Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem38 Structured version   Unicode version

Theorem stoweidlem38 27763
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 10045 . . . . 5  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
32adantr 452 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  /  M )  e.  RR )
4 fzfid 11312 . . . . 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 27740 . . . . . . 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 12528 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  e.  RR )
12 1re 9090 . . . . . . 7  |-  1  e.  RR
1312a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  RR )
14 0le1 9551 . . . . . . 7  |-  0  <_  1
1514a1i 11 . . . . . 6  |-  ( ph  ->  0  <_  1 )
161nnred 10015 . . . . . 6  |-  ( ph  ->  M  e.  RR )
171nngt0d 10043 . . . . . 6  |-  ( ph  ->  0  <  M )
18 divge0 9879 . . . . . 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 12574 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  0  <_ 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) )
243, 11, 20, 23mulge0d 9603 . . 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 27755 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
2724, 26breqtrrd 4238 . 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 12578 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  sum_ i  e.  ( 1 ... M
) 1 )
32 fzfid 11312 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
33 ax-1cn 9048 . . . . . . . . 9  |-  1  e.  CC
34 fsumconst 12573 . . . . . . . . 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 10274 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
37 hashfz1 11630 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
3836, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3938oveq1d 6096 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  x.  1 )  =  ( M  x.  1 ) )
401nncnd 10016 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
4140mulid1d 9105 . . . . . . . 8  |-  ( ph  ->  ( M  x.  1 )  =  M )
4235, 39, 413eqtrd 2472 . . . . . . 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 4236 . . . . 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 9550 . . . . . . . 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 9878 . . . . . . 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 9863 . . . . . 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 4232 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  ( ( 1  /  M )  x.  M
) )
5733a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
581nnne0d 10044 . . . . . 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 9687 . . . 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 4236 . 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 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   {crab 2709   class class class wbr 4212    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081   Fincfn 7109   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    < clt 9120    <_ cle 9121    / cdiv 9677   NNcn 10000   NN0cn0 10221   ...cfz 11043   #chash 11618   sum_csu 12479
This theorem is referenced by:  stoweidlem44  27769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480
  Copyright terms: Public domain W3C validator