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Theorem stoweidlem5 27415
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 9016 . . . . . 6  |-  1  e.  RR
43rehalfcli 10141 . . . . 5  |-  ( 1  /  2 )  e.  RR
5 halfgt0 10113 . . . . 5  |-  0  <  ( 1  /  2
)
64, 5elrpii 10540 . . . 4  |-  ( 1  /  2 )  e.  RR+
7 ifcl 3711 . . . 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 2464 . 2  |-  ( ph  ->  D  e.  RR+ )
109rpred 10573 . . 3  |-  ( ph  ->  D  e.  RR )
114a1i 11 . . 3  |-  ( ph  ->  ( 1  /  2
)  e.  RR )
123a1i 11 . . 3  |-  ( ph  ->  1  e.  RR )
132rpred 10573 . . . . 5  |-  ( ph  ->  C  e.  RR )
14 min2 10702 . . . . 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 4179 . . 3  |-  ( ph  ->  D  <_  ( 1  /  2 ) )
17 halflt1 10114 . . . 4  |-  ( 1  /  2 )  <  1
1817a1i 11 . . 3  |-  ( ph  ->  ( 1  /  2
)  <  1 )
1910, 11, 12, 16, 18lelttrd 9153 . 2  |-  ( ph  ->  D  <  1 )
20 stoweidlem5.1 . . 3  |-  F/ t
ph
218rpred 10573 . . . . . . 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 3284 . . . . . . 7  |-  ( (
ph  /\  t  e.  Q )  ->  t  e.  T )
2825, 27ffvelrnd 5803 . . . . . 6  |-  ( (
ph  /\  t  e.  Q )  ->  ( P `  t )  e.  RR )
29 min1 10701 . . . . . . . 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 2740 . . . . . 6  |-  ( (
ph  /\  t  e.  Q )  ->  C  <_  ( P `  t
) )
3422, 23, 28, 31, 33letrd 9152 . . . . 5  |-  ( (
ph  /\  t  e.  Q )  ->  if ( C  <_  ( 1  /  2 ) ,  C ,  ( 1  /  2 ) )  <_  ( P `  t ) )
351, 34syl5eqbr 4179 . . . 4  |-  ( (
ph  /\  t  e.  Q )  ->  D  <_  ( P `  t
) )
3635ex 424 . . 3  |-  ( ph  ->  ( t  e.  Q  ->  D  <_  ( P `  t ) ) )
3720, 36ralrimi 2723 . 2  |-  ( ph  ->  A. t  e.  Q  D  <_  ( P `  t ) )
38 eleq1 2440 . . . . 5  |-  ( d  =  D  ->  (
d  e.  RR+  <->  D  e.  RR+ ) )
39 breq1 4149 . . . . 5  |-  ( d  =  D  ->  (
d  <  1  <->  D  <  1 ) )
40 breq1 4149 . . . . . 6  |-  ( d  =  D  ->  (
d  <_  ( P `  t )  <->  D  <_  ( P `  t ) ) )
4140ralbidv 2662 . . . . 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 2973 . . 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 1717   A.wral 2642    C_ wss 3256   ifcif 3675   class class class wbr 4146   -->wf 5383   ` cfv 5387  (class class class)co 6013   RRcr 8915   1c1 8917    < clt 9046    <_ cle 9047    / cdiv 9602   2c2 9974   RR+crp 10537
This theorem is referenced by:  stoweidlem28  27438
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-2 9983  df-rp 10538
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