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Theorem pntlemd 22728
Description: Lemma for pnt 22748. 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 11345 . . . 4  |-  ( 0 (,) 1 )  C_  RR
2 pntlem1.l . . . 4  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
31, 2sseldi 3342 . . 3  |-  ( ph  ->  L  e.  RR )
4 eliooord 11343 . . . . 5  |-  ( L  e.  ( 0 (,) 1 )  ->  (
0  <  L  /\  L  <  1 ) )
52, 4syl 16 . . . 4  |-  ( ph  ->  ( 0  <  L  /\  L  <  1
) )
65simpld 456 . . 3  |-  ( ph  ->  0  <  L )
73, 6elrpd 11013 . 2  |-  ( ph  ->  L  e.  RR+ )
8 pntlem1.d . . 3  |-  D  =  ( A  +  1 )
9 pntlem1.a . . . 4  |-  ( ph  ->  A  e.  RR+ )
10 1rp 10983 . . . 4  |-  1  e.  RR+
11 rpaddcl 10999 . . . 4  |-  ( ( A  e.  RR+  /\  1  e.  RR+ )  ->  ( A  +  1 )  e.  RR+ )
129, 10, 11sylancl 655 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  RR+ )
138, 12syl5eqel 2517 . 2  |-  ( ph  ->  D  e.  RR+ )
14 pntlem1.f . . 3  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
15 1re 9373 . . . . . . . 8  |-  1  e.  RR
16 ltaddrp 11011 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
1715, 9, 16sylancr 656 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
189rpcnd 11017 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
19 ax-1cn 9328 . . . . . . . . 9  |-  1  e.  CC
20 addcom 9543 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
2118, 19, 20sylancl 655 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
228, 21syl5eq 2477 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
2317, 22breqtrrd 4306 . . . . . 6  |-  ( ph  ->  1  <  D )
2413recgt1d 11029 . . . . . 6  |-  ( ph  ->  ( 1  <  D  <->  ( 1  /  D )  <  1 ) )
2523, 24mpbid 210 . . . . 5  |-  ( ph  ->  ( 1  /  D
)  <  1 )
2613rprecred 11026 . . . . . 6  |-  ( ph  ->  ( 1  /  D
)  e.  RR )
27 difrp 11012 . . . . . 6  |-  ( ( ( 1  /  D
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  /  D )  <  1  <->  ( 1  -  ( 1  /  D ) )  e.  RR+ ) )
2826, 15, 27sylancl 655 . . . . 5  |-  ( ph  ->  ( ( 1  /  D )  <  1  <->  ( 1  -  ( 1  /  D ) )  e.  RR+ ) )
2925, 28mpbid 210 . . . 4  |-  ( ph  ->  ( 1  -  (
1  /  D ) )  e.  RR+ )
30 3nn0 10585 . . . . . . . . 9  |-  3  e.  NN0
31 2nn 10467 . . . . . . . . 9  |-  2  e.  NN
3230, 31decnncl 10756 . . . . . . . 8  |- ; 3 2  e.  NN
33 nnrp 10988 . . . . . . . 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 11000 . . . . . . 7  |-  ( (; 3
2  e.  RR+  /\  B  e.  RR+ )  ->  (; 3 2  x.  B )  e.  RR+ )
3734, 35, 36sylancr 656 . . . . . 6  |-  ( ph  ->  (; 3 2  x.  B
)  e.  RR+ )
387, 37rpdivcld 11032 . . . . 5  |-  ( ph  ->  ( L  /  (; 3 2  x.  B ) )  e.  RR+ )
39 2z 10666 . . . . . 6  |-  2  e.  ZZ
40 rpexpcl 11868 . . . . . 6  |-  ( ( D  e.  RR+  /\  2  e.  ZZ )  ->  ( D ^ 2 )  e.  RR+ )
4113, 39, 40sylancl 655 . . . . 5  |-  ( ph  ->  ( D ^ 2 )  e.  RR+ )
4238, 41rpdivcld 11032 . . . 4  |-  ( ph  ->  ( ( L  / 
(; 3 2  x.  B
) )  /  ( D ^ 2 ) )  e.  RR+ )
4329, 42rpmulcld 11031 . . 3  |-  ( ph  ->  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )  e.  RR+ )
4414, 43syl5eqel 2517 . 2  |-  ( ph  ->  F  e.  RR+ )
457, 13, 443jca 1161 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 958    = wceq 1362    e. wcel 1755   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    - cmin 9583    / cdiv 9981   NNcn 10310   2c2 10359   3c3 10360   ZZcz 10634  ;cdc 10743   RR+crp 10979   (,)cioo 11288   ^cexp 11849  ψcchp 22315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-rp 10980  df-ioo 11292  df-seq 11791  df-exp 11850
This theorem is referenced by:  pntlemc  22729  pntlema  22730  pntlemb  22731  pntlemq  22735  pntlemr  22736  pntlemj  22737  pntlemf  22739  pntlemo  22741  pntleml  22745
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