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Theorem pntlemc 22728
Description: Lemma for pnt 22747. 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 22727 . . . . 5  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
109simp2d 994 . . . 4  |-  ( ph  ->  D  e.  RR+ )
112, 10rpdivcld 11031 . . 3  |-  ( ph  ->  ( U  /  D
)  e.  RR+ )
121, 11syl5eqel 2517 . 2  |-  ( ph  ->  E  e.  RR+ )
13 pntlem1.k . . 3  |-  K  =  ( exp `  ( B  /  E ) )
145, 12rpdivcld 11031 . . . . 5  |-  ( ph  ->  ( B  /  E
)  e.  RR+ )
1514rpred 11014 . . . 4  |-  ( ph  ->  ( B  /  E
)  e.  RR )
1615rpefcld 13371 . . 3  |-  ( ph  ->  ( exp `  ( B  /  E ) )  e.  RR+ )
1713, 16syl5eqel 2517 . 2  |-  ( ph  ->  K  e.  RR+ )
1812rpred 11014 . . . 4  |-  ( ph  ->  E  e.  RR )
1912rpgt0d 11017 . . . 4  |-  ( ph  ->  0  <  E )
202rpred 11014 . . . . . . . 8  |-  ( ph  ->  U  e.  RR )
214rpred 11014 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
2210rpred 11014 . . . . . . . 8  |-  ( ph  ->  D  e.  RR )
23 pntlem1.u2 . . . . . . . 8  |-  ( ph  ->  U  <_  A )
2421ltp1d 10250 . . . . . . . . 9  |-  ( ph  ->  A  <  ( A  +  1 ) )
2524, 7syl6breqr 4320 . . . . . . . 8  |-  ( ph  ->  A  <  D )
2620, 21, 22, 23, 25lelttrd 9516 . . . . . . 7  |-  ( ph  ->  U  <  D )
2710rpcnd 11016 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
2827mulid1d 9390 . . . . . . 7  |-  ( ph  ->  ( D  x.  1 )  =  D )
2926, 28breqtrrd 4306 . . . . . 6  |-  ( ph  ->  U  <  ( D  x.  1 ) )
30 1red 9388 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
3120, 30, 10ltdivmuld 11061 . . . . . 6  |-  ( ph  ->  ( ( U  /  D )  <  1  <->  U  <  ( D  x.  1 ) ) )
3229, 31mpbird 232 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  1 )
331, 32syl5eqbr 4313 . . . 4  |-  ( ph  ->  E  <  1 )
34 0xr 9417 . . . . 5  |-  0  e.  RR*
35 1re 9372 . . . . . 6  |-  1  e.  RR
3635rexri 9423 . . . . 5  |-  1  e.  RR*
37 elioo2 11328 . . . . 5  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) ) )
3834, 36, 37mp2an 665 . . . 4  |-  ( E  e.  ( 0 (,) 1 )  <->  ( E  e.  RR  /\  0  < 
E  /\  E  <  1 ) )
3918, 19, 33, 38syl3anbrc 1165 . . 3  |-  ( ph  ->  E  e.  ( 0 (,) 1 ) )
40 efgt1 13382 . . . . 5  |-  ( ( B  /  E )  e.  RR+  ->  1  < 
( exp `  ( B  /  E ) ) )
4114, 40syl 16 . . . 4  |-  ( ph  ->  1  <  ( exp `  ( B  /  E
) ) )
4241, 13syl6breqr 4320 . . 3  |-  ( ph  ->  1  <  K )
43 ltaddrp 11010 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR+ )  -> 
1  <  ( 1  +  A ) )
4435, 4, 43sylancr 656 . . . . . . 7  |-  ( ph  ->  1  <  ( 1  +  A ) )
452rpcnne0d 11023 . . . . . . . 8  |-  ( ph  ->  ( U  e.  CC  /\  U  =/=  0 ) )
46 divid 10008 . . . . . . . 8  |-  ( ( U  e.  CC  /\  U  =/=  0 )  -> 
( U  /  U
)  =  1 )
4745, 46syl 16 . . . . . . 7  |-  ( ph  ->  ( U  /  U
)  =  1 )
484rpcnd 11016 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
49 ax-1cn 9327 . . . . . . . . 9  |-  1  e.  CC
50 addcom 9542 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
5148, 49, 50sylancl 655 . . . . . . . 8  |-  ( ph  ->  ( A  +  1 )  =  ( 1  +  A ) )
527, 51syl5eq 2477 . . . . . . 7  |-  ( ph  ->  D  =  ( 1  +  A ) )
5344, 47, 523brtr4d 4310 . . . . . 6  |-  ( ph  ->  ( U  /  U
)  <  D )
5420, 2, 10, 53ltdiv23d 11070 . . . . 5  |-  ( ph  ->  ( U  /  D
)  <  U )
551, 54syl5eqbr 4313 . . . 4  |-  ( ph  ->  E  <  U )
56 difrp 11011 . . . . 5  |-  ( ( E  e.  RR  /\  U  e.  RR )  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5718, 20, 56syl2anc 654 . . . 4  |-  ( ph  ->  ( E  <  U  <->  ( U  -  E )  e.  RR+ ) )
5855, 57mpbid 210 . . 3  |-  ( ph  ->  ( U  -  E
)  e.  RR+ )
5939, 42, 583jca 1161 . 2  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
6012, 17, 593jca 1161 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270    + caddc 9272    x. cmul 9274   RR*cxr 9404    < clt 9405    <_ cle 9406    - cmin 9582    / cdiv 9980   2c2 10358   3c3 10359  ;cdc 10742   RR+crp 10978   (,)cioo 11287   ^cexp 11848   expce 13329  ψcchp 22314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-rp 10979  df-ioo 11291  df-ico 11293  df-fz 11424  df-fzo 11532  df-fl 11625  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-sum 13147  df-ef 13335
This theorem is referenced by:  pntlema  22729  pntlemb  22730  pntlemg  22731  pntlemh  22732  pntlemq  22734  pntlemr  22735  pntlemj  22736  pntlemi  22737  pntlemf  22738  pntlemo  22740  pntleme  22741  pntlemp  22743
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