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Theorem stoweidlem38 29676
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 10354 . . . . 5  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
32adantr 462 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  /  M )  e.  RR )
4 fzfid 11778 . . . . 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 29653 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( ( G `  i ) `  S
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  S )  /\  (
( G `  i
) `  S )  <_  1 ) )
98simp1d 993 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  e.  RR )
109an32s 795 . . . . 5  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  e.  RR )
114, 10fsumrecl 13194 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  e.  RR )
12 1re 9372 . . . . . . 7  |-  1  e.  RR
1312a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  RR )
14 0le1 9850 . . . . . . 7  |-  0  <_  1
1514a1i 11 . . . . . 6  |-  ( ph  ->  0  <_  1 )
161nnred 10324 . . . . . 6  |-  ( ph  ->  M  e.  RR )
171nngt0d 10352 . . . . . 6  |-  ( ph  ->  0  <  M )
18 divge0 10185 . . . . . 6  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <_  ( 1  /  M ) )
1913, 15, 16, 17, 18syl22anc 1212 . . . . 5  |-  ( ph  ->  0  <_  ( 1  /  M ) )
2019adantr 462 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( 1  /  M
) )
218simp2d 994 . . . . . 6  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  0  <_  ( ( G `  i ) `  S
) )
2221an32s 795 . . . . 5  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( ( G `  i ) `  S
) )
234, 10, 22fsumge0 13240 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  0  <_ 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) )
243, 11, 20, 23mulge0d 9903 . . 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 29668 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
2724, 26breqtrrd 4306 . 2  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( P `  S
) )
2812a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  1  e.  RR )
298simp3d 995 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  <_  1 )
3029an32s 795 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  <_  1 )
314, 10, 28, 30fsumle 13244 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  sum_ i  e.  ( 1 ... M
) 1 )
32 fzfid 11778 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
33 ax-1cn 9327 . . . . . . . . 9  |-  1  e.  CC
34 fsumconst 13239 . . . . . . . . 9  |-  ( ( ( 1 ... M
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
3532, 33, 34sylancl 655 . . . . . . . 8  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
361nnnn0d 10623 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
37 hashfz1 12100 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
3836, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3938oveq1d 6095 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  x.  1 )  =  ( M  x.  1 ) )
401nncnd 10325 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
4140mulid1d 9390 . . . . . . . 8  |-  ( ph  ->  ( M  x.  1 )  =  M )
4235, 39, 413eqtrd 2469 . . . . . . 7  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  M )
4342adantr 462 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) 1  =  M )
4431, 43breqtrd 4304 . . . . 5  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  M
)
4516adantr 462 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  M  e.  RR )
4612a1i 11 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  1  e.  RR )
47 0lt1 9849 . . . . . . . 8  |-  0  <  1
4847a1i 11 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  0  <  1 )
4916, 17jca 529 . . . . . . . 8  |-  ( ph  ->  ( M  e.  RR  /\  0  <  M ) )
5049adantr 462 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  ( M  e.  RR  /\  0  <  M ) )
51 divgt0 10184 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <  ( 1  /  M ) )
5246, 48, 50, 51syl21anc 1210 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  0  <  ( 1  /  M
) )
53 lemul2 10169 . . . . . 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 1215 . . . . 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 210 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) )  <_ 
( ( 1  /  M )  x.  M
) )
5626, 55eqbrtrd 4300 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  ( ( 1  /  M )  x.  M
) )
5733a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
581nnne0d 10353 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
5957, 40, 583jca 1161 . . . . 5  |-  ( ph  ->  ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
6059adantr 462 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
61 divcan1 9990 . . . 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 4304 . 2  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  1 )
6427, 63jca 529 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596   A.wral 2705   {crab 2709   class class class wbr 4280    e. cmpt 4338   -->wf 5402   ` cfv 5406  (class class class)co 6080   Fincfn 7298   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270    x. cmul 9274    < clt 9405    <_ cle 9406    / cdiv 9980   NNcn 10309   NN0cn0 10566   ...cfz 11423   #chash 12086   sum_csu 13146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-n0 10567  df-z 10634  df-uz 10849  df-rp 10979  df-ico 11293  df-fz 11424  df-fzo 11532  df-seq 11790  df-exp 11849  df-hash 12087  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-clim 12949  df-sum 13147
This theorem is referenced by:  stoweidlem44  29682
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