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Theorem stoweidlem37 27653
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 27646 . . 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 27552 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i )  e.  Q )
10 fveq1 5686 . . . . . . . . . . 11  |-  ( h  =  ( G `  i )  ->  (
h `  Z )  =  ( ( G `
 i ) `  Z ) )
1110eqeq1d 2412 . . . . . . . . . 10  |-  ( h  =  ( G `  i )  ->  (
( h `  Z
)  =  0  <->  (
( G `  i
) `  Z )  =  0 ) )
12 fveq1 5686 . . . . . . . . . . . . 13  |-  ( h  =  ( G `  i )  ->  (
h `  t )  =  ( ( G `
 i ) `  t ) )
1312breq2d 4184 . . . . . . . . . . . 12  |-  ( h  =  ( G `  i )  ->  (
0  <_  ( h `  t )  <->  0  <_  ( ( G `  i
) `  t )
) )
1412breq1d 4182 . . . . . . . . . . . 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 2686 . . . . . . . . . 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 3054 . . . . . . . 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 12452 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  sum_ i  e.  ( 1 ... M
) 0 )
23 fzfi 11266 . . . . 5  |-  ( 1 ... M )  e. 
Fin
24 olc 374 . . . . 5  |-  ( ( 1 ... M )  e.  Fin  ->  (
( 1 ... M
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... M )  e. 
Fin ) )
25 sumz 12471 . . . . 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 2452 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  0 )
2827oveq2d 6056 . 2  |-  ( ph  ->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  Z )
)  =  ( ( 1  /  M )  x.  0 ) )
294nncnd 9972 . . . 4  |-  ( ph  ->  M  e.  CC )
304nnne0d 10000 . . . 4  |-  ( ph  ->  M  =/=  0 )
3129, 30reccld 9739 . . 3  |-  ( ph  ->  ( 1  /  M
)  e.  CC )
3231mul01d 9221 . 2  |-  ( ph  ->  ( ( 1  /  M )  x.  0 )  =  0 )
338, 28, 323eqtrd 2440 1  |-  ( ph  ->  ( P `  Z
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    <_ cle 9077    / cdiv 9633   NNcn 9956   ZZ>=cuz 10444   ...cfz 10999   sum_csu 12434
This theorem is referenced by:  stoweidlem44  27660
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435
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