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Theorem stoweidlem38 31295
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
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 10572 . . . . 5  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
32adantr 465 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  /  M )  e.  RR )
4 fzfid 12041 . . . . 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 31272 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( ( G `  i ) `  S
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  S )  /\  (
( G `  i
) `  S )  <_  1 ) )
98simp1d 1003 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  e.  RR )
109an32s 802 . . . . 5  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  e.  RR )
114, 10fsumrecl 13507 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  e.  RR )
12 1red 9602 . . . . . 6  |-  ( ph  ->  1  e.  RR )
13 0le1 10067 . . . . . . 7  |-  0  <_  1
1413a1i 11 . . . . . 6  |-  ( ph  ->  0  <_  1 )
151nnred 10542 . . . . . 6  |-  ( ph  ->  M  e.  RR )
161nngt0d 10570 . . . . . 6  |-  ( ph  ->  0  <  M )
17 divge0 10402 . . . . . 6  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <_  ( 1  /  M ) )
1812, 14, 15, 16, 17syl22anc 1224 . . . . 5  |-  ( ph  ->  0  <_  ( 1  /  M ) )
1918adantr 465 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( 1  /  M
) )
208simp2d 1004 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  0  <_  ( ( G `  i ) `  S
) )
2120an32s 802 . . . . 5  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( ( G `  i ) `  S
) )
224, 10, 21fsumge0 13560 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  0  <_ 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) )
233, 11, 19, 22mulge0d 10120 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
) )
24 stoweidlem38.2 . . . 4  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
255, 24, 1, 6, 7stoweidlem30 31287 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
2623, 25breqtrrd 4468 . 2  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( P `  S
) )
27 1red 9602 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  1  e.  RR )
288simp3d 1005 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  <_  1 )
2928an32s 802 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  <_  1 )
304, 10, 27, 29fsumle 13564 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  sum_ i  e.  ( 1 ... M
) 1 )
31 fzfid 12041 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
32 ax-1cn 9541 . . . . . . . . 9  |-  1  e.  CC
33 fsumconst 13556 . . . . . . . . 9  |-  ( ( ( 1 ... M
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
3431, 32, 33sylancl 662 . . . . . . . 8  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
351nnnn0d 10843 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
36 hashfz1 12376 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
3735, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3837oveq1d 6292 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  x.  1 )  =  ( M  x.  1 ) )
391nncnd 10543 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
4039mulid1d 9604 . . . . . . . 8  |-  ( ph  ->  ( M  x.  1 )  =  M )
4134, 38, 403eqtrd 2507 . . . . . . 7  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  M )
4241adantr 465 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) 1  =  M )
4330, 42breqtrd 4466 . . . . 5  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  M
)
4415adantr 465 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  M  e.  RR )
45 1red 9602 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  1  e.  RR )
46 0lt1 10066 . . . . . . . 8  |-  0  <  1
4746a1i 11 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  0  <  1 )
4815, 16jca 532 . . . . . . . 8  |-  ( ph  ->  ( M  e.  RR  /\  0  <  M ) )
4948adantr 465 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  ( M  e.  RR  /\  0  <  M ) )
50 divgt0 10401 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <  ( 1  /  M ) )
5145, 47, 49, 50syl21anc 1222 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  0  <  ( 1  /  M
) )
52 lemul2 10386 . . . . . 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
) ) )
5311, 44, 3, 51, 52syl112anc 1227 . . . . 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 ) ) )
5443, 53mpbid 210 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) )  <_ 
( ( 1  /  M )  x.  M
) )
5525, 54eqbrtrd 4462 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  ( ( 1  /  M )  x.  M
) )
5632a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
571nnne0d 10571 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
5856, 39, 573jca 1171 . . . . 5  |-  ( ph  ->  ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
5958adantr 465 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
60 divcan1 10207 . . . 4  |-  ( ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  (
( 1  /  M
)  x.  M )  =  1 )
6159, 60syl 16 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  x.  M )  =  1 )
6255, 61breqtrd 4466 . 2  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  1 )
6326, 62jca 532 1  |-  ( (
ph  /\  S  e.  T )  ->  (
0  <_  ( P `  S )  /\  ( P `  S )  <_  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   {crab 2813   class class class wbr 4442    |-> cmpt 4500   -->wf 5577   ` cfv 5581  (class class class)co 6277   Fincfn 7508   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    x. cmul 9488    < clt 9619    <_ cle 9620    / cdiv 10197   NNcn 10527   NN0cn0 10786   ...cfz 11663   #chash 12362   sum_csu 13459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  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 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-ico 11526  df-fz 11664  df-fzo 11784  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-clim 13262  df-sum 13460
This theorem is referenced by:  stoweidlem44  31301
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