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Theorem stoweidlem37 27456
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 27449 . . 3  |-  ( (
ph  /\  Z  e.  T )  ->  ( P `  Z )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
) ) )
81, 7mpdan 650 . 2  |-  ( ph  ->  ( P `  Z
)  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
) ) )
95fnvinran 27355 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i )  e.  Q )
10 fveq1 5669 . . . . . . . . . . 11  |-  ( h  =  ( G `  i )  ->  (
h `  Z )  =  ( ( G `
 i ) `  Z ) )
1110eqeq1d 2397 . . . . . . . . . 10  |-  ( h  =  ( G `  i )  ->  (
( h `  Z
)  =  0  <->  (
( G `  i
) `  Z )  =  0 ) )
12 fveq1 5669 . . . . . . . . . . . . 13  |-  ( h  =  ( G `  i )  ->  (
h `  t )  =  ( ( G `
 i ) `  t ) )
1312breq2d 4167 . . . . . . . . . . . 12  |-  ( h  =  ( G `  i )  ->  (
0  <_  ( h `  t )  <->  0  <_  ( ( G `  i
) `  t )
) )
1412breq1d 4165 . . . . . . . . . . . 12  |-  ( h  =  ( G `  i )  ->  (
( h `  t
)  <_  1  <->  ( ( G `  i ) `  t )  <_  1
) )
1513, 14anbi12d 692 . . . . . . . . . . 11  |-  ( h  =  ( G `  i )  ->  (
( 0  <_  (
h `  t )  /\  ( h `  t
)  <_  1 )  <-> 
( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) )
1615ralbidv 2671 . . . . . . . . . 10  |-  ( 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 692 . . . . . . . . 9  |-  ( 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 3039 . . . . . . . 8  |-  ( ( 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 189 . . . . . . 7  |-  ( (
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
) ) ) )
2019simprd 450 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( ( G `  i ) `  Z
)  =  0  /\ 
A. t  e.  T  ( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) )
2120simpld 446 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  Z )  =  0 )
2221sumeq2dv 12426 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  sum_ i  e.  ( 1 ... M
) 0 )
23 fzfi 11240 . . . . 5  |-  ( 1 ... M )  e. 
Fin
24 olc 374 . . . . 5  |-  ( ( 1 ... M )  e.  Fin  ->  (
( 1 ... M
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... M )  e. 
Fin ) )
25 sumz 12445 . . . . 5  |-  ( ( ( 1 ... M
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... M )  e. 
Fin )  ->  sum_ i  e.  ( 1 ... M
) 0  =  0 )
2623, 24, 25mp2b 10 . . . 4  |-  sum_ i  e.  ( 1 ... M
) 0  =  0
2722, 26syl6eq 2437 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  0 )
2827oveq2d 6038 . 2  |-  ( ph  ->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  Z )
)  =  ( ( 1  /  M )  x.  0 ) )
294nncnd 9950 . . . 4  |-  ( ph  ->  M  e.  CC )
304nnne0d 9978 . . . 4  |-  ( ph  ->  M  =/=  0 )
3129, 30reccld 9717 . . 3  |-  ( ph  ->  ( 1  /  M
)  e.  CC )
3231mul01d 9199 . 2  |-  ( ph  ->  ( ( 1  /  M )  x.  0 )  =  0 )
338, 28, 323eqtrd 2425 1  |-  ( ph  ->  ( P `  Z
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655    C_ wss 3265   class class class wbr 4155    e. cmpt 4209   -->wf 5392   ` cfv 5396  (class class class)co 6022   Fincfn 7047   RRcr 8924   0cc0 8925   1c1 8926    x. cmul 8930    <_ cle 9056    / cdiv 9611   NNcn 9934   ZZ>=cuz 10422   ...cfz 10977   sum_csu 12408
This theorem is referenced by:  stoweidlem44  27463
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-fz 10978  df-fzo 11068  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-sum 12409
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