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Theorem stoweidlem38 37170
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 10542 . . . . 5  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
32adantr 463 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  /  M )  e.  RR )
4 fzfid 12037 . . . . 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 37147 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( ( G `  i ) `  S
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  S )  /\  (
( G `  i
) `  S )  <_  1 ) )
98simp1d 1009 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  e.  RR )
109an32s 805 . . . . 5  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  e.  RR )
114, 10fsumrecl 13612 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  e.  RR )
12 1red 9561 . . . . . 6  |-  ( ph  ->  1  e.  RR )
13 0le1 10036 . . . . . . 7  |-  0  <_  1
1413a1i 11 . . . . . 6  |-  ( ph  ->  0  <_  1 )
151nnred 10511 . . . . . 6  |-  ( ph  ->  M  e.  RR )
161nngt0d 10540 . . . . . 6  |-  ( ph  ->  0  <  M )
17 divge0 10372 . . . . . 6  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <_  ( 1  /  M ) )
1812, 14, 15, 16, 17syl22anc 1231 . . . . 5  |-  ( ph  ->  0  <_  ( 1  /  M ) )
1918adantr 463 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( 1  /  M
) )
208simp2d 1010 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  0  <_  ( ( G `  i ) `  S
) )
2120an32s 805 . . . . 5  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( ( G `  i ) `  S
) )
224, 10, 21fsumge0 13667 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  0  <_ 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) )
233, 11, 19, 22mulge0d 10089 . . 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 37162 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
2623, 25breqtrrd 4420 . 2  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( P `  S
) )
27 1red 9561 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  1  e.  RR )
288simp3d 1011 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  <_  1 )
2928an32s 805 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  <_  1 )
304, 10, 27, 29fsumle 13671 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  sum_ i  e.  ( 1 ... M
) 1 )
31 fzfid 12037 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
32 ax-1cn 9500 . . . . . . . . 9  |-  1  e.  CC
33 fsumconst 13663 . . . . . . . . 9  |-  ( ( ( 1 ... M
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
3431, 32, 33sylancl 660 . . . . . . . 8  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
351nnnn0d 10813 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
36 hashfz1 12373 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
3735, 36syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3837oveq1d 6249 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  x.  1 )  =  ( M  x.  1 ) )
391nncnd 10512 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
4039mulid1d 9563 . . . . . . . 8  |-  ( ph  ->  ( M  x.  1 )  =  M )
4134, 38, 403eqtrd 2447 . . . . . . 7  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  M )
4241adantr 463 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) 1  =  M )
4330, 42breqtrd 4418 . . . . 5  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  M
)
4415adantr 463 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  M  e.  RR )
45 1red 9561 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  1  e.  RR )
46 0lt1 10035 . . . . . . . 8  |-  0  <  1
4746a1i 11 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  0  <  1 )
4815, 16jca 530 . . . . . . . 8  |-  ( ph  ->  ( M  e.  RR  /\  0  <  M ) )
4948adantr 463 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  ( M  e.  RR  /\  0  <  M ) )
50 divgt0 10371 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <  ( 1  /  M ) )
5145, 47, 49, 50syl21anc 1229 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  0  <  ( 1  /  M
) )
52 lemul2 10356 . . . . . 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 1234 . . . . 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 4414 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  ( ( 1  /  M )  x.  M
) )
5632a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
571nnne0d 10541 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
5856, 39, 573jca 1177 . . . . 5  |-  ( ph  ->  ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
5958adantr 463 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
60 divcan1 10177 . . . 4  |-  ( ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  (
( 1  /  M
)  x.  M )  =  1 )
6159, 60syl 17 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  x.  M )  =  1 )
6255, 61breqtrd 4418 . 2  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  1 )
6326, 62jca 530 1  |-  ( (
ph  /\  S  e.  T )  ->  (
0  <_  ( P `  S )  /\  ( P `  S )  <_  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   {crab 2757   class class class wbr 4394    |-> cmpt 4452   -->wf 5521   ` cfv 5525  (class class class)co 6234   Fincfn 7474   CCcc 9440   RRcr 9441   0cc0 9442   1c1 9443    x. cmul 9447    < clt 9578    <_ cle 9579    / cdiv 10167   NNcn 10496   NN0cn0 10756   ...cfz 11643   #chash 12359   sum_csu 13564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-ico 11506  df-fz 11644  df-fzo 11768  df-seq 12062  df-exp 12121  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-sum 13565
This theorem is referenced by:  stoweidlem44  37176
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