MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pntlemc Unicode version

Theorem pntlemc 20744
Description: Lemma for pnt 20763. 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 20743 . . . . 5  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
109simp2d 968 . . . 4  |-  ( ph  ->  D  e.  RR+ )
112, 10rpdivcld 10407 . . 3  |-  ( ph  ->  ( U  /  D
)  e.  RR+ )
121, 11syl5eqel 2367 . 2  |-  ( ph  ->  E  e.  RR+ )
13 pntlem1.k . . 3  |-  K  =  ( exp `  ( B  /  E ) )
145, 12rpdivcld 10407 . . . . 5  |-  ( ph  ->  ( B  /  E
)  e.  RR+ )
1514rpred 10390 . . . 4  |-  ( ph  ->  ( B  /  E
)  e.  RR )
1615rpefcld 12385 . . 3  |-  ( ph  ->  ( exp `  ( B  /  E ) )  e.  RR+ )
1713, 16syl5eqel 2367 . 2  |-  ( ph  ->  K  e.  RR+ )
1812rpred 10390 . . . 4  |-  ( ph  ->  E  e.  RR )
1912rpgt0d 10393 . . . 4  |-  ( ph  ->  0  <  E )
202rpred 10390 . . . . . . . 8  |-  ( ph  ->  U  e.  RR )
214rpred 10390 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2210rpred 10390 . . . . . . . 8  |-  ( ph  ->  D  e.  RR )
23 pntlem1.u2 . . . . . . . 8  |-  ( ph  ->  U  <_  A )
2421ltp1d 9687 . . . . . . . . 9  |-  ( ph  ->  A  <  ( A  +  1 ) )
2524, 7syl6breqr 4063 . . . . . . . 8  |-  ( ph  ->  A  <  D )
2620, 21, 22, 23, 25lelttrd 8974 . . . . . . 7  |-  ( ph  ->  U  <  D )
2710rpcnd 10392 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
2827mulid1d 8852 . . . . . . 7  |-  ( ph  ->  ( D  x.  1 )  =  D )
2926, 28breqtrrd 4049 . . . . . 6  |-  ( ph  ->  U  <  ( D  x.  1 ) )
30 1re 8837 . . . . . . . 8  |-  1  e.  RR
3130a1i 10 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
3220, 31, 10ltdivmuld 10437 . . . . . 6  |-  ( ph  ->  ( ( U  /  D )  <  1  <->  U  <  ( D  x.  1 ) ) )
3329, 32mpbird 223 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  1 )
341, 33syl5eqbr 4056 . . . 4  |-  ( ph  ->  E  <  1 )
35 0xr 8878 . . . . 5  |-  0  e.  RR*
36 rexr 8877 . . . . . 6  |-  ( 1  e.  RR  ->  1  e.  RR* )
3730, 36ax-mp 8 . . . . 5  |-  1  e.  RR*
38 elioo2 10697 . . . . 5  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) ) )
3935, 37, 38mp2an 653 . . . 4  |-  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) )
4018, 19, 34, 39syl3anbrc 1136 . . 3  |-  ( ph  ->  E  e.  ( 0 (,) 1 ) )
41 efgt1 12396 . . . . 5  |-  ( ( B  /  E )  e.  RR+  ->  1  < 
( exp `  ( B  /  E ) ) )
4214, 41syl 15 . . . 4  |-  ( ph  ->  1  <  ( exp `  ( B  /  E
) ) )
4342, 13syl6breqr 4063 . . 3  |-  ( ph  ->  1  <  K )
44 ltaddrp 10386 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
4530, 4, 44sylancr 644 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
462rpcnne0d 10399 . . . . . . . 8  |-  ( ph  ->  ( U  e.  CC  /\  U  =/=  0 ) )
47 divid 9451 . . . . . . . 8  |-  ( ( U  e.  CC  /\  U  =/=  0 )  -> 
( U  /  U
)  =  1 )
4846, 47syl 15 . . . . . . 7  |-  ( ph  ->  ( U  /  U
)  =  1 )
494rpcnd 10392 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
50 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
51 addcom 8998 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
5249, 50, 51sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
537, 52syl5eq 2327 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
5445, 48, 533brtr4d 4053 . . . . . 6  |-  ( ph  ->  ( U  /  U
)  <  D )
5520, 2, 10, 54ltdiv23d 10446 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  U )
561, 55syl5eqbr 4056 . . . 4  |-  ( ph  ->  E  <  U )
57 difrp 10387 . . . . 5  |-  ( ( E  e.  RR  /\  U  e.  RR )  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5818, 20, 57syl2anc 642 . . . 4  |-  ( ph  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5956, 58mpbid 201 . . 3  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
6040, 43, 593jca 1132 . 2  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
6112, 17, 603jca 1132 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   3c3 9796  ;cdc 10124   RR+crp 10354   (,)cioo 10656   ^cexp 11104   expce 12343  ψcchp 20330
This theorem is referenced by:  pntlema  20745  pntlemb  20746  pntlemg  20747  pntlemh  20748  pntlemq  20750  pntlemr  20751  pntlemj  20752  pntlemi  20753  pntlemf  20754  pntlemo  20756  pntleme  20757  pntlemp  20759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-ioo 10660  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349
  Copyright terms: Public domain W3C validator