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Theorem stoweidlem30 27881
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 nfcv 2432 . . 3  |-  F/_ s S
2 nfv 1609 . . 3  |-  F/ s ( ( ph  /\  S  e.  T )  ->  ( P `  S
)  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
3 eleq1 2356 . . . . 5  |-  ( s  =  S  ->  (
s  e.  T  <->  S  e.  T ) )
43anbi2d 684 . . . 4  |-  ( s  =  S  ->  (
( ph  /\  s  e.  T )  <->  ( ph  /\  S  e.  T ) ) )
5 fveq2 5541 . . . . 5  |-  ( s  =  S  ->  ( P `  s )  =  ( P `  S ) )
6 fveq2 5541 . . . . . . 7  |-  ( s  =  S  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  S ) )
76sumeq2sdv 12193 . . . . . 6  |-  ( s  =  S  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
)
87oveq2d 5890 . . . . 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 )
) )
95, 8eqeq12d 2310 . . . 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 ) ) ) )
104, 9imbi12d 311 . . 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 ) ) ) ) )
11 simpr 447 . . . . . 6  |-  ( (
ph  /\  s  e.  T )  ->  s  e.  T )
12 1re 8853 . . . . . . . . . 10  |-  1  e.  RR
1312a1i 10 . . . . . . . . 9  |-  ( ph  ->  1  e.  RR )
14 stoweidlem30.3 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN )
15 nnre 9769 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  RR )
1614, 15syl 15 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
17 nnne0 9794 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  =/=  0 )
1814, 17syl 15 . . . . . . . . 9  |-  ( ph  ->  M  =/=  0 )
1913, 16, 18redivcld 9604 . . . . . . . 8  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
2019adantr 451 . . . . . . 7  |-  ( (
ph  /\  s  e.  T )  ->  (
1  /  M )  e.  RR )
21 fzfid 11051 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
1 ... M )  e. 
Fin )
22 simpll 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  ph )
23 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  i  e.  ( 1 ... M
) )
2422, 23jca 518 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  ( ph  /\  i  e.  ( 1 ... M ) ) )
25 simplr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  s  e.  T )
2624, 25jca 518 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T ) )
27 stoweidlem30.1 . . . . . . . . . . 11  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
28 stoweidlem30.4 . . . . . . . . . . 11  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
29 stoweidlem30.5 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
3027, 28, 29stoweidlem15 27866 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( ( G `  i ) `  s
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  s )  /\  (
( G `  i
) `  s )  <_  1 ) )
3130simp1d 967 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( G `  i
) `  s )  e.  RR )
3226, 31syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  s )  e.  RR )
3321, 32fsumrecl 12223 . . . . . . 7  |-  ( (
ph  /\  s  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  e.  RR )
3420, 33remulcld 8879 . . . . . 6  |-  ( (
ph  /\  s  e.  T )  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s ) )  e.  RR )
3511, 34jca 518 . . . . 5  |-  ( (
ph  /\  s  e.  T )  ->  (
s  e.  T  /\  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)  e.  RR ) )
36 nfcv 2432 . . . . . 6  |-  F/_ t
s
37 nfcv 2432 . . . . . 6  |-  F/_ t
( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
38 fveq2 5541 . . . . . . . 8  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
3938sumeq2sdv 12193 . . . . . . 7  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
4039oveq2d 5890 . . . . . 6  |-  ( t  =  s  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
) )
41 stoweidlem30.2 . . . . . 6  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
4236, 37, 40, 41fvmptf 5632 . . . . 5  |-  ( ( 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 ) ) )
4335, 42syl 15 . . . 4  |-  ( (
ph  /\  s  e.  T )  ->  ( P `  s )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) ) )
4443a1i 10 . . 3  |-  ( s  e.  T  ->  (
( ph  /\  s  e.  T )  ->  ( P `  s )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) ) ) )
451, 2, 10, 44vtoclgaf 2861 . 2  |-  ( S  e.  T  ->  (
( ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) ) )
4645anabsi7 792 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 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   class class class wbr 4039    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    <_ cle 8884    / cdiv 9439   NNcn 9762   ...cfz 10798   sum_csu 12174
This theorem is referenced by:  stoweidlem37  27888  stoweidlem38  27889  stoweidlem44  27895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175
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