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Theorem stoweidlem37 37187
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 . . 3  |-  ( ph  ->  Z  e.  T )
2 stoweidlem37.1 . . . 4  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
3 stoweidlem37.2 . . . 4  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
4 stoweidlem37.3 . . . 4  |-  ( ph  ->  M  e.  NN )
5 stoweidlem37.4 . . . 4  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
6 stoweidlem37.5 . . . 4  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
72, 3, 4, 5, 6stoweidlem30 37180 . . 3  |-  ( (
ph  /\  Z  e.  T )  ->  ( P `  Z )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
) ) )
81, 7mpdan 666 . 2  |-  ( ph  ->  ( P `  Z
)  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
) ) )
95fnvinran 36769 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i )  e.  Q )
10 fveq1 5848 . . . . . . . . . 10  |-  ( h  =  ( G `  i )  ->  (
h `  Z )  =  ( ( G `
 i ) `  Z ) )
1110eqeq1d 2404 . . . . . . . . 9  |-  ( h  =  ( G `  i )  ->  (
( h `  Z
)  =  0  <->  (
( G `  i
) `  Z )  =  0 ) )
12 fveq1 5848 . . . . . . . . . . . 12  |-  ( h  =  ( G `  i )  ->  (
h `  t )  =  ( ( G `
 i ) `  t ) )
1312breq2d 4407 . . . . . . . . . . 11  |-  ( h  =  ( G `  i )  ->  (
0  <_  ( h `  t )  <->  0  <_  ( ( G `  i
) `  t )
) )
1412breq1d 4405 . . . . . . . . . . 11  |-  ( h  =  ( G `  i )  ->  (
( h `  t
)  <_  1  <->  ( ( G `  i ) `  t )  <_  1
) )
1513, 14anbi12d 709 . . . . . . . . . 10  |-  ( h  =  ( G `  i )  ->  (
( 0  <_  (
h `  t )  /\  ( h `  t
)  <_  1 )  <-> 
( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) )
1615ralbidv 2843 . . . . . . . . 9  |-  ( 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 ) ) )
1711, 16anbi12d 709 . . . . . . . 8  |-  ( 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 ) ) ) )
1817, 2elrab2 3209 . . . . . . 7  |-  ( ( G `  i )  e.  Q  <->  ( ( G `  i )  e.  A  /\  (
( ( G `  i ) `  Z
)  =  0  /\ 
A. t  e.  T  ( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) ) )
199, 18sylib 196 . . . . . 6  |-  ( (
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
) ) ) )
2019simprld 757 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  Z )  =  0 )
2120sumeq2dv 13674 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  sum_ i  e.  ( 1 ... M
) 0 )
22 fzfi 12123 . . . . 5  |-  ( 1 ... M )  e. 
Fin
23 olc 382 . . . . 5  |-  ( ( 1 ... M )  e.  Fin  ->  (
( 1 ... M
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... M )  e. 
Fin ) )
24 sumz 13693 . . . . 5  |-  ( ( ( 1 ... M
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... M )  e. 
Fin )  ->  sum_ i  e.  ( 1 ... M
) 0  =  0 )
2522, 23, 24mp2b 10 . . . 4  |-  sum_ i  e.  ( 1 ... M
) 0  =  0
2621, 25syl6eq 2459 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  0 )
2726oveq2d 6294 . 2  |-  ( ph  ->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  Z )
)  =  ( ( 1  /  M )  x.  0 ) )
284nncnd 10592 . . . 4  |-  ( ph  ->  M  e.  CC )
294nnne0d 10621 . . . 4  |-  ( ph  ->  M  =/=  0 )
3028, 29reccld 10354 . . 3  |-  ( ph  ->  ( 1  /  M
)  e.  CC )
3130mul01d 9813 . 2  |-  ( ph  ->  ( ( 1  /  M )  x.  0 )  =  0 )
328, 27, 313eqtrd 2447 1  |-  ( ph  ->  ( P `  Z
)  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   {crab 2758    C_ wss 3414   class class class wbr 4395    |-> cmpt 4453   -->wf 5565   ` cfv 5569  (class class class)co 6278   Fincfn 7554   RRcr 9521   0cc0 9522   1c1 9523    x. cmul 9527    <_ cle 9659    / cdiv 10247   NNcn 10576   ZZ>=cuz 11127   ...cfz 11726   sum_csu 13657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658
This theorem is referenced by:  stoweidlem44  37194
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