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Theorem pntlemc 20576
Description: Lemma for pnt 20595. 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 20575 . . . . 5  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
109simp2d 973 . . . 4  |-  ( ph  ->  D  e.  RR+ )
112, 10rpdivcld 10286 . . 3  |-  ( ph  ->  ( U  /  D
)  e.  RR+ )
121, 11syl5eqel 2337 . 2  |-  ( ph  ->  E  e.  RR+ )
13 pntlem1.k . . 3  |-  K  =  ( exp `  ( B  /  E ) )
145, 12rpdivcld 10286 . . . . 5  |-  ( ph  ->  ( B  /  E
)  e.  RR+ )
1514rpred 10269 . . . 4  |-  ( ph  ->  ( B  /  E
)  e.  RR )
1615rpefcld 12259 . . 3  |-  ( ph  ->  ( exp `  ( B  /  E ) )  e.  RR+ )
1713, 16syl5eqel 2337 . 2  |-  ( ph  ->  K  e.  RR+ )
1812rpred 10269 . . . 4  |-  ( ph  ->  E  e.  RR )
1912rpgt0d 10272 . . . 4  |-  ( ph  ->  0  <  E )
202rpred 10269 . . . . . . . 8  |-  ( ph  ->  U  e.  RR )
214rpred 10269 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2210rpred 10269 . . . . . . . 8  |-  ( ph  ->  D  e.  RR )
23 pntlem1.u2 . . . . . . . 8  |-  ( ph  ->  U  <_  A )
2421ltp1d 9567 . . . . . . . . 9  |-  ( ph  ->  A  <  ( A  +  1 ) )
2524, 7syl6breqr 3960 . . . . . . . 8  |-  ( ph  ->  A  <  D )
2620, 21, 22, 23, 25lelttrd 8854 . . . . . . 7  |-  ( ph  ->  U  <  D )
2710rpcnd 10271 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
2827mulid1d 8732 . . . . . . 7  |-  ( ph  ->  ( D  x.  1 )  =  D )
2926, 28breqtrrd 3946 . . . . . 6  |-  ( ph  ->  U  <  ( D  x.  1 ) )
30 1re 8717 . . . . . . . 8  |-  1  e.  RR
3130a1i 12 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
3220, 31, 10ltdivmuld 10316 . . . . . 6  |-  ( ph  ->  ( ( U  /  D )  <  1  <->  U  <  ( D  x.  1 ) ) )
3329, 32mpbird 225 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  1 )
341, 33syl5eqbr 3953 . . . 4  |-  ( ph  ->  E  <  1 )
35 0xr 8758 . . . . 5  |-  0  e.  RR*
36 rexr 8757 . . . . . 6  |-  ( 1  e.  RR  ->  1  e.  RR* )
3730, 36ax-mp 10 . . . . 5  |-  1  e.  RR*
38 elioo2 10575 . . . . 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 12270 . . . . 5  |-  ( ( B  /  E )  e.  RR+  ->  1  < 
( exp `  ( B  /  E ) ) )
4214, 41syl 17 . . . 4  |-  ( ph  ->  1  <  ( exp `  ( B  /  E
) ) )
4342, 13syl6breqr 3960 . . 3  |-  ( ph  ->  1  <  K )
44 ltaddrp 10265 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
4530, 4, 44sylancr 647 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
462rpcnne0d 10278 . . . . . . . 8  |-  ( ph  ->  ( U  e.  CC  /\  U  =/=  0 ) )
47 divid 9331 . . . . . . . 8  |-  ( ( U  e.  CC  /\  U  =/=  0 )  -> 
( U  /  U
)  =  1 )
4846, 47syl 17 . . . . . . 7  |-  ( ph  ->  ( U  /  U
)  =  1 )
494rpcnd 10271 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
50 ax-1cn 8675 . . . . . . . . 9  |-  1  e.  CC
51 addcom 8878 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
5249, 50, 51sylancl 646 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
537, 52syl5eq 2297 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
5445, 48, 533brtr4d 3950 . . . . . 6  |-  ( ph  ->  ( U  /  U
)  <  D )
5520, 2, 10, 54ltdiv23d 10325 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  U )
561, 55syl5eqbr 3953 . . . 4  |-  ( ph  ->  E  <  U )
57 difrp 10266 . . . . 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 2412   class class class wbr 3920    e. cmpt 3974   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622   RR*cxr 8746    < clt 8747    <_ cle 8748    - cmin 8917    / cdiv 9303   2c2 9675   3c3 9676  ;cdc 10003   RR+crp 10233   (,)cioo 10534   ^cexp 10982   expce 12217  ψcchp 20162
This theorem is referenced by:  pntlema  20577  pntlemb  20578  pntlemg  20579  pntlemh  20580  pntlemq  20582  pntlemr  20583  pntlemj  20584  pntlemi  20585  pntlemf  20586  pntlemo  20588  pntleme  20589  pntlemp  20591
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-pm 6661  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-rp 10234  df-ioo 10538  df-ico 10540  df-fz 10661  df-fzo 10749  df-fl 10803  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-shft 11439  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-limsup 11822  df-clim 11839  df-rlim 11840  df-sum 12036  df-ef 12223
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