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Theorem stoweidlem38 27889
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 1re 8853 . . . . . . . . . 10  |-  1  e.  RR
21a1i 10 . . . . . . . . 9  |-  ( ph  ->  1  e.  RR )
3 stoweidlem38.3 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN )
4 nnre 9769 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  e.  RR )
53, 4syl 15 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
6 nnne0 9794 . . . . . . . . . 10  |-  ( M  e.  NN  ->  M  =/=  0 )
73, 6syl 15 . . . . . . . . 9  |-  ( ph  ->  M  =/=  0 )
82, 5, 73jca 1132 . . . . . . . 8  |-  ( ph  ->  ( 1  e.  RR  /\  M  e.  RR  /\  M  =/=  0 ) )
9 redivcl 9495 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
1  /  M )  e.  RR )
108, 9syl 15 . . . . . . 7  |-  ( ph  ->  ( 1  /  M
)  e.  RR )
1110adantr 451 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  (
1  /  M )  e.  RR )
12 0le1 9313 . . . . . . . . . . 11  |-  0  <_  1
1312a1i 10 . . . . . . . . . 10  |-  ( ph  ->  0  <_  1 )
142, 13jca 518 . . . . . . . . 9  |-  ( ph  ->  ( 1  e.  RR  /\  0  <_  1 ) )
15 nngt0 9791 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  0  <  M )
163, 15syl 15 . . . . . . . . . 10  |-  ( ph  ->  0  <  M )
175, 16jca 518 . . . . . . . . 9  |-  ( ph  ->  ( M  e.  RR  /\  0  <  M ) )
1814, 17jca 518 . . . . . . . 8  |-  ( ph  ->  ( ( 1  e.  RR  /\  0  <_ 
1 )  /\  ( M  e.  RR  /\  0  <  M ) ) )
19 divge0 9641 . . . . . . . 8  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <_  ( 1  /  M ) )
2018, 19syl 15 . . . . . . 7  |-  ( ph  ->  0  <_  ( 1  /  M ) )
2120adantr 451 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( 1  /  M
) )
2211, 21jca 518 . . . . 5  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  e.  RR  /\  0  <_  ( 1  /  M ) ) )
23 fzfid 11051 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  (
1 ... M )  e. 
Fin )
24 simpll 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  ph )
25 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  i  e.  ( 1 ... M
) )
2624, 25jca 518 . . . . . . . . 9  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  ( ph  /\  i  e.  ( 1 ... M ) ) )
27 simplr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  S  e.  T )
2826, 27jca 518 . . . . . . . 8  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T ) )
29 stoweidlem38.1 . . . . . . . . . 10  |-  Q  =  { h  e.  A  |  ( ( h `
 Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) ) }
30 stoweidlem38.4 . . . . . . . . . 10  |-  ( ph  ->  G : ( 1 ... M ) --> Q )
31 stoweidlem38.5 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
3229, 30, 31stoweidlem15 27866 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( ( G `  i ) `  S
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  S )  /\  (
( G `  i
) `  S )  <_  1 ) )
3332simp1d 967 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  e.  RR )
3428, 33syl 15 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  e.  RR )
3523, 34fsumrecl 12223 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  e.  RR )
36 fzfi 11050 . . . . . . . . . . 11  |-  ( 1 ... M )  e. 
Fin
3736olci 380 . . . . . . . . . 10  |-  ( ( 1 ... M ) 
C_  ( ZZ>= `  0
)  \/  ( 1 ... M )  e. 
Fin )
38 sumz 12211 . . . . . . . . . 10  |-  ( ( ( 1 ... M
)  C_  ( ZZ>= ` 
0 )  \/  (
1 ... M )  e. 
Fin )  ->  sum_ i  e.  ( 1 ... M
) 0  =  0 )
3937, 38ax-mp 8 . . . . . . . . 9  |-  sum_ i  e.  ( 1 ... M
) 0  =  0
4039a1i 10 . . . . . . . 8  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 0  =  0 )
4140adantr 451 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) 0  =  0 )
4236a1i 10 . . . . . . . 8  |-  ( (
ph  /\  S  e.  T )  ->  (
1 ... M )  e. 
Fin )
43 0re 8854 . . . . . . . . 9  |-  0  e.  RR
4443a1i 10 . . . . . . . 8  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  0  e.  RR )
4532simp2d 968 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  0  <_  ( ( G `  i ) `  S
) )
4628, 45syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  0  <_  ( ( G `  i ) `  S
) )
4742, 44, 34, 46fsumle 12273 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) 0  <_  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) )
4841, 47eqbrtrrd 4061 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  0  <_ 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) )
4935, 48jca 518 . . . . 5  |-  ( (
ph  /\  S  e.  T )  ->  ( sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )  e.  RR  /\  0  <_  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
) )
5022, 49jca 518 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
( ( 1  /  M )  e.  RR  /\  0  <_  ( 1  /  M ) )  /\  ( sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  e.  RR  /\  0  <_  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) ) ) )
51 mulge0 9307 . . . 4  |-  ( ( ( ( 1  /  M )  e.  RR  /\  0  <_  ( 1  /  M ) )  /\  ( sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  e.  RR  /\  0  <_  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) ) )  ->  0  <_  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) ) )
5250, 51syl 15 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )
) )
53 stoweidlem38.2 . . . 4  |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
5429, 53, 3, 30, 31stoweidlem30 27881 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  =  ( ( 1  /  M )  x. 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  S
) ) )
5552, 54breqtrrd 4065 . 2  |-  ( (
ph  /\  S  e.  T )  ->  0  <_  ( P `  S
) )
561a1i 10 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  1  e.  RR )
57 simp3 957 . . . . . . . . . 10  |-  ( ( ( ( G `  i ) `  S
)  e.  RR  /\  0  <_  ( ( G `
 i ) `  S )  /\  (
( G `  i
) `  S )  <_  1 )  ->  (
( G `  i
) `  S )  <_  1 )
5832, 57syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  i  e.  ( 1 ... M
) )  /\  S  e.  T )  ->  (
( G `  i
) `  S )  <_  1 )
5928, 58syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  <_  1 )
6059idi 2 . . . . . . 7  |-  ( ( ( ph  /\  S  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  S )  <_  1 )
6123, 34, 56, 60fsumle 12273 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  sum_ i  e.  ( 1 ... M
) 1 )
6236a1i 10 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
63 ax-1cn 8811 . . . . . . . . . . 11  |-  1  e.  CC
6463a1i 10 . . . . . . . . . 10  |-  ( ph  ->  1  e.  CC )
6562, 64jca 518 . . . . . . . . 9  |-  ( ph  ->  ( ( 1 ... M )  e.  Fin  /\  1  e.  CC ) )
66 fsumconst 12268 . . . . . . . . 9  |-  ( ( ( 1 ... M
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
6765, 66syl 15 . . . . . . . 8  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  ( (
# `  ( 1 ... M ) )  x.  1 ) )
68 nnnn0 9988 . . . . . . . . . . . 12  |-  ( M  e.  NN  ->  M  e.  NN0 )
693, 68syl 15 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN0 )
70 hashfz1 11361 . . . . . . . . . . 11  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
7169, 70syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
7271oveq1d 5889 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  x.  1 )  =  ( M  x.  1 ) )
73 nncn 9770 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  M  e.  CC )
743, 73syl 15 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
75 mulid1 8851 . . . . . . . . . 10  |-  ( M  e.  CC  ->  ( M  x.  1 )  =  M )
7674, 75syl 15 . . . . . . . . 9  |-  ( ph  ->  ( M  x.  1 )  =  M )
7772, 76eqtrd 2328 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  x.  1 )  =  M )
7867, 77eqtrd 2328 . . . . . . 7  |-  ( ph  -> 
sum_ i  e.  ( 1 ... M ) 1  =  M )
7978adantr 451 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) 1  =  M )
8061, 79breqtrd 4063 . . . . 5  |-  ( (
ph  /\  S  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S )  <_  M
)
815adantr 451 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  M  e.  RR )
821a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  S  e.  T )  ->  1  e.  RR )
83 0lt1 9312 . . . . . . . . . . . 12  |-  0  <  1
8483a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  S  e.  T )  ->  0  <  1 )
8582, 84jca 518 . . . . . . . . . 10  |-  ( (
ph  /\  S  e.  T )  ->  (
1  e.  RR  /\  0  <  1 ) )
8617adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  S  e.  T )  ->  ( M  e.  RR  /\  0  <  M ) )
8785, 86jca 518 . . . . . . . . 9  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  e.  RR  /\  0  <  1 )  /\  ( M  e.  RR  /\  0  < 
M ) ) )
88 divgt0 9640 . . . . . . . . 9  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( M  e.  RR  /\  0  < 
M ) )  -> 
0  <  ( 1  /  M ) )
8987, 88syl 15 . . . . . . . 8  |-  ( (
ph  /\  S  e.  T )  ->  0  <  ( 1  /  M
) )
9011, 89jca 518 . . . . . . 7  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  e.  RR  /\  0  <  ( 1  /  M ) ) )
9135, 81, 903jca 1132 . . . . . 6  |-  ( (
ph  /\  S  e.  T )  ->  ( sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  S )  e.  RR  /\  M  e.  RR  /\  ( ( 1  /  M )  e.  RR  /\  0  <  ( 1  /  M
) ) ) )
92 lemul2 9625 . . . . . 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
) ) )
9391, 92syl 15 . . . . 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 ) ) )
9480, 93mpbid 201 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  S ) )  <_ 
( ( 1  /  M )  x.  M
) )
9554, 94eqbrtrd 4059 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  ( ( 1  /  M )  x.  M
) )
9664, 74, 73jca 1132 . . . . 5  |-  ( ph  ->  ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
9796adantr 451 . . . 4  |-  ( (
ph  /\  S  e.  T )  ->  (
1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 ) )
98 divcan1 9449 . . . 4  |-  ( ( 1  e.  CC  /\  M  e.  CC  /\  M  =/=  0 )  ->  (
( 1  /  M
)  x.  M )  =  1 )
9997, 98syl 15 . . 3  |-  ( (
ph  /\  S  e.  T )  ->  (
( 1  /  M
)  x.  M )  =  1 )
10095, 99breqtrd 4063 . 2  |-  ( (
ph  /\  S  e.  T )  ->  ( P `  S )  <_  1 )
10155, 100jca 518 1  |-  ( (
ph  /\  S  e.  T )  ->  (
0  <_  ( P `  S )  /\  ( P `  S )  <_  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZ>=cuz 10246   ...cfz 10798   #chash 11353   sum_csu 12174
This theorem is referenced by:  stoweidlem44  27895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175
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