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Theorem stoweidlem37 27786
Description: This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p,  ( G `  i ) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem37.1  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
stoweidlem37.2  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
stoweidlem37.3  |-  ( ph  ->  M  e.  NN )
stoweidlem37.4  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
stoweidlem37.5  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
stoweidlem37.6  |-  ( ph  ->  Z  e.  T )
Assertion
Ref Expression
stoweidlem37  |-  ( ph  ->  ( P `  Z
)  =  0 )
Distinct variable groups:    f, i, T    A, f    f, G    ph, f, i    h, i, t, T    A, h    h, G, t    h, Z, i, t    i, M, t
Allowed substitution hints:    ph( t, h)    A( t, i)    P( t, f, h, i)    Q( t, f, h, i)    G( i)    M( f, h)    Z( f)

Proof of Theorem stoweidlem37
StepHypRef Expression
1 stoweidlem37.6 . . . 4  |-  ( ph  ->  Z  e.  T )
2 stoweidlem37.1 . . . . 5  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
3 stoweidlem37.2 . . . . 5  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
4 stoweidlem37.3 . . . . 5  |-  ( ph  ->  M  e.  NN )
5 stoweidlem37.4 . . . . 5  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
6 stoweidlem37.5 . . . . 5  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
72, 3, 4, 5, 6stoweidlem30 27779 . . . 4  |-  ( (
ph  /\  Z  e.  T )  ->  ( P `  Z )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
) ) )
81, 7mpdan 649 . . 3  |-  ( ph  ->  ( P `  Z
)  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
) ) )
95adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  G : ( 1 ... M ) --> Q )
10 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  i  e.  ( 1 ... M
) )
119, 10jca 518 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G : ( 1 ... M ) --> Q  /\  i  e.  ( 1 ... M ) ) )
12 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( G : ( 1 ... M ) --> Q  /\  i  e.  ( 1 ... M ) )  ->  ( G `  i )  e.  Q
)
1311, 12syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i )  e.  Q )
142eleq2i 2347 . . . . . . . . . . 11  |-  ( ( G `  i )  e.  Q  <->  ( G `  i )  e.  {
h  e.  A  | 
( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_ 
( h `  t
)  /\  ( h `  t )  <_  1
) ) } )
15 fveq1 5524 . . . . . . . . . . . . . 14  |-  ( h  =  ( G `  i )  ->  (
h `  Z )  =  ( ( G `
 i ) `  Z ) )
1615eqeq1d 2291 . . . . . . . . . . . . 13  |-  ( h  =  ( G `  i )  ->  (
( h `  Z
)  =  0  <->  (
( G `  i
) `  Z )  =  0 ) )
17 fveq1 5524 . . . . . . . . . . . . . . . 16  |-  ( h  =  ( G `  i )  ->  (
h `  t )  =  ( ( G `
 i ) `  t ) )
1817breq2d 4035 . . . . . . . . . . . . . . 15  |-  ( h  =  ( G `  i )  ->  (
0  <_  ( h `  t )  <->  0  <_  ( ( G `  i
) `  t )
) )
1917breq1d 4033 . . . . . . . . . . . . . . 15  |-  ( h  =  ( G `  i )  ->  (
( h `  t
)  <_  1  <->  ( ( G `  i ) `  t )  <_  1
) )
2018, 19anbi12d 691 . . . . . . . . . . . . . 14  |-  ( h  =  ( G `  i )  ->  (
( 0  <_  (
h `  t )  /\  ( h `  t
)  <_  1 )  <-> 
( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) )
2120ralbidv 2563 . . . . . . . . . . . . 13  |-  ( h  =  ( G `  i )  ->  ( A. t  e.  T  ( 0  <_  (
h `  t )  /\  ( h `  t
)  <_  1 )  <->  A. t  e.  T  ( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) )
2216, 21anbi12d 691 . . . . . . . . . . . 12  |-  ( h  =  ( G `  i )  ->  (
( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_ 
( h `  t
)  /\  ( h `  t )  <_  1
) )  <->  ( (
( G `  i
) `  Z )  =  0  /\  A. t  e.  T  (
0  <_  ( ( G `  i ) `  t )  /\  (
( G `  i
) `  t )  <_  1 ) ) ) )
2322elrab 2923 . . . . . . . . . . 11  |-  ( ( G `  i )  e.  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  (
0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }  <-> 
( ( G `  i )  e.  A  /\  ( ( ( G `
 i ) `  Z )  =  0  /\  A. t  e.  T  ( 0  <_ 
( ( G `  i ) `  t
)  /\  ( ( G `  i ) `  t )  <_  1
) ) ) )
2414, 23bitri 240 . . . . . . . . . 10  |-  ( ( G `  i )  e.  Q  <->  ( ( G `  i )  e.  A  /\  (
( ( G `  i ) `  Z
)  =  0  /\ 
A. t  e.  T  ( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) ) )
2513, 24sylib 188 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
)  e.  A  /\  ( ( ( G `
 i ) `  Z )  =  0  /\  A. t  e.  T  ( 0  <_ 
( ( G `  i ) `  t
)  /\  ( ( G `  i ) `  t )  <_  1
) ) ) )
2625simprd 449 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( ( G `  i ) `  Z
)  =  0  /\ 
A. t  e.  T  ( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) )
2726simpld 445 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  Z )  =  0 )
2827ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  0 )
29 sumeq2 12167 . . . . . 6  |-  ( A. i  e.  ( 1 ... M ) ( ( G `  i
) `  Z )  =  0  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  Z )  =  sum_ i  e.  ( 1 ... M ) 0 )
3028, 29syl 15 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  sum_ i  e.  ( 1 ... M
) 0 )
31 fzfi 11034 . . . . . 6  |-  ( 1 ... M )  e. 
Fin
32 olc 373 . . . . . 6  |-  ( ( 1 ... M )  e.  Fin  ->  (
( 1 ... M
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... M )  e. 
Fin ) )
33 sumz 12195 . . . . . 6  |-  ( ( ( 1 ... M
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... M )  e. 
Fin )  ->  sum_ i  e.  ( 1 ... M
) 0  =  0 )
3431, 32, 33mp2b 9 . . . . 5  |-  sum_ i  e.  ( 1 ... M
) 0  =  0
3530, 34syl6eq 2331 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  0 )
3635oveq2d 5874 . . 3  |-  ( ph  ->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  Z )
)  =  ( ( 1  /  M )  x.  0 ) )
378, 36eqtrd 2315 . 2  |-  ( ph  ->  ( P `  Z
)  =  ( ( 1  /  M )  x.  0 ) )
38 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
3938a1i 10 . . . . 5  |-  ( ph  ->  1  e.  CC )
40 nncn 9754 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  CC )
414, 40syl 15 . . . . 5  |-  ( ph  ->  M  e.  CC )
42 nnne0 9778 . . . . . 6  |-  ( M  e.  NN  ->  M  =/=  0 )
434, 42syl 15 . . . . 5  |-  ( ph  ->  M  =/=  0 )
4439, 41, 433jca 1132 . . . 4  |-  ( ph  ->  ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
45 divcl 9430 . . . 4  |-  ( ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  (
1  /  M )  e.  CC )
4644, 45syl 15 . . 3  |-  ( ph  ->  ( 1  /  M
)  e.  CC )
47 mul01 8991 . . 3  |-  ( ( 1  /  M )  e.  CC  ->  (
( 1  /  M
)  x.  0 )  =  0 )
4846, 47syl 15 . 2  |-  ( ph  ->  ( ( 1  /  M )  x.  0 )  =  0 )
4937, 48eqtrd 2315 1  |-  ( ph  ->  ( P `  Z
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    <_ cle 8868    / cdiv 9423   NNcn 9746   ZZ>=cuz 10230   ...cfz 10782   sum_csu 12158
This theorem is referenced by:  stoweidlem44  27793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
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