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Theorem stoweidlem5 27621
Description: There exists a δ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < δ < 1 , p >= δ on  T  \  U. Here  D is used to represent δ in the paper and  Q to represent  T 
\  U in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem5.1  |-  F/ t
ph
stoweidlem5.2  |-  D  =  if ( C  <_ 
( 1  /  2
) ,  C , 
( 1  /  2
) )
stoweidlem5.3  |-  ( ph  ->  P : T --> RR )
stoweidlem5.4  |-  ( ph  ->  Q  C_  T )
stoweidlem5.5  |-  ( ph  ->  C  e.  RR+ )
stoweidlem5.6  |-  ( ph  ->  A. t  e.  Q  C  <_  ( P `  t ) )
Assertion
Ref Expression
stoweidlem5  |-  ( ph  ->  E. d ( d  e.  RR+  /\  d  <  1  /\  A. t  e.  Q  d  <_  ( P `  t ) ) )
Distinct variable groups:    t, d, D    P, d    Q, d
Allowed substitution hints:    ph( t, d)    C( t, d)    P( t)    Q( t)    T( t, d)

Proof of Theorem stoweidlem5
StepHypRef Expression
1 stoweidlem5.2 . . 3  |-  D  =  if ( C  <_ 
( 1  /  2
) ,  C , 
( 1  /  2
) )
2 stoweidlem5.5 . . . 4  |-  ( ph  ->  C  e.  RR+ )
3 1re 9046 . . . . . 6  |-  1  e.  RR
43rehalfcli 10172 . . . . 5  |-  ( 1  /  2 )  e.  RR
5 halfgt0 10144 . . . . 5  |-  0  <  ( 1  /  2
)
64, 5elrpii 10571 . . . 4  |-  ( 1  /  2 )  e.  RR+
7 ifcl 3735 . . . 4  |-  ( ( C  e.  RR+  /\  (
1  /  2 )  e.  RR+ )  ->  if ( C  <_  ( 1  /  2 ) ,  C ,  ( 1  /  2 ) )  e.  RR+ )
82, 6, 7sylancl 644 . . 3  |-  ( ph  ->  if ( C  <_ 
( 1  /  2
) ,  C , 
( 1  /  2
) )  e.  RR+ )
91, 8syl5eqel 2488 . 2  |-  ( ph  ->  D  e.  RR+ )
109rpred 10604 . . 3  |-  ( ph  ->  D  e.  RR )
114a1i 11 . . 3  |-  ( ph  ->  ( 1  /  2
)  e.  RR )
123a1i 11 . . 3  |-  ( ph  ->  1  e.  RR )
132rpred 10604 . . . . 5  |-  ( ph  ->  C  e.  RR )
14 min2 10733 . . . . 5  |-  ( ( C  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  if ( C  <_  ( 1  / 
2 ) ,  C ,  ( 1  / 
2 ) )  <_ 
( 1  /  2
) )
1513, 4, 14sylancl 644 . . . 4  |-  ( ph  ->  if ( C  <_ 
( 1  /  2
) ,  C , 
( 1  /  2
) )  <_  (
1  /  2 ) )
161, 15syl5eqbr 4205 . . 3  |-  ( ph  ->  D  <_  ( 1  /  2 ) )
17 halflt1 10145 . . . 4  |-  ( 1  /  2 )  <  1
1817a1i 11 . . 3  |-  ( ph  ->  ( 1  /  2
)  <  1 )
1910, 11, 12, 16, 18lelttrd 9184 . 2  |-  ( ph  ->  D  <  1 )
20 stoweidlem5.1 . . 3  |-  F/ t
ph
218rpred 10604 . . . . . . 7  |-  ( ph  ->  if ( C  <_ 
( 1  /  2
) ,  C , 
( 1  /  2
) )  e.  RR )
2221adantr 452 . . . . . 6  |-  ( (
ph  /\  t  e.  Q )  ->  if ( C  <_  ( 1  /  2 ) ,  C ,  ( 1  /  2 ) )  e.  RR )
2313adantr 452 . . . . . 6  |-  ( (
ph  /\  t  e.  Q )  ->  C  e.  RR )
24 stoweidlem5.3 . . . . . . . 8  |-  ( ph  ->  P : T --> RR )
2524adantr 452 . . . . . . 7  |-  ( (
ph  /\  t  e.  Q )  ->  P : T --> RR )
26 stoweidlem5.4 . . . . . . . 8  |-  ( ph  ->  Q  C_  T )
2726sselda 3308 . . . . . . 7  |-  ( (
ph  /\  t  e.  Q )  ->  t  e.  T )
2825, 27ffvelrnd 5830 . . . . . 6  |-  ( (
ph  /\  t  e.  Q )  ->  ( P `  t )  e.  RR )
29 min1 10732 . . . . . . . 8  |-  ( ( C  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  if ( C  <_  ( 1  / 
2 ) ,  C ,  ( 1  / 
2 ) )  <_  C )
3013, 4, 29sylancl 644 . . . . . . 7  |-  ( ph  ->  if ( C  <_ 
( 1  /  2
) ,  C , 
( 1  /  2
) )  <_  C
)
3130adantr 452 . . . . . 6  |-  ( (
ph  /\  t  e.  Q )  ->  if ( C  <_  ( 1  /  2 ) ,  C ,  ( 1  /  2 ) )  <_  C )
32 stoweidlem5.6 . . . . . . 7  |-  ( ph  ->  A. t  e.  Q  C  <_  ( P `  t ) )
3332r19.21bi 2764 . . . . . 6  |-  ( (
ph  /\  t  e.  Q )  ->  C  <_  ( P `  t
) )
3422, 23, 28, 31, 33letrd 9183 . . . . 5  |-  ( (
ph  /\  t  e.  Q )  ->  if ( C  <_  ( 1  /  2 ) ,  C ,  ( 1  /  2 ) )  <_  ( P `  t ) )
351, 34syl5eqbr 4205 . . . 4  |-  ( (
ph  /\  t  e.  Q )  ->  D  <_  ( P `  t
) )
3635ex 424 . . 3  |-  ( ph  ->  ( t  e.  Q  ->  D  <_  ( P `  t ) ) )
3720, 36ralrimi 2747 . 2  |-  ( ph  ->  A. t  e.  Q  D  <_  ( P `  t ) )
38 eleq1 2464 . . . . 5  |-  ( d  =  D  ->  (
d  e.  RR+  <->  D  e.  RR+ ) )
39 breq1 4175 . . . . 5  |-  ( d  =  D  ->  (
d  <  1  <->  D  <  1 ) )
40 breq1 4175 . . . . . 6  |-  ( d  =  D  ->  (
d  <_  ( P `  t )  <->  D  <_  ( P `  t ) ) )
4140ralbidv 2686 . . . . 5  |-  ( d  =  D  ->  ( A. t  e.  Q  d  <_  ( P `  t )  <->  A. t  e.  Q  D  <_  ( P `  t ) ) )
4238, 39, 413anbi123d 1254 . . . 4  |-  ( d  =  D  ->  (
( d  e.  RR+  /\  d  <  1  /\ 
A. t  e.  Q  d  <_  ( P `  t ) )  <->  ( D  e.  RR+  /\  D  <  1  /\  A. t  e.  Q  D  <_  ( P `  t ) ) ) )
4342spcegv 2997 . . 3  |-  ( D  e.  RR+  ->  ( ( D  e.  RR+  /\  D  <  1  /\  A. t  e.  Q  D  <_  ( P `  t ) )  ->  E. d
( d  e.  RR+  /\  d  <  1  /\ 
A. t  e.  Q  d  <_  ( P `  t ) ) ) )
449, 43syl 16 . 2  |-  ( ph  ->  ( ( D  e.  RR+  /\  D  <  1  /\  A. t  e.  Q  D  <_  ( P `  t ) )  ->  E. d ( d  e.  RR+  /\  d  <  1  /\  A. t  e.  Q  d  <_  ( P `  t ) ) ) )
459, 19, 37, 44mp3and 1282 1  |-  ( ph  ->  E. d ( d  e.  RR+  /\  d  <  1  /\  A. t  e.  Q  d  <_  ( P `  t ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547   F/wnf 1550    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   ifcif 3699   class class class wbr 4172   -->wf 5409   ` cfv 5413  (class class class)co 6040   RRcr 8945   1c1 8947    < clt 9076    <_ cle 9077    / cdiv 9633   2c2 10005   RR+crp 10568
This theorem is referenced by:  stoweidlem28  27644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-rp 10569
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