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Theorem pntlemd 21148
Description: Lemma for pnt 21168. 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 10897 . . . 4  |-  ( 0 (,) 1 )  C_  RR
2 pntlem1.l . . . 4  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
31, 2sseldi 3282 . . 3  |-  ( ph  ->  L  e.  RR )
4 eliooord 10895 . . . . 5  |-  ( L  e.  ( 0 (,) 1 )  ->  (
0  <  L  /\  L  <  1 ) )
52, 4syl 16 . . . 4  |-  ( ph  ->  ( 0  <  L  /\  L  <  1
) )
65simpld 446 . . 3  |-  ( ph  ->  0  <  L )
73, 6elrpd 10571 . 2  |-  ( ph  ->  L  e.  RR+ )
8 pntlem1.d . . 3  |-  D  =  ( A  +  1 )
9 pntlem1.a . . . 4  |-  ( ph  ->  A  e.  RR+ )
10 1rp 10541 . . . 4  |-  1  e.  RR+
11 rpaddcl 10557 . . . 4  |-  ( ( A  e.  RR+  /\  1  e.  RR+ )  ->  ( A  +  1 )  e.  RR+ )
129, 10, 11sylancl 644 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  RR+ )
138, 12syl5eqel 2464 . 2  |-  ( ph  ->  D  e.  RR+ )
14 pntlem1.f . . 3  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
15 1re 9016 . . . . . . . 8  |-  1  e.  RR
16 ltaddrp 10569 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
1715, 9, 16sylancr 645 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
189rpcnd 10575 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
19 ax-1cn 8974 . . . . . . . . 9  |-  1  e.  CC
20 addcom 9177 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
2118, 19, 20sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
228, 21syl5eq 2424 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
2317, 22breqtrrd 4172 . . . . . 6  |-  ( ph  ->  1  <  D )
2413recgt1d 10587 . . . . . 6  |-  ( ph  ->  ( 1  <  D  <->  ( 1  /  D )  <  1 ) )
2523, 24mpbid 202 . . . . 5  |-  ( ph  ->  ( 1  /  D
)  <  1 )
2613rprecred 10584 . . . . . 6  |-  ( ph  ->  ( 1  /  D
)  e.  RR )
27 difrp 10570 . . . . . 6  |-  ( ( ( 1  /  D
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  /  D )  <  1  <->  ( 1  -  ( 1  /  D ) )  e.  RR+ ) )
2826, 15, 27sylancl 644 . . . . 5  |-  ( ph  ->  ( ( 1  /  D )  <  1  <->  ( 1  -  ( 1  /  D ) )  e.  RR+ ) )
2925, 28mpbid 202 . . . 4  |-  ( ph  ->  ( 1  -  (
1  /  D ) )  e.  RR+ )
30 3nn0 10164 . . . . . . . . 9  |-  3  e.  NN0
31 2nn 10058 . . . . . . . . 9  |-  2  e.  NN
3230, 31decnncl 10320 . . . . . . . 8  |- ; 3 2  e.  NN
33 nnrp 10546 . . . . . . . 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 10558 . . . . . . 7  |-  ( (; 3
2  e.  RR+  /\  B  e.  RR+ )  ->  (; 3 2  x.  B )  e.  RR+ )
3734, 35, 36sylancr 645 . . . . . 6  |-  ( ph  ->  (; 3 2  x.  B
)  e.  RR+ )
387, 37rpdivcld 10590 . . . . 5  |-  ( ph  ->  ( L  /  (; 3 2  x.  B ) )  e.  RR+ )
39 2z 10237 . . . . . 6  |-  2  e.  ZZ
40 rpexpcl 11320 . . . . . 6  |-  ( ( D  e.  RR+  /\  2  e.  ZZ )  ->  ( D ^ 2 )  e.  RR+ )
4113, 39, 40sylancl 644 . . . . 5  |-  ( ph  ->  ( D ^ 2 )  e.  RR+ )
4238, 41rpdivcld 10590 . . . 4  |-  ( ph  ->  ( ( L  / 
(; 3 2  x.  B
) )  /  ( D ^ 2 ) )  e.  RR+ )
4329, 42rpmulcld 10589 . . 3  |-  ( ph  ->  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )  e.  RR+ )
4414, 43syl5eqel 2464 . 2  |-  ( ph  ->  F  e.  RR+ )
457, 13, 443jca 1134 1  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146    e. cmpt 4200   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    < clt 9046    - cmin 9216    / cdiv 9602   NNcn 9925   2c2 9974   3c3 9975   ZZcz 10207  ;cdc 10307   RR+crp 10537   (,)cioo 10841   ^cexp 11302  ψcchp 20735
This theorem is referenced by:  pntlemc  21149  pntlema  21150  pntlemb  21151  pntlemq  21155  pntlemr  21156  pntlemj  21157  pntlemf  21159  pntlemo  21161  pntleml  21165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-rp 10538  df-ioo 10845  df-seq 11244  df-exp 11303
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