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Theorem pntlemc 20856
Description: Lemma for pnt 20875. 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 20855 . . . . 5  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
109simp2d 968 . . . 4  |-  ( ph  ->  D  e.  RR+ )
112, 10rpdivcld 10499 . . 3  |-  ( ph  ->  ( U  /  D
)  e.  RR+ )
121, 11syl5eqel 2442 . 2  |-  ( ph  ->  E  e.  RR+ )
13 pntlem1.k . . 3  |-  K  =  ( exp `  ( B  /  E ) )
145, 12rpdivcld 10499 . . . . 5  |-  ( ph  ->  ( B  /  E
)  e.  RR+ )
1514rpred 10482 . . . 4  |-  ( ph  ->  ( B  /  E
)  e.  RR )
1615rpefcld 12482 . . 3  |-  ( ph  ->  ( exp `  ( B  /  E ) )  e.  RR+ )
1713, 16syl5eqel 2442 . 2  |-  ( ph  ->  K  e.  RR+ )
1812rpred 10482 . . . 4  |-  ( ph  ->  E  e.  RR )
1912rpgt0d 10485 . . . 4  |-  ( ph  ->  0  <  E )
202rpred 10482 . . . . . . . 8  |-  ( ph  ->  U  e.  RR )
214rpred 10482 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2210rpred 10482 . . . . . . . 8  |-  ( ph  ->  D  e.  RR )
23 pntlem1.u2 . . . . . . . 8  |-  ( ph  ->  U  <_  A )
2421ltp1d 9777 . . . . . . . . 9  |-  ( ph  ->  A  <  ( A  +  1 ) )
2524, 7syl6breqr 4144 . . . . . . . 8  |-  ( ph  ->  A  <  D )
2620, 21, 22, 23, 25lelttrd 9064 . . . . . . 7  |-  ( ph  ->  U  <  D )
2710rpcnd 10484 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
2827mulid1d 8942 . . . . . . 7  |-  ( ph  ->  ( D  x.  1 )  =  D )
2926, 28breqtrrd 4130 . . . . . 6  |-  ( ph  ->  U  <  ( D  x.  1 ) )
30 1re 8927 . . . . . . . 8  |-  1  e.  RR
3130a1i 10 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
3220, 31, 10ltdivmuld 10529 . . . . . 6  |-  ( ph  ->  ( ( U  /  D )  <  1  <->  U  <  ( D  x.  1 ) ) )
3329, 32mpbird 223 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  1 )
341, 33syl5eqbr 4137 . . . 4  |-  ( ph  ->  E  <  1 )
35 0xr 8968 . . . . 5  |-  0  e.  RR*
36 rexr 8967 . . . . . 6  |-  ( 1  e.  RR  ->  1  e.  RR* )
3730, 36ax-mp 8 . . . . 5  |-  1  e.  RR*
38 elioo2 10789 . . . . 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 12493 . . . . 5  |-  ( ( B  /  E )  e.  RR+  ->  1  < 
( exp `  ( B  /  E ) ) )
4214, 41syl 15 . . . 4  |-  ( ph  ->  1  <  ( exp `  ( B  /  E
) ) )
4342, 13syl6breqr 4144 . . 3  |-  ( ph  ->  1  <  K )
44 ltaddrp 10478 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
4530, 4, 44sylancr 644 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
462rpcnne0d 10491 . . . . . . . 8  |-  ( ph  ->  ( U  e.  CC  /\  U  =/=  0 ) )
47 divid 9541 . . . . . . . 8  |-  ( ( U  e.  CC  /\  U  =/=  0 )  -> 
( U  /  U
)  =  1 )
4846, 47syl 15 . . . . . . 7  |-  ( ph  ->  ( U  /  U
)  =  1 )
494rpcnd 10484 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
50 ax-1cn 8885 . . . . . . . . 9  |-  1  e.  CC
51 addcom 9088 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
5249, 50, 51sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
537, 52syl5eq 2402 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
5445, 48, 533brtr4d 4134 . . . . . 6  |-  ( ph  ->  ( U  /  U
)  <  D )
5520, 2, 10, 54ltdiv23d 10538 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  U )
561, 55syl5eqbr 4137 . . . 4  |-  ( ph  ->  E  <  U )
57 difrp 10479 . . . . 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 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4104    e. cmpt 4158   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    + caddc 8830    x. cmul 8832   RR*cxr 8956    < clt 8957    <_ cle 8958    - cmin 9127    / cdiv 9513   2c2 9885   3c3 9886  ;cdc 10216   RR+crp 10446   (,)cioo 10748   ^cexp 11197   expce 12440  ψcchp 20442
This theorem is referenced by:  pntlema  20857  pntlemb  20858  pntlemg  20859  pntlemh  20860  pntlemq  20862  pntlemr  20863  pntlemj  20864  pntlemi  20865  pntlemf  20866  pntlemo  20868  pntleme  20869  pntlemp  20871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-pm 6863  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-rp 10447  df-ioo 10752  df-ico 10754  df-fz 10875  df-fzo 10963  df-fl 11017  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-shft 11658  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-sum 12256  df-ef 12446
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