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Theorem stoweidlem30 31973
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 2529 . . . . 5  |-  ( s  =  S  ->  (
s  e.  T  <->  S  e.  T ) )
21anbi2d 703 . . . 4  |-  ( s  =  S  ->  (
( ph  /\  s  e.  T )  <->  ( ph  /\  S  e.  T ) ) )
3 fveq2 5872 . . . . 5  |-  ( s  =  S  ->  ( P `  s )  =  ( P `  S ) )
4 fveq2 5872 . . . . . . 7  |-  ( s  =  S  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  S ) )
54sumeq2sdv 13537 . . . . . 6  |-  ( s  =  S  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
)
65oveq2d 6312 . . . . 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 2479 . . . 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 320 . . 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 461 . . . 4  |-  ( (
ph  /\  s  e.  T )  ->  s  e.  T )
10 stoweidlem30.3 . . . . . . 7  |-  ( ph  ->  M  e.  NN )
1110nnrecred 10602 . . . . . 6  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
1211adantr 465 . . . . 5  |-  ( (
ph  /\  s  e.  T )  ->  (
1  /  M )  e.  RR )
13 fzfid 12085 . . . . . 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 31958 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( ( G `  i ) `  s
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  s )  /\  (
( G `  i
) `  s )  <_  1 ) )
1817simp1d 1008 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  s  e.  T )  ->  (
( G `  i
) `  s )  e.  RR )
1918an32s 804 . . . . . 6  |-  ( ( ( ph  /\  s  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  s )  e.  RR )
2013, 19fsumrecl 13567 . . . . 5  |-  ( (
ph  /\  s  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  e.  RR )
2112, 20remulcld 9641 . . . 4  |-  ( (
ph  /\  s  e.  T )  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s ) )  e.  RR )
22 fveq2 5872 . . . . . . 7  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
2322sumeq2sdv 13537 . . . . . 6  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
2423oveq2d 6312 . . . . 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 5954 . . . 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 661 . . 3  |-  ( (
ph  /\  s  e.  T )  ->  ( P `  s )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) ) )
288, 27vtoclg 3167 . 2  |-  ( S  e.  T  ->  (
( ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) ) )
2928anabsi7 819 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 369    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   class class class wbr 4456    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    <_ cle 9646    / cdiv 10227   NNcn 10556   ...cfz 11697   sum_csu 13519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520
This theorem is referenced by:  stoweidlem37  31980  stoweidlem38  31981  stoweidlem44  31987
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