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Theorem pntlemc 22970
Description: Lemma for pnt 22989. 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 22969 . . . . 5  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
109simp2d 1001 . . . 4  |-  ( ph  ->  D  e.  RR+ )
112, 10rpdivcld 11148 . . 3  |-  ( ph  ->  ( U  /  D
)  e.  RR+ )
121, 11syl5eqel 2543 . 2  |-  ( ph  ->  E  e.  RR+ )
13 pntlem1.k . . 3  |-  K  =  ( exp `  ( B  /  E ) )
145, 12rpdivcld 11148 . . . . 5  |-  ( ph  ->  ( B  /  E
)  e.  RR+ )
1514rpred 11131 . . . 4  |-  ( ph  ->  ( B  /  E
)  e.  RR )
1615rpefcld 13500 . . 3  |-  ( ph  ->  ( exp `  ( B  /  E ) )  e.  RR+ )
1713, 16syl5eqel 2543 . 2  |-  ( ph  ->  K  e.  RR+ )
1812rpred 11131 . . . 4  |-  ( ph  ->  E  e.  RR )
1912rpgt0d 11134 . . . 4  |-  ( ph  ->  0  <  E )
202rpred 11131 . . . . . . . 8  |-  ( ph  ->  U  e.  RR )
214rpred 11131 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2210rpred 11131 . . . . . . . 8  |-  ( ph  ->  D  e.  RR )
23 pntlem1.u2 . . . . . . . 8  |-  ( ph  ->  U  <_  A )
2421ltp1d 10367 . . . . . . . . 9  |-  ( ph  ->  A  <  ( A  +  1 ) )
2524, 7syl6breqr 4433 . . . . . . . 8  |-  ( ph  ->  A  <  D )
2620, 21, 22, 23, 25lelttrd 9633 . . . . . . 7  |-  ( ph  ->  U  <  D )
2710rpcnd 11133 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
2827mulid1d 9507 . . . . . . 7  |-  ( ph  ->  ( D  x.  1 )  =  D )
2926, 28breqtrrd 4419 . . . . . 6  |-  ( ph  ->  U  <  ( D  x.  1 ) )
30 1red 9505 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
3120, 30, 10ltdivmuld 11178 . . . . . 6  |-  ( ph  ->  ( ( U  /  D )  <  1  <->  U  <  ( D  x.  1 ) ) )
3229, 31mpbird 232 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  1 )
331, 32syl5eqbr 4426 . . . 4  |-  ( ph  ->  E  <  1 )
34 0xr 9534 . . . . 5  |-  0  e.  RR*
35 1re 9489 . . . . . 6  |-  1  e.  RR
3635rexri 9540 . . . . 5  |-  1  e.  RR*
37 elioo2 11445 . . . . 5  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) ) )
3834, 36, 37mp2an 672 . . . 4  |-  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) )
3918, 19, 33, 38syl3anbrc 1172 . . 3  |-  ( ph  ->  E  e.  ( 0 (,) 1 ) )
40 efgt1 13511 . . . . 5  |-  ( ( B  /  E )  e.  RR+  ->  1  < 
( exp `  ( B  /  E ) ) )
4114, 40syl 16 . . . 4  |-  ( ph  ->  1  <  ( exp `  ( B  /  E
) ) )
4241, 13syl6breqr 4433 . . 3  |-  ( ph  ->  1  <  K )
43 ltaddrp 11127 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
4435, 4, 43sylancr 663 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
452rpcnne0d 11140 . . . . . . . 8  |-  ( ph  ->  ( U  e.  CC  /\  U  =/=  0 ) )
46 divid 10125 . . . . . . . 8  |-  ( ( U  e.  CC  /\  U  =/=  0 )  -> 
( U  /  U
)  =  1 )
4745, 46syl 16 . . . . . . 7  |-  ( ph  ->  ( U  /  U
)  =  1 )
484rpcnd 11133 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
49 ax-1cn 9444 . . . . . . . . 9  |-  1  e.  CC
50 addcom 9659 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
5148, 49, 50sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
527, 51syl5eq 2504 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
5344, 47, 523brtr4d 4423 . . . . . 6  |-  ( ph  ->  ( U  /  U
)  <  D )
5420, 2, 10, 53ltdiv23d 11187 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  U )
551, 54syl5eqbr 4426 . . . 4  |-  ( ph  ->  E  <  U )
56 difrp 11128 . . . . 5  |-  ( ( E  e.  RR  /\  U  e.  RR )  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5718, 20, 56syl2anc 661 . . . 4  |-  ( ph  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5855, 57mpbid 210 . . 3  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
5939, 42, 583jca 1168 . 2  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
6012, 17, 593jca 1168 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393    |-> cmpt 4451   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391   RR*cxr 9521    < clt 9522    <_ cle 9523    - cmin 9699    / cdiv 10097   2c2 10475   3c3 10476  ;cdc 10859   RR+crp 11095   (,)cioo 11404   ^cexp 11975   expce 13458  ψcchp 22556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-rp 11096  df-ioo 11408  df-ico 11410  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464
This theorem is referenced by:  pntlema  22971  pntlemb  22972  pntlemg  22973  pntlemh  22974  pntlemq  22976  pntlemr  22977  pntlemj  22978  pntlemi  22979  pntlemf  22980  pntlemo  22982  pntleme  22983  pntlemp  22985
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