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Theorem stoweidlem30 27440
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 2440 . . . . 5  |-  ( s  =  S  ->  (
s  e.  T  <->  S  e.  T ) )
21anbi2d 685 . . . 4  |-  ( s  =  S  ->  (
( ph  /\  s  e.  T )  <->  ( ph  /\  S  e.  T ) ) )
3 fveq2 5661 . . . . 5  |-  ( s  =  S  ->  ( P `  s )  =  ( P `  S ) )
4 fveq2 5661 . . . . . . 7  |-  ( s  =  S  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  S ) )
54sumeq2sdv 12418 . . . . . 6  |-  ( s  =  S  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
)
65oveq2d 6029 . . . . 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 2394 . . . 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 312 . . 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 448 . . . 4  |-  ( (
ph  /\  s  e.  T )  ->  s  e.  T )
10 stoweidlem30.3 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
1110nnrecred 9970 . . . . . 6  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
1211adantr 452 . . . . 5  |-  ( (
ph  /\  s  e.  T )  ->  (
1  /  M )  e.  RR )
13 fzfid 11232 . . . . . 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 27425 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( ( G `  i ) `  s
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  s )  /\  (
( G `  i
) `  s )  <_  1 ) )
1817simp1d 969 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( G `  i
) `  s )  e.  RR )
1918an32s 780 . . . . . 6  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  s )  e.  RR )
2013, 19fsumrecl 12448 . . . . 5  |-  ( (
ph  /\  s  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  e.  RR )
2112, 20remulcld 9042 . . . 4  |-  ( (
ph  /\  s  e.  T )  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s ) )  e.  RR )
22 fveq2 5661 . . . . . . 7  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
2322sumeq2sdv 12418 . . . . . 6  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
2423oveq2d 6029 . . . . 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 5736 . . . 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 643 . . 3  |-  ( (
ph  /\  s  e.  T )  ->  ( P `  s )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) ) )
288, 27vtoclg 2947 . 2  |-  ( S  e.  T  ->  (
( ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) ) )
2928anabsi7 793 1  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646   class class class wbr 4146    e. cmpt 4200   -->wf 5383   ` cfv 5387  (class class class)co 6013   RRcr 8915   0cc0 8916   1c1 8917    x. cmul 8921    <_ cle 9047    / cdiv 9602   NNcn 9925   ...cfz 10968   sum_csu 12399
This theorem is referenced by:  stoweidlem37  27447  stoweidlem38  27448  stoweidlem44  27454
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400
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