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Theorem stoweidlem38 27654
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 GlaucoSiliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem38.1  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
stoweidlem38.2  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
stoweidlem38.3  |-  ( ph  ->  M  e.  NN )
stoweidlem38.4  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
stoweidlem38.5  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
Assertion
Ref Expression
stoweidlem38  |-  ( (
ph  /\  S  e.  T )  ->  (
0  <_  ( P `  S )  /\  ( P `  S )  <_  1 ) )
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 stoweidlem38
StepHypRef Expression
1 stoweidlem38.3 . . . . . 6  |-  ( ph  ->  M  e.  NN )
21nnrecred 10001 . . . . 5  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
32adantr 452 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  /  M )  e.  RR )
4 fzfid 11267 . . . . 5  |-  ( (
ph  /\  S  e.  T )  ->  (
1 ... M )  e. 
Fin )
5 stoweidlem38.1 . . . . . . . 8  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
6 stoweidlem38.4 . . . . . . . 8  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
7 stoweidlem38.5 . . . . . . . 8  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
85, 6, 7stoweidlem15 27631 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( ( G `  i ) `  S
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  S )  /\  (
( G `  i
) `  S )  <_  1 ) )
98simp1d 969 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  e.  RR )
109an32s 780 . . . . 5  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  e.  RR )
114, 10fsumrecl 12483 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  e.  RR )
12 1re 9046 . . . . . . 7  |-  1  e.  RR
1312a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  RR )
14 0le1 9507 . . . . . . 7  |-  0  <_  1
1514a1i 11 . . . . . 6  |-  ( ph  ->  0  <_  1 )
161nnred 9971 . . . . . 6  |-  ( ph  ->  M  e.  RR )
171nngt0d 9999 . . . . . 6  |-  ( ph  ->  0  <  M )
18 divge0 9835 . . . . . 6  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <_  ( 1  /  M ) )
1913, 15, 16, 17, 18syl22anc 1185 . . . . 5  |-  ( ph  ->  0  <_  ( 1  /  M ) )
2019adantr 452 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( 1  /  M
) )
218simp2d 970 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  0  <_  ( ( G `  i ) `  S
) )
2221an32s 780 . . . . 5  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( ( G `  i ) `  S
) )
234, 10, 22fsumge0 12529 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  0  <_ 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) )
243, 11, 20, 23mulge0d 9559 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
) )
25 stoweidlem38.2 . . . 4  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
265, 25, 1, 6, 7stoweidlem30 27646 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
2724, 26breqtrrd 4198 . 2  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( P `  S
) )
2812a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  1  e.  RR )
298simp3d 971 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  <_  1 )
3029an32s 780 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  <_  1 )
314, 10, 28, 30fsumle 12533 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  sum_ i  e.  ( 1 ... M
) 1 )
32 fzfid 11267 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
33 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
34 fsumconst 12528 . . . . . . . . 9  |-  ( ( ( 1 ... M
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
3532, 33, 34sylancl 644 . . . . . . . 8  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
361nnnn0d 10230 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
37 hashfz1 11585 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
3836, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3938oveq1d 6055 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  x.  1 )  =  ( M  x.  1 ) )
401nncnd 9972 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
4140mulid1d 9061 . . . . . . . 8  |-  ( ph  ->  ( M  x.  1 )  =  M )
4235, 39, 413eqtrd 2440 . . . . . . 7  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  M )
4342adantr 452 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) 1  =  M )
4431, 43breqtrd 4196 . . . . 5  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  M
)
4516adantr 452 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  M  e.  RR )
4612a1i 11 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  1  e.  RR )
47 0lt1 9506 . . . . . . . 8  |-  0  <  1
4847a1i 11 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  0  <  1 )
4916, 17jca 519 . . . . . . . 8  |-  ( ph  ->  ( M  e.  RR  /\  0  <  M ) )
5049adantr 452 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  ( M  e.  RR  /\  0  <  M ) )
51 divgt0 9834 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <  ( 1  /  M ) )
5246, 48, 50, 51syl21anc 1183 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  0  <  ( 1  /  M
) )
53 lemul2 9819 . . . . . 6  |-  ( (
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
)  e.  RR  /\  M  e.  RR  /\  (
( 1  /  M
)  e.  RR  /\  0  <  ( 1  /  M ) ) )  ->  ( sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  M  <->  ( ( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) )  <_ 
( ( 1  /  M )  x.  M
) ) )
5411, 45, 3, 52, 53syl112anc 1188 . . . . 5  |-  ( (
ph  /\  S  e.  T )  ->  ( sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )  <_  M  <->  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) )  <_  (
( 1  /  M
)  x.  M ) ) )
5544, 54mpbid 202 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) )  <_ 
( ( 1  /  M )  x.  M
) )
5626, 55eqbrtrd 4192 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  ( ( 1  /  M )  x.  M
) )
5733a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
581nnne0d 10000 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
5957, 40, 583jca 1134 . . . . 5  |-  ( ph  ->  ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
6059adantr 452 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
61 divcan1 9643 . . . 4  |-  ( ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  (
( 1  /  M
)  x.  M )  =  1 )
6260, 61syl 16 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  x.  M )  =  1 )
6356, 62breqtrd 4196 . 2  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  1 )
6427, 63jca 519 1  |-  ( (
ph  /\  S  e.  T )  ->  (
0  <_  ( P `  S )  /\  ( P `  S )  <_  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   {crab 2670   class class class wbr 4172    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076    <_ cle 9077    / cdiv 9633   NNcn 9956   NN0cn0 10177   ...cfz 10999   #chash 11573   sum_csu 12434
This theorem is referenced by:  stoweidlem44  27660
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435
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