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Theorem pntlemc 24481
Description: Lemma for pnt 24500. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  U is α,  E is ε, and  K is K. (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 ) ) )
pntlem1.u  |-  ( ph  ->  U  e.  RR+ )
pntlem1.u2  |-  ( ph  ->  U  <_  A )
pntlem1.e  |-  E  =  ( U  /  D
)
pntlem1.k  |-  K  =  ( exp `  ( B  /  E ) )
Assertion
Ref Expression
pntlemc  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
Distinct variable group:    E, a
Allowed substitution hints:    ph( a)    A( a)    B( a)    D( a)    R( a)    U( a)    F( a)    K( a)    L( a)

Proof of Theorem pntlemc
StepHypRef Expression
1 pntlem1.e . . 3  |-  E  =  ( U  /  D
)
2 pntlem1.u . . . 4  |-  ( ph  ->  U  e.  RR+ )
3 pntlem1.r . . . . . 6  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
4 pntlem1.a . . . . . 6  |-  ( ph  ->  A  e.  RR+ )
5 pntlem1.b . . . . . 6  |-  ( ph  ->  B  e.  RR+ )
6 pntlem1.l . . . . . 6  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
7 pntlem1.d . . . . . 6  |-  D  =  ( A  +  1 )
8 pntlem1.f . . . . . 6  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
93, 4, 5, 6, 7, 8pntlemd 24480 . . . . 5  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
109simp2d 1027 . . . 4  |-  ( ph  ->  D  e.  RR+ )
112, 10rpdivcld 11386 . . 3  |-  ( ph  ->  ( U  /  D
)  e.  RR+ )
121, 11syl5eqel 2543 . 2  |-  ( ph  ->  E  e.  RR+ )
13 pntlem1.k . . 3  |-  K  =  ( exp `  ( B  /  E ) )
145, 12rpdivcld 11386 . . . . 5  |-  ( ph  ->  ( B  /  E
)  e.  RR+ )
1514rpred 11369 . . . 4  |-  ( ph  ->  ( B  /  E
)  e.  RR )
1615rpefcld 14207 . . 3  |-  ( ph  ->  ( exp `  ( B  /  E ) )  e.  RR+ )
1713, 16syl5eqel 2543 . 2  |-  ( ph  ->  K  e.  RR+ )
1812rpred 11369 . . . 4  |-  ( ph  ->  E  e.  RR )
1912rpgt0d 11372 . . . 4  |-  ( ph  ->  0  <  E )
202rpred 11369 . . . . . . . 8  |-  ( ph  ->  U  e.  RR )
214rpred 11369 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2210rpred 11369 . . . . . . . 8  |-  ( ph  ->  D  e.  RR )
23 pntlem1.u2 . . . . . . . 8  |-  ( ph  ->  U  <_  A )
2421ltp1d 10564 . . . . . . . . 9  |-  ( ph  ->  A  <  ( A  +  1 ) )
2524, 7syl6breqr 4456 . . . . . . . 8  |-  ( ph  ->  A  <  D )
2620, 21, 22, 23, 25lelttrd 9818 . . . . . . 7  |-  ( ph  ->  U  <  D )
2710rpcnd 11371 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
2827mulid1d 9685 . . . . . . 7  |-  ( ph  ->  ( D  x.  1 )  =  D )
2926, 28breqtrrd 4442 . . . . . 6  |-  ( ph  ->  U  <  ( D  x.  1 ) )
30 1red 9683 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
3120, 30, 10ltdivmuld 11417 . . . . . 6  |-  ( ph  ->  ( ( U  /  D )  <  1  <->  U  <  ( D  x.  1 ) ) )
3229, 31mpbird 240 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  1 )
331, 32syl5eqbr 4449 . . . 4  |-  ( ph  ->  E  <  1 )
34 0xr 9712 . . . . 5  |-  0  e.  RR*
35 1re 9667 . . . . . 6  |-  1  e.  RR
3635rexri 9718 . . . . 5  |-  1  e.  RR*
37 elioo2 11705 . . . . 5  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) ) )
3834, 36, 37mp2an 683 . . . 4  |-  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) )
3918, 19, 33, 38syl3anbrc 1198 . . 3  |-  ( ph  ->  E  e.  ( 0 (,) 1 ) )
40 efgt1 14218 . . . . 5  |-  ( ( B  /  E )  e.  RR+  ->  1  < 
( exp `  ( B  /  E ) ) )
4114, 40syl 17 . . . 4  |-  ( ph  ->  1  <  ( exp `  ( B  /  E
) ) )
4241, 13syl6breqr 4456 . . 3  |-  ( ph  ->  1  <  K )
43 ltaddrp 11364 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
4435, 4, 43sylancr 674 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
452rpcnne0d 11378 . . . . . . . 8  |-  ( ph  ->  ( U  e.  CC  /\  U  =/=  0 ) )
46 divid 10324 . . . . . . . 8  |-  ( ( U  e.  CC  /\  U  =/=  0 )  -> 
( U  /  U
)  =  1 )
4745, 46syl 17 . . . . . . 7  |-  ( ph  ->  ( U  /  U
)  =  1 )
484rpcnd 11371 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
49 ax-1cn 9622 . . . . . . . . 9  |-  1  e.  CC
50 addcom 9844 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
5148, 49, 50sylancl 673 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
527, 51syl5eq 2507 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
5344, 47, 523brtr4d 4446 . . . . . 6  |-  ( ph  ->  ( U  /  U
)  <  D )
5420, 2, 10, 53ltdiv23d 11431 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  U )
551, 54syl5eqbr 4449 . . . 4  |-  ( ph  ->  E  <  U )
56 difrp 11365 . . . . 5  |-  ( ( E  e.  RR  /\  U  e.  RR )  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5718, 20, 56syl2anc 671 . . . 4  |-  ( ph  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5855, 57mpbid 215 . . 3  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
5939, 42, 583jca 1194 . 2  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
6012, 17, 593jca 1194 1  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   class class class wbr 4415    |-> cmpt 4474   ` cfv 5600  (class class class)co 6314   CCcc 9562   RRcr 9563   0cc0 9564   1c1 9565    + caddc 9567    x. cmul 9569   RR*cxr 9699    < clt 9700    <_ cle 9701    - cmin 9885    / cdiv 10296   2c2 10686   3c3 10687  ;cdc 11079   RR+crp 11330   (,)cioo 11663   ^cexp 12303   expce 14162  ψcchp 24067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642  ax-addf 9643  ax-mulf 9644
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-pm 7500  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-sup 7981  df-inf 7982  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-rp 11331  df-ioo 11667  df-ico 11669  df-fz 11813  df-fzo 11946  df-fl 12059  df-seq 12245  df-exp 12304  df-fac 12491  df-bc 12519  df-hash 12547  df-shft 13178  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-limsup 13574  df-clim 13600  df-rlim 13601  df-sum 13801  df-ef 14169
This theorem is referenced by:  pntlema  24482  pntlemb  24483  pntlemg  24484  pntlemh  24485  pntlemq  24487  pntlemr  24488  pntlemj  24489  pntlemi  24490  pntlemf  24491  pntlemo  24493  pntleme  24494  pntlemp  24496
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