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Theorem pntlemc 20706
Description: Lemma for pnt 20725. 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 20705 . . . . 5  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
109simp2d 973 . . . 4  |-  ( ph  ->  D  e.  RR+ )
112, 10rpdivcld 10374 . . 3  |-  ( ph  ->  ( U  /  D
)  e.  RR+ )
121, 11syl5eqel 2342 . 2  |-  ( ph  ->  E  e.  RR+ )
13 pntlem1.k . . 3  |-  K  =  ( exp `  ( B  /  E ) )
145, 12rpdivcld 10374 . . . . 5  |-  ( ph  ->  ( B  /  E
)  e.  RR+ )
1514rpred 10357 . . . 4  |-  ( ph  ->  ( B  /  E
)  e.  RR )
1615rpefcld 12347 . . 3  |-  ( ph  ->  ( exp `  ( B  /  E ) )  e.  RR+ )
1713, 16syl5eqel 2342 . 2  |-  ( ph  ->  K  e.  RR+ )
1812rpred 10357 . . . 4  |-  ( ph  ->  E  e.  RR )
1912rpgt0d 10360 . . . 4  |-  ( ph  ->  0  <  E )
202rpred 10357 . . . . . . . 8  |-  ( ph  ->  U  e.  RR )
214rpred 10357 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2210rpred 10357 . . . . . . . 8  |-  ( ph  ->  D  e.  RR )
23 pntlem1.u2 . . . . . . . 8  |-  ( ph  ->  U  <_  A )
2421ltp1d 9655 . . . . . . . . 9  |-  ( ph  ->  A  <  ( A  +  1 ) )
2524, 7syl6breqr 4037 . . . . . . . 8  |-  ( ph  ->  A  <  D )
2620, 21, 22, 23, 25lelttrd 8942 . . . . . . 7  |-  ( ph  ->  U  <  D )
2710rpcnd 10359 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
2827mulid1d 8820 . . . . . . 7  |-  ( ph  ->  ( D  x.  1 )  =  D )
2926, 28breqtrrd 4023 . . . . . 6  |-  ( ph  ->  U  <  ( D  x.  1 ) )
30 1re 8805 . . . . . . . 8  |-  1  e.  RR
3130a1i 12 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
3220, 31, 10ltdivmuld 10404 . . . . . 6  |-  ( ph  ->  ( ( U  /  D )  <  1  <->  U  <  ( D  x.  1 ) ) )
3329, 32mpbird 225 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  1 )
341, 33syl5eqbr 4030 . . . 4  |-  ( ph  ->  E  <  1 )
35 0xr 8846 . . . . 5  |-  0  e.  RR*
36 rexr 8845 . . . . . 6  |-  ( 1  e.  RR  ->  1  e.  RR* )
3730, 36ax-mp 10 . . . . 5  |-  1  e.  RR*
38 elioo2 10663 . . . . 5  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) ) )
3935, 37, 38mp2an 656 . . . 4  |-  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) )
4018, 19, 34, 39syl3anbrc 1141 . . 3  |-  ( ph  ->  E  e.  ( 0 (,) 1 ) )
41 efgt1 12358 . . . . 5  |-  ( ( B  /  E )  e.  RR+  ->  1  < 
( exp `  ( B  /  E ) ) )
4214, 41syl 17 . . . 4  |-  ( ph  ->  1  <  ( exp `  ( B  /  E
) ) )
4342, 13syl6breqr 4037 . . 3  |-  ( ph  ->  1  <  K )
44 ltaddrp 10353 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
4530, 4, 44sylancr 647 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
462rpcnne0d 10366 . . . . . . . 8  |-  ( ph  ->  ( U  e.  CC  /\  U  =/=  0 ) )
47 divid 9419 . . . . . . . 8  |-  ( ( U  e.  CC  /\  U  =/=  0 )  -> 
( U  /  U
)  =  1 )
4846, 47syl 17 . . . . . . 7  |-  ( ph  ->  ( U  /  U
)  =  1 )
494rpcnd 10359 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
50 ax-1cn 8763 . . . . . . . . 9  |-  1  e.  CC
51 addcom 8966 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
5249, 50, 51sylancl 646 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
537, 52syl5eq 2302 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
5445, 48, 533brtr4d 4027 . . . . . 6  |-  ( ph  ->  ( U  /  U
)  <  D )
5520, 2, 10, 54ltdiv23d 10413 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  U )
561, 55syl5eqbr 4030 . . . 4  |-  ( ph  ->  E  <  U )
57 difrp 10354 . . . . 5  |-  ( ( E  e.  RR  /\  U  e.  RR )  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5818, 20, 57syl2anc 645 . . . 4  |-  ( ph  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5956, 58mpbid 203 . . 3  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
6040, 43, 593jca 1137 . 2  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
6112, 17, 603jca 1137 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997    e. cmpt 4051   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710   RR*cxr 8834    < clt 8835    <_ cle 8836    - cmin 9005    / cdiv 9391   2c2 9763   3c3 9764  ;cdc 10091   RR+crp 10321   (,)cioo 10622   ^cexp 11070   expce 12305  ψcchp 20292
This theorem is referenced by:  pntlema  20707  pntlemb  20708  pntlemg  20709  pntlemh  20710  pntlemq  20712  pntlemr  20713  pntlemj  20714  pntlemi  20715  pntlemf  20716  pntlemo  20718  pntleme  20719  pntlemp  20721
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9933  df-z 9992  df-dec 10092  df-uz 10198  df-rp 10322  df-ioo 10626  df-ico 10628  df-fz 10749  df-fzo 10837  df-fl 10891  df-seq 11013  df-exp 11071  df-fac 11255  df-bc 11282  df-hash 11304  df-shft 11527  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-limsup 11910  df-clim 11927  df-rlim 11928  df-sum 12124  df-ef 12311
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