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Theorem stoweidlem37 37892
Description: This lemma is used to prove the existence of a function p as in Lemma 1 of [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
stoweidlem37.1  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
stoweidlem37.2  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
stoweidlem37.3  |-  ( ph  ->  M  e.  NN )
stoweidlem37.4  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
stoweidlem37.5  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
stoweidlem37.6  |-  ( ph  ->  Z  e.  T )
Assertion
Ref Expression
stoweidlem37  |-  ( ph  ->  ( P `  Z
)  =  0 )
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, t    i, M, t
Allowed substitution hints:    ph( t, h)    A( t, i)    P( t, f, h, i)    Q( t, f, h, i)    G( i)    M( f, h)    Z( f)

Proof of Theorem stoweidlem37
StepHypRef Expression
1 stoweidlem37.6 . . 3  |-  ( ph  ->  Z  e.  T )
2 stoweidlem37.1 . . . 4  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
3 stoweidlem37.2 . . . 4  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
4 stoweidlem37.3 . . . 4  |-  ( ph  ->  M  e.  NN )
5 stoweidlem37.4 . . . 4  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
6 stoweidlem37.5 . . . 4  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
72, 3, 4, 5, 6stoweidlem30 37885 . . 3  |-  ( (
ph  /\  Z  e.  T )  ->  ( P `  Z )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
) ) )
81, 7mpdan 673 . 2  |-  ( ph  ->  ( P `  Z
)  =  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
) ) )
95fnvinran 37329 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i )  e.  Q )
10 fveq1 5862 . . . . . . . . . 10  |-  ( h  =  ( G `  i )  ->  (
h `  Z )  =  ( ( G `
 i ) `  Z ) )
1110eqeq1d 2452 . . . . . . . . 9  |-  ( h  =  ( G `  i )  ->  (
( h `  Z
)  =  0  <->  (
( G `  i
) `  Z )  =  0 ) )
12 fveq1 5862 . . . . . . . . . . . 12  |-  ( h  =  ( G `  i )  ->  (
h `  t )  =  ( ( G `
 i ) `  t ) )
1312breq2d 4413 . . . . . . . . . . 11  |-  ( h  =  ( G `  i )  ->  (
0  <_  ( h `  t )  <->  0  <_  ( ( G `  i
) `  t )
) )
1412breq1d 4411 . . . . . . . . . . 11  |-  ( h  =  ( G `  i )  ->  (
( h `  t
)  <_  1  <->  ( ( G `  i ) `  t )  <_  1
) )
1513, 14anbi12d 716 . . . . . . . . . 10  |-  ( h  =  ( G `  i )  ->  (
( 0  <_  (
h `  t )  /\  ( h `  t
)  <_  1 )  <-> 
( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) )
1615ralbidv 2826 . . . . . . . . 9  |-  ( h  =  ( G `  i )  ->  ( A. t  e.  T  ( 0  <_  (
h `  t )  /\  ( h `  t
)  <_  1 )  <->  A. t  e.  T  ( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) )
1711, 16anbi12d 716 . . . . . . . 8  |-  ( h  =  ( G `  i )  ->  (
( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_ 
( h `  t
)  /\  ( h `  t )  <_  1
) )  <->  ( (
( G `  i
) `  Z )  =  0  /\  A. t  e.  T  (
0  <_  ( ( G `  i ) `  t )  /\  (
( G `  i
) `  t )  <_  1 ) ) ) )
1817, 2elrab2 3197 . . . . . . 7  |-  ( ( G `  i )  e.  Q  <->  ( ( G `  i )  e.  A  /\  (
( ( G `  i ) `  Z
)  =  0  /\ 
A. t  e.  T  ( 0  <_  (
( G `  i
) `  t )  /\  ( ( G `  i ) `  t
)  <_  1 ) ) ) )
199, 18sylib 200 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
)  e.  A  /\  ( ( ( G `
 i ) `  Z )  =  0  /\  A. t  e.  T  ( 0  <_ 
( ( G `  i ) `  t
)  /\  ( ( G `  i ) `  t )  <_  1
) ) ) )
2019simprld 764 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  Z )  =  0 )
2120sumeq2dv 13762 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  sum_ i  e.  ( 1 ... M
) 0 )
22 fzfi 12182 . . . . 5  |-  ( 1 ... M )  e. 
Fin
23 olc 386 . . . . 5  |-  ( ( 1 ... M )  e.  Fin  ->  (
( 1 ... M
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... M )  e. 
Fin ) )
24 sumz 13781 . . . . 5  |-  ( ( ( 1 ... M
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... M )  e. 
Fin )  ->  sum_ i  e.  ( 1 ... M
) 0  =  0 )
2522, 23, 24mp2b 10 . . . 4  |-  sum_ i  e.  ( 1 ... M
) 0  =  0
2621, 25syl6eq 2500 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  Z
)  =  0 )
2726oveq2d 6304 . 2  |-  ( ph  ->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  Z )
)  =  ( ( 1  /  M )  x.  0 ) )
284nncnd 10622 . . . 4  |-  ( ph  ->  M  e.  CC )
294nnne0d 10651 . . . 4  |-  ( ph  ->  M  =/=  0 )
3028, 29reccld 10373 . . 3  |-  ( ph  ->  ( 1  /  M
)  e.  CC )
3130mul01d 9829 . 2  |-  ( ph  ->  ( ( 1  /  M )  x.  0 )  =  0 )
328, 27, 313eqtrd 2488 1  |-  ( ph  ->  ( P `  Z
)  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736   {crab 2740    C_ wss 3403   class class class wbr 4401    |-> cmpt 4460   -->wf 5577   ` cfv 5581  (class class class)co 6288   Fincfn 7566   RRcr 9535   0cc0 9536   1c1 9537    x. cmul 9541    <_ cle 9673    / cdiv 10266   NNcn 10606   ZZ>=cuz 11156   ...cfz 11781   sum_csu 13745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746
This theorem is referenced by:  stoweidlem44  37899
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