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Theorem stoweidlem30 37928
Description: This lemma is used to prove the existence of a function p as in Lemma 1 [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
stoweidlem30.1  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
stoweidlem30.2  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
stoweidlem30.3  |-  ( ph  ->  M  e.  NN )
stoweidlem30.4  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
stoweidlem30.5  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
Assertion
Ref Expression
stoweidlem30  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
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, M, t    S, i
Allowed substitution hints:    ph( t, h)    A( t, i)    P( t, f, h, i)    Q( t, f, h, i)    S( t, f, h)    G( i)    M( f, h)    Z( t,
f, i)

Proof of Theorem stoweidlem30
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 eleq1 2527 . . . . 5  |-  ( s  =  S  ->  (
s  e.  T  <->  S  e.  T ) )
21anbi2d 715 . . . 4  |-  ( s  =  S  ->  (
( ph  /\  s  e.  T )  <->  ( ph  /\  S  e.  T ) ) )
3 fveq2 5887 . . . . 5  |-  ( s  =  S  ->  ( P `  s )  =  ( P `  S ) )
4 fveq2 5887 . . . . . . 7  |-  ( s  =  S  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  S ) )
54sumeq2sdv 13818 . . . . . 6  |-  ( s  =  S  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
)
65oveq2d 6330 . . . . 5  |-  ( s  =  S  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s ) )  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
) )
73, 6eqeq12d 2476 . . . 4  |-  ( s  =  S  ->  (
( P `  s
)  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) )  <->  ( P `  S )  =  ( ( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) ) ) )
82, 7imbi12d 326 . . 3  |-  ( s  =  S  ->  (
( ( ph  /\  s  e.  T )  ->  ( P `  s
)  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) ) )  <->  ( ( ph  /\  S  e.  T
)  ->  ( P `  S )  =  ( ( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) ) ) ) )
9 simpr 467 . . . 4  |-  ( (
ph  /\  s  e.  T )  ->  s  e.  T )
10 stoweidlem30.3 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
1110nnrecred 10682 . . . . . 6  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
1211adantr 471 . . . . 5  |-  ( (
ph  /\  s  e.  T )  ->  (
1  /  M )  e.  RR )
13 fzfid 12217 . . . . . 6  |-  ( (
ph  /\  s  e.  T )  ->  (
1 ... M )  e. 
Fin )
14 stoweidlem30.1 . . . . . . . . 9  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
15 stoweidlem30.4 . . . . . . . . 9  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
16 stoweidlem30.5 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
1714, 15, 16stoweidlem15 37912 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( ( G `  i ) `  s
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  s )  /\  (
( G `  i
) `  s )  <_  1 ) )
1817simp1d 1026 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( G `  i
) `  s )  e.  RR )
1918an32s 818 . . . . . 6  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  s )  e.  RR )
2013, 19fsumrecl 13848 . . . . 5  |-  ( (
ph  /\  s  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  e.  RR )
2112, 20remulcld 9696 . . . 4  |-  ( (
ph  /\  s  e.  T )  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s ) )  e.  RR )
22 fveq2 5887 . . . . . . 7  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
2322sumeq2sdv 13818 . . . . . 6  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
2423oveq2d 6330 . . . . 5  |-  ( t  =  s  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
) )
25 stoweidlem30.2 . . . . 5  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
2624, 25fvmptg 5968 . . . 4  |-  ( ( s  e.  T  /\  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)  e.  RR )  ->  ( P `  s )  =  ( ( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s ) ) )
279, 21, 26syl2anc 671 . . 3  |-  ( (
ph  /\  s  e.  T )  ->  ( P `  s )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) ) )
288, 27vtoclg 3118 . 2  |-  ( S  e.  T  ->  (
( ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) ) )
2928anabsi7 833 1  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748   {crab 2752   class class class wbr 4415    |-> cmpt 4474   -->wf 5596   ` cfv 5600  (class class class)co 6314   RRcr 9563   0cc0 9564   1c1 9565    x. cmul 9569    <_ cle 9701    / cdiv 10296   NNcn 10636   ...cfz 11812   sum_csu 13800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-sup 7981  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-rp 11331  df-fz 11813  df-fzo 11946  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-clim 13600  df-sum 13801
This theorem is referenced by:  stoweidlem37  37935  stoweidlem38  37936  stoweidlem44  37942
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