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Theorem stoweidlem5 27730
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 9090 . . . . . 6  |-  1  e.  RR
43rehalfcli 10216 . . . . 5  |-  ( 1  /  2 )  e.  RR
5 halfgt0 10188 . . . . 5  |-  0  <  ( 1  /  2
)
64, 5elrpii 10615 . . . 4  |-  ( 1  /  2 )  e.  RR+
7 ifcl 3775 . . . 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 2520 . 2  |-  ( ph  ->  D  e.  RR+ )
109rpred 10648 . . 3  |-  ( ph  ->  D  e.  RR )
114a1i 11 . . 3  |-  ( ph  ->  ( 1  /  2
)  e.  RR )
123a1i 11 . . 3  |-  ( ph  ->  1  e.  RR )
132rpred 10648 . . . . 5  |-  ( ph  ->  C  e.  RR )
14 min2 10777 . . . . 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 4245 . . 3  |-  ( ph  ->  D  <_  ( 1  /  2 ) )
17 halflt1 10189 . . . 4  |-  ( 1  /  2 )  <  1
1817a1i 11 . . 3  |-  ( ph  ->  ( 1  /  2
)  <  1 )
1910, 11, 12, 16, 18lelttrd 9228 . 2  |-  ( ph  ->  D  <  1 )
20 stoweidlem5.1 . . 3  |-  F/ t
ph
218rpred 10648 . . . . . . 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 3348 . . . . . . 7  |-  ( (
ph  /\  t  e.  Q )  ->  t  e.  T )
2825, 27ffvelrnd 5871 . . . . . 6  |-  ( (
ph  /\  t  e.  Q )  ->  ( P `  t )  e.  RR )
29 min1 10776 . . . . . . . 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 2804 . . . . . 6  |-  ( (
ph  /\  t  e.  Q )  ->  C  <_  ( P `  t
) )
3422, 23, 28, 31, 33letrd 9227 . . . . 5  |-  ( (
ph  /\  t  e.  Q )  ->  if ( C  <_  ( 1  /  2 ) ,  C ,  ( 1  /  2 ) )  <_  ( P `  t ) )
351, 34syl5eqbr 4245 . . . 4  |-  ( (
ph  /\  t  e.  Q )  ->  D  <_  ( P `  t
) )
3635ex 424 . . 3  |-  ( ph  ->  ( t  e.  Q  ->  D  <_  ( P `  t ) ) )
3720, 36ralrimi 2787 . 2  |-  ( ph  ->  A. t  e.  Q  D  <_  ( P `  t ) )
38 eleq1 2496 . . . . 5  |-  ( d  =  D  ->  (
d  e.  RR+  <->  D  e.  RR+ ) )
39 breq1 4215 . . . . 5  |-  ( d  =  D  ->  (
d  <  1  <->  D  <  1 ) )
40 breq1 4215 . . . . . 6  |-  ( d  =  D  ->  (
d  <_  ( P `  t )  <->  D  <_  ( P `  t ) ) )
4140ralbidv 2725 . . . . 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 3037 . . 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 1550   F/wnf 1553    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ifcif 3739   class class class wbr 4212   -->wf 5450   ` cfv 5454  (class class class)co 6081   RRcr 8989   1c1 8991    < clt 9120    <_ cle 9121    / cdiv 9677   2c2 10049   RR+crp 10612
This theorem is referenced by:  stoweidlem28  27753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058  df-rp 10613
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