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Theorem pntlemd 23644
Description: Lemma for pnt 23664. 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 11590 . . . 4  |-  ( 0 (,) 1 )  C_  RR
2 pntlem1.l . . . 4  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
31, 2sseldi 3484 . . 3  |-  ( ph  ->  L  e.  RR )
4 eliooord 11588 . . . . 5  |-  ( L  e.  ( 0 (,) 1 )  ->  (
0  <  L  /\  L  <  1 ) )
52, 4syl 16 . . . 4  |-  ( ph  ->  ( 0  <  L  /\  L  <  1
) )
65simpld 459 . . 3  |-  ( ph  ->  0  <  L )
73, 6elrpd 11258 . 2  |-  ( ph  ->  L  e.  RR+ )
8 pntlem1.d . . 3  |-  D  =  ( A  +  1 )
9 pntlem1.a . . . 4  |-  ( ph  ->  A  e.  RR+ )
10 1rp 11228 . . . 4  |-  1  e.  RR+
11 rpaddcl 11244 . . . 4  |-  ( ( A  e.  RR+  /\  1  e.  RR+ )  ->  ( A  +  1 )  e.  RR+ )
129, 10, 11sylancl 662 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  RR+ )
138, 12syl5eqel 2533 . 2  |-  ( ph  ->  D  e.  RR+ )
14 pntlem1.f . . 3  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
15 1re 9593 . . . . . . . 8  |-  1  e.  RR
16 ltaddrp 11256 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
1715, 9, 16sylancr 663 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
189rpcnd 11262 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
19 ax-1cn 9548 . . . . . . . . 9  |-  1  e.  CC
20 addcom 9764 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
2118, 19, 20sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
228, 21syl5eq 2494 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
2317, 22breqtrrd 4459 . . . . . 6  |-  ( ph  ->  1  <  D )
2413recgt1d 11274 . . . . . 6  |-  ( ph  ->  ( 1  <  D  <->  ( 1  /  D )  <  1 ) )
2523, 24mpbid 210 . . . . 5  |-  ( ph  ->  ( 1  /  D
)  <  1 )
2613rprecred 11271 . . . . . 6  |-  ( ph  ->  ( 1  /  D
)  e.  RR )
27 difrp 11257 . . . . . 6  |-  ( ( ( 1  /  D
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  /  D )  <  1  <->  ( 1  -  ( 1  /  D ) )  e.  RR+ ) )
2826, 15, 27sylancl 662 . . . . 5  |-  ( ph  ->  ( ( 1  /  D )  <  1  <->  ( 1  -  ( 1  /  D ) )  e.  RR+ ) )
2925, 28mpbid 210 . . . 4  |-  ( ph  ->  ( 1  -  (
1  /  D ) )  e.  RR+ )
30 3nn0 10814 . . . . . . . . 9  |-  3  e.  NN0
31 2nn 10694 . . . . . . . . 9  |-  2  e.  NN
3230, 31decnncl 10992 . . . . . . . 8  |- ; 3 2  e.  NN
33 nnrp 11233 . . . . . . . 8  |-  (; 3 2  e.  NN  -> ; 3
2  e.  RR+ )
3432, 33ax-mp 5 . . . . . . 7  |- ; 3 2  e.  RR+
35 pntlem1.b . . . . . . 7  |-  ( ph  ->  B  e.  RR+ )
36 rpmulcl 11245 . . . . . . 7  |-  ( (; 3
2  e.  RR+  /\  B  e.  RR+ )  ->  (; 3 2  x.  B )  e.  RR+ )
3734, 35, 36sylancr 663 . . . . . 6  |-  ( ph  ->  (; 3 2  x.  B
)  e.  RR+ )
387, 37rpdivcld 11277 . . . . 5  |-  ( ph  ->  ( L  /  (; 3 2  x.  B ) )  e.  RR+ )
39 2z 10897 . . . . . 6  |-  2  e.  ZZ
40 rpexpcl 12159 . . . . . 6  |-  ( ( D  e.  RR+  /\  2  e.  ZZ )  ->  ( D ^ 2 )  e.  RR+ )
4113, 39, 40sylancl 662 . . . . 5  |-  ( ph  ->  ( D ^ 2 )  e.  RR+ )
4238, 41rpdivcld 11277 . . . 4  |-  ( ph  ->  ( ( L  / 
(; 3 2  x.  B
) )  /  ( D ^ 2 ) )  e.  RR+ )
4329, 42rpmulcld 11276 . . 3  |-  ( ph  ->  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )  e.  RR+ )
4414, 43syl5eqel 2533 . 2  |-  ( ph  ->  F  e.  RR+ )
457, 13, 443jca 1175 1  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   class class class wbr 4433    |-> cmpt 4491   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    < clt 9626    - cmin 9805    / cdiv 10207   NNcn 10537   2c2 10586   3c3 10587   ZZcz 10865  ;cdc 10979   RR+crp 11224   (,)cioo 11533   ^cexp 12140  ψcchp 23231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-rp 11225  df-ioo 11537  df-seq 12082  df-exp 12141
This theorem is referenced by:  pntlemc  23645  pntlema  23646  pntlemb  23647  pntlemq  23651  pntlemr  23652  pntlemj  23653  pntlemf  23655  pntlemo  23657  pntleml  23661
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