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Theorem pntlemc 21279
Description: Lemma for pnt 21298. 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 21278 . . . . 5  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
109simp2d 970 . . . 4  |-  ( ph  ->  D  e.  RR+ )
112, 10rpdivcld 10655 . . 3  |-  ( ph  ->  ( U  /  D
)  e.  RR+ )
121, 11syl5eqel 2519 . 2  |-  ( ph  ->  E  e.  RR+ )
13 pntlem1.k . . 3  |-  K  =  ( exp `  ( B  /  E ) )
145, 12rpdivcld 10655 . . . . 5  |-  ( ph  ->  ( B  /  E
)  e.  RR+ )
1514rpred 10638 . . . 4  |-  ( ph  ->  ( B  /  E
)  e.  RR )
1615rpefcld 12696 . . 3  |-  ( ph  ->  ( exp `  ( B  /  E ) )  e.  RR+ )
1713, 16syl5eqel 2519 . 2  |-  ( ph  ->  K  e.  RR+ )
1812rpred 10638 . . . 4  |-  ( ph  ->  E  e.  RR )
1912rpgt0d 10641 . . . 4  |-  ( ph  ->  0  <  E )
202rpred 10638 . . . . . . . 8  |-  ( ph  ->  U  e.  RR )
214rpred 10638 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2210rpred 10638 . . . . . . . 8  |-  ( ph  ->  D  e.  RR )
23 pntlem1.u2 . . . . . . . 8  |-  ( ph  ->  U  <_  A )
2421ltp1d 9931 . . . . . . . . 9  |-  ( ph  ->  A  <  ( A  +  1 ) )
2524, 7syl6breqr 4244 . . . . . . . 8  |-  ( ph  ->  A  <  D )
2620, 21, 22, 23, 25lelttrd 9218 . . . . . . 7  |-  ( ph  ->  U  <  D )
2710rpcnd 10640 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
2827mulid1d 9095 . . . . . . 7  |-  ( ph  ->  ( D  x.  1 )  =  D )
2926, 28breqtrrd 4230 . . . . . 6  |-  ( ph  ->  U  <  ( D  x.  1 ) )
30 1re 9080 . . . . . . . 8  |-  1  e.  RR
3130a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
3220, 31, 10ltdivmuld 10685 . . . . . 6  |-  ( ph  ->  ( ( U  /  D )  <  1  <->  U  <  ( D  x.  1 ) ) )
3329, 32mpbird 224 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  1 )
341, 33syl5eqbr 4237 . . . 4  |-  ( ph  ->  E  <  1 )
35 0xr 9121 . . . . 5  |-  0  e.  RR*
3630rexri 9127 . . . . 5  |-  1  e.  RR*
37 elioo2 10947 . . . . 5  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) ) )
3835, 36, 37mp2an 654 . . . 4  |-  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) )
3918, 19, 34, 38syl3anbrc 1138 . . 3  |-  ( ph  ->  E  e.  ( 0 (,) 1 ) )
40 efgt1 12707 . . . . 5  |-  ( ( B  /  E )  e.  RR+  ->  1  < 
( exp `  ( B  /  E ) ) )
4114, 40syl 16 . . . 4  |-  ( ph  ->  1  <  ( exp `  ( B  /  E
) ) )
4241, 13syl6breqr 4244 . . 3  |-  ( ph  ->  1  <  K )
43 ltaddrp 10634 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
4430, 4, 43sylancr 645 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
452rpcnne0d 10647 . . . . . . . 8  |-  ( ph  ->  ( U  e.  CC  /\  U  =/=  0 ) )
46 divid 9695 . . . . . . . 8  |-  ( ( U  e.  CC  /\  U  =/=  0 )  -> 
( U  /  U
)  =  1 )
4745, 46syl 16 . . . . . . 7  |-  ( ph  ->  ( U  /  U
)  =  1 )
484rpcnd 10640 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
49 ax-1cn 9038 . . . . . . . . 9  |-  1  e.  CC
50 addcom 9242 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
5148, 49, 50sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
527, 51syl5eq 2479 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
5344, 47, 523brtr4d 4234 . . . . . 6  |-  ( ph  ->  ( U  /  U
)  <  D )
5420, 2, 10, 53ltdiv23d 10694 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  U )
551, 54syl5eqbr 4237 . . . 4  |-  ( ph  ->  E  <  U )
56 difrp 10635 . . . . 5  |-  ( ( E  e.  RR  /\  U  e.  RR )  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5718, 20, 56syl2anc 643 . . . 4  |-  ( ph  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5855, 57mpbid 202 . . 3  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
5939, 42, 583jca 1134 . 2  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
6012, 17, 593jca 1134 1  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8978   RRcr 8979   0cc0 8980   1c1 8981    + caddc 8983    x. cmul 8985   RR*cxr 9109    < clt 9110    <_ cle 9111    - cmin 9281    / cdiv 9667   2c2 10039   3c3 10040  ;cdc 10372   RR+crp 10602   (,)cioo 10906   ^cexp 11372   expce 12654  ψcchp 20865
This theorem is referenced by:  pntlema  21280  pntlemb  21281  pntlemg  21282  pntlemh  21283  pntlemq  21285  pntlemr  21286  pntlemj  21287  pntlemi  21288  pntlemf  21289  pntlemo  21291  pntleme  21292  pntlemp  21294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-addf 9059  ax-mulf 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7469  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-rp 10603  df-ioo 10910  df-ico 10912  df-fz 11034  df-fzo 11126  df-fl 11192  df-seq 11314  df-exp 11373  df-fac 11557  df-bc 11584  df-hash 11609  df-shft 11872  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-limsup 12255  df-clim 12272  df-rlim 12273  df-sum 12470  df-ef 12660
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