31decnncl 10137 . . . . . . . 8  |- ; 3 2  e.  NN
33 nnrp 10363 . . . . . . . 8  |-  (; 3 2  e.  NN  -> ; 3
2  e.  RR+ )
3432, 33ax-mp 8 . . . . . . 7  |- ; 3 2  e.  RR+
35 pntlem1.b . . . . . . 7  |-  ( ph  ->  B  e.  RR+ )
36 rpmulcl 10375 . . . . . . 7  |-  ( (; 3
2  e.  RR+  /\  B  e.  RR+ )  ->  (; 3 2  x.  B )  e.  RR+ )
3734, 35, 36sylancr 644 . . . . . 6  |-  ( ph  ->  (; 3 2  x.  B
)  e.  RR+ )
387, 37rpdivcld 10407 . . . . 5  |-  ( ph  ->  ( L  /  (; 3 2  x.  B ) )  e.  RR+ )
39 2z 10054 . . . . . 6  |-  2  e.  ZZ
40 rpexpcl 11122 . . . . . 6  |-  ( ( D  e.  RR+  /\  2  e.  ZZ )  ->  ( D ^ 2 )  e.  RR+ )
4113, 39, 40sylancl 643 . . . . 5  |-  ( ph  ->  ( D ^ 2 )  e.  RR+ )
4238, 41rpdivcld 10407 . . . 4  |-  ( ph  ->  ( ( L  / 
(; 3 2  x.  B
) )  /  ( D ^ 2 ) )  e.  RR+ )
4329, 42rpmulcld 10406 . . 3  |-  ( ph  ->  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )  e.  RR+ )
4414, 43syl5eqel 2367 . 2  |-  ( ph  ->  F  e.  RR+ )
457, 13, 443jca 1132 1  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
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    x. cmul 8742    < clt 8867    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   3c3 9796   ZZcz 10024  ;cdc 10124   RR+crp 10354   (,)cioo 10656   ^cexp 11104  ψcchp 20330
This theorem is referenced by:  pntlemc  20744  pntlema  20745  pntlemb  20746  pntlemq  20750  pntlemr  20751  pntlemj  20752  pntlemf  20754  pntlemo  20756  pntleml  20760
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-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-ioo 10660  df-seq 11047  df-exp 11105
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