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Theorem stoweidlem30 37832
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 2495 . . . . 5  |-  ( s  =  S  ->  (
s  e.  T  <->  S  e.  T ) )
21anbi2d 708 . . . 4  |-  ( s  =  S  ->  (
( ph  /\  s  e.  T )  <->  ( ph  /\  S  e.  T ) ) )
3 fveq2 5882 . . . . 5  |-  ( s  =  S  ->  ( P `  s )  =  ( P `  S ) )
4 fveq2 5882 . . . . . . 7  |-  ( s  =  S  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  S ) )
54sumeq2sdv 13770 . . . . . 6  |-  ( s  =  S  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
)
65oveq2d 6322 . . . . 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 2444 . . . 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 321 . . 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 462 . . . 4  |-  ( (
ph  /\  s  e.  T )  ->  s  e.  T )
10 stoweidlem30.3 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
1110nnrecred 10663 . . . . . 6  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
1211adantr 466 . . . . 5  |-  ( (
ph  /\  s  e.  T )  ->  (
1  /  M )  e.  RR )
13 fzfid 12193 . . . . . 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 37816 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( ( G `  i ) `  s
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  s )  /\  (
( G `  i
) `  s )  <_  1 ) )
1817simp1d 1017 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( G `  i
) `  s )  e.  RR )
1918an32s 811 . . . . . 6  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  s )  e.  RR )
2013, 19fsumrecl 13800 . . . . 5  |-  ( (
ph  /\  s  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  e.  RR )
2112, 20remulcld 9679 . . . 4  |-  ( (
ph  /\  s  e.  T )  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s ) )  e.  RR )
22 fveq2 5882 . . . . . . 7  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
2322sumeq2sdv 13770 . . . . . 6  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
2423oveq2d 6322 . . . . 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 5963 . . . 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 665 . . 3  |-  ( (
ph  /\  s  e.  T )  ->  ( P `  s )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) ) )
288, 27vtoclg 3139 . 2  |-  ( S  e.  T  ->  (
( ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) ) )
2928anabsi7 826 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 370    = wceq 1437    e. wcel 1872   A.wral 2771   {crab 2775   class class class wbr 4423    |-> cmpt 4482   -->wf 5597   ` cfv 5601  (class class class)co 6306   RRcr 9546   0cc0 9547   1c1 9548    x. cmul 9552    <_ cle 9684    / cdiv 10277   NNcn 10617   ...cfz 11792   sum_csu 13752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-oadd 7198  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-sup 7966  df-oi 8035  df-card 8382  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-3 10677  df-n0 10878  df-z 10946  df-uz 11168  df-rp 11311  df-fz 11793  df-fzo 11924  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-clim 13552  df-sum 13753
This theorem is referenced by:  stoweidlem37  37839  stoweidlem38  37840  stoweidlem44  37846
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