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Theorem pntlemd 20739
Description: Lemma for pnt 20759. 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 10708 . . . 4  |-  ( 0 (,) 1 )  C_  RR
2 pntlem1.l . . . 4  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
31, 2sseldi 3179 . . 3  |-  ( ph  ->  L  e.  RR )
4 eliooord 10706 . . . . 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 10384 . 2  |-  ( ph  ->  L  e.  RR+ )
8 pntlem1.d . . 3  |-  D  =  ( A  +  1 )
9 pntlem1.a . . . 4  |-  ( ph  ->  A  e.  RR+ )
10 1rp 10354 . . . 4  |-  1  e.  RR+
11 rpaddcl 10370 . . . 4  |-  ( ( A  e.  RR+  /\  1  e.  RR+ )  ->  ( A  +  1 )  e.  RR+ )
129, 10, 11sylancl 643 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  RR+ )
138, 12syl5eqel 2368 . 2  |-  ( ph  ->  D  e.  RR+ )
14 pntlem1.f . . 3  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
15 1re 8833 . . . . . . . 8  |-  1  e.  RR
16 ltaddrp 10382 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
1715, 9, 16sylancr 644 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
189rpcnd 10388 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
19 ax-1cn 8791 . . . . . . . . 9  |-  1  e.  CC
20 addcom 8994 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
2118, 19, 20sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
228, 21syl5eq 2328 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
2317, 22breqtrrd 4050 . . . . . 6  |-  ( ph  ->  1  <  D )
2413recgt1d 10400 . . . . . 6  |-  ( ph  ->  ( 1  <  D  <->  ( 1  /  D )  <  1 ) )
2523, 24mpbid 201 . . . . 5  |-  ( ph  ->  ( 1  /  D
)  <  1 )
2613rprecred 10397 . . . . . 6  |-  ( ph  ->  ( 1  /  D
)  e.  RR )
27 difrp 10383 . . . . . 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 9979 . . . . . . . . 9  |-  3  e.  NN0
31 2nn 9873 . . . . . . . . 9  |-  2  e.  NN
3230, 31decnncl 10133 . . . . . . . 8  |- ; 3 2  e.  NN
33 nnrp 10359 . . . . . . . 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 10371 . . . . . . 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 10403 . . . . 5  |-  ( ph  ->  ( L  /  (; 3 2  x.  B ) )  e.  RR+ )
39 2z 10050 . . . . . 6  |-  2  e.  ZZ
40 rpexpcl 11118 . . . . . 6  |-  ( ( D  e.  RR+  /\  2  e.  ZZ )  ->  ( D ^ 2 )  e.  RR+ )
4113, 39, 40sylancl 643 . . . . 5  |-  ( ph  ->  ( D ^ 2 )  e.  RR+ )
4238, 41rpdivcld 10403 . . . 4  |-  ( ph  ->  ( ( L  / 
(; 3 2  x.  B
) )  /  ( D ^ 2 ) )  e.  RR+ )
4329, 42rpmulcld 10402 . . 3  |-  ( ph  ->  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )  e.  RR+ )
4414, 43syl5eqel 2368 . 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 1685   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5820   CCcc 8731   RRcr 8732   0cc0 8733   1c1 8734    + caddc 8736    x. cmul 8738    < clt 8863    - cmin 9033    / cdiv 9419   NNcn 9742   2c2 9791   3c3 9792   ZZcz 10020  ;cdc 10120   RR+crp 10350   (,)cioo 10652   ^cexp 11100  ψcchp 20326
This theorem is referenced by:  pntlemc  20740  pntlema  20741  pntlemb  20742  pntlemq  20746  pntlemr  20747  pntlemj  20748  pntlemf  20750  pntlemo  20752  pntleml  20756
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-rp 10351  df-ioo 10656  df-seq 11043  df-exp 11101
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