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Theorem stoweidlem62 37225
Description: This theorem proves the Stone Weierstrass theorem for the non-trivial case in which T is nonempty. The proof follows [BrosowskiDeutsh] p. 89 (through page 92). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem62.1  |-  F/_ t F
stoweidlem62.2  |-  F/ f
ph
stoweidlem62.3  |-  F/ t
ph
stoweidlem62.4  |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
stoweidlem62.5  |-  K  =  ( topGen `  ran  (,) )
stoweidlem62.6  |-  T  = 
U. J
stoweidlem62.7  |-  ( ph  ->  J  e.  Comp )
stoweidlem62.8  |-  C  =  ( J  Cn  K
)
stoweidlem62.9  |-  ( ph  ->  A  C_  C )
stoweidlem62.10  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem62.11  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem62.12  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
stoweidlem62.13  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
stoweidlem62.14  |-  ( ph  ->  F  e.  C )
stoweidlem62.15  |-  ( ph  ->  E  e.  RR+ )
stoweidlem62.16  |-  ( ph  ->  T  =/=  (/) )
stoweidlem62.17  |-  ( ph  ->  E  <  ( 1  /  3 ) )
Assertion
Ref Expression
stoweidlem62  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
Distinct variable groups:    f, g,
t, A    f, q,
r, x, t, A   
f, E, g, t   
f, F, g    f, H, g    f, J, r, t    T, f, g, t    ph, f, g    E, q, r, x    H, q, r, x    T, q, r, x    ph, q,
r, x    t, K    x, F
Allowed substitution hints:    ph( t)    C( x, t, f, g, r, q)    F( t, r, q)    H( t)    J( x, g, q)    K( x, f, g, r, q)

Proof of Theorem stoweidlem62
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 stoweidlem62.4 . . . . 5  |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
2 nfmpt1 4486 . . . . 5  |-  F/_ t
( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
31, 2nfcxfr 2564 . . . 4  |-  F/_ t H
4 stoweidlem62.3 . . . 4  |-  F/ t
ph
5 stoweidlem62.5 . . . 4  |-  K  =  ( topGen `  ran  (,) )
6 stoweidlem62.7 . . . 4  |-  ( ph  ->  J  e.  Comp )
7 stoweidlem62.6 . . . 4  |-  T  = 
U. J
8 stoweidlem62.16 . . . 4  |-  ( ph  ->  T  =/=  (/) )
9 stoweidlem62.8 . . . 4  |-  C  =  ( J  Cn  K
)
10 stoweidlem62.9 . . . 4  |-  ( ph  ->  A  C_  C )
11 eleq1 2476 . . . . . . 7  |-  ( g  =  h  ->  (
g  e.  A  <->  h  e.  A ) )
12113anbi3d 1309 . . . . . 6  |-  ( g  =  h  ->  (
( ph  /\  f  e.  A  /\  g  e.  A )  <->  ( ph  /\  f  e.  A  /\  h  e.  A )
) )
13 fveq1 5850 . . . . . . . . 9  |-  ( g  =  h  ->  (
g `  t )  =  ( h `  t ) )
1413oveq2d 6296 . . . . . . . 8  |-  ( g  =  h  ->  (
( f `  t
)  +  ( g `
 t ) )  =  ( ( f `
 t )  +  ( h `  t
) ) )
1514mpteq2dv 4484 . . . . . . 7  |-  ( g  =  h  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( f `  t )  +  ( h `  t ) ) ) )
1615eleq1d 2473 . . . . . 6  |-  ( g  =  h  ->  (
( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A  <->  ( t  e.  T  |->  ( ( f `  t
)  +  ( h `
 t ) ) )  e.  A ) )
1712, 16imbi12d 320 . . . . 5  |-  ( g  =  h  ->  (
( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A
)  <->  ( ( ph  /\  f  e.  A  /\  h  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( h `  t ) ) )  e.  A
) ) )
18 stoweidlem62.10 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
1917, 18chvarv 2043 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  h  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( h `  t ) ) )  e.  A )
2013oveq2d 6296 . . . . . . . 8  |-  ( g  =  h  ->  (
( f `  t
)  x.  ( g `
 t ) )  =  ( ( f `
 t )  x.  ( h `  t
) ) )
2120mpteq2dv 4484 . . . . . . 7  |-  ( g  =  h  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( f `  t )  x.  ( h `  t ) ) ) )
2221eleq1d 2473 . . . . . 6  |-  ( g  =  h  ->  (
( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) )  e.  A  <->  ( t  e.  T  |->  ( ( f `  t
)  x.  ( h `
 t ) ) )  e.  A ) )
2312, 22imbi12d 320 . . . . 5  |-  ( g  =  h  ->  (
( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) )  e.  A
)  <->  ( ( ph  /\  f  e.  A  /\  h  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
h `  t )
) )  e.  A
) ) )
24 stoweidlem62.11 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
2523, 24chvarv 2043 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  h  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( h `  t ) ) )  e.  A )
26 stoweidlem62.12 . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
27 stoweidlem62.13 . . . 4  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
28 stoweidlem62.1 . . . . . 6  |-  F/_ t F
2928nfrn 5068 . . . . . . 7  |-  F/_ t ran  F
30 nfcv 2566 . . . . . . 7  |-  F/_ t RR
31 nfcv 2566 . . . . . . 7  |-  F/_ t `'  <
3229, 30, 31nfsup 7946 . . . . . 6  |-  F/_ t sup ( ran  F ,  RR ,  `'  <  )
33 eqid 2404 . . . . . 6  |-  ( T  X.  { -u sup ( ran  F ,  RR ,  `'  <  ) } )  =  ( T  X.  { -u sup ( ran  F ,  RR ,  `'  <  ) } )
34 cmptop 20190 . . . . . . 7  |-  ( J  e.  Comp  ->  J  e. 
Top )
356, 34syl 17 . . . . . 6  |-  ( ph  ->  J  e.  Top )
36 stoweidlem62.14 . . . . . 6  |-  ( ph  ->  F  e.  C )
3736, 9syl6eleq 2502 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3828, 4, 7, 5, 6, 37, 8stoweidlem29 37192 . . . . . . 7  |-  ( ph  ->  ( sup ( ran 
F ,  RR ,  `'  <  )  e.  ran  F  /\  sup ( ran 
F ,  RR ,  `'  <  )  e.  RR  /\ 
A. t  e.  T  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  t ) ) )
3938simp2d 1012 . . . . . 6  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4028, 32, 4, 7, 33, 5, 35, 9, 36, 39stoweidlem47 37210 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  e.  C )
411, 40syl5eqel 2496 . . . 4  |-  ( ph  ->  H  e.  C )
4238simp3d 1013 . . . . . . . . 9  |-  ( ph  ->  A. t  e.  T  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  t ) )
4342r19.21bi 2775 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  sup ( ran  F ,  RR ,  `'  <  )  <_ 
( F `  t
) )
445, 7, 9, 36fcnre 36793 . . . . . . . . . 10  |-  ( ph  ->  F : T --> RR )
4544fnvinran 36782 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
4639adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4745, 46subge0d 10184 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
0  <_  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <->  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  t ) ) )
4843, 47mpbird 234 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  0  <_  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
49 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
5045, 46resubcld 10030 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
511fvmpt2 5943 . . . . . . . 8  |-  ( ( t  e.  T  /\  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )  -> 
( H `  t
)  =  ( ( F `  t )  -  sup ( ran 
F ,  RR ,  `'  <  ) ) )
5249, 50, 51syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  ( ( F `
 t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
5348, 52breqtrrd 4423 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  0  <_  ( H `  t
) )
5453ex 434 . . . . 5  |-  ( ph  ->  ( t  e.  T  ->  0  <_  ( H `  t ) ) )
554, 54ralrimi 2806 . . . 4  |-  ( ph  ->  A. t  e.  T 
0  <_  ( H `  t ) )
56 stoweidlem62.15 . . . . 5  |-  ( ph  ->  E  e.  RR+ )
5756rphalfcld 11318 . . . 4  |-  ( ph  ->  ( E  /  2
)  e.  RR+ )
5856rpred 11306 . . . . . 6  |-  ( ph  ->  E  e.  RR )
5958rehalfcld 10828 . . . . 5  |-  ( ph  ->  ( E  /  2
)  e.  RR )
60 3re 10652 . . . . . . 7  |-  3  e.  RR
61 3ne0 10673 . . . . . . 7  |-  3  =/=  0
6260, 61rereccli 10352 . . . . . 6  |-  ( 1  /  3 )  e.  RR
6362a1i 11 . . . . 5  |-  ( ph  ->  ( 1  /  3
)  e.  RR )
64 rphalflt 11294 . . . . . 6  |-  ( E  e.  RR+  ->  ( E  /  2 )  < 
E )
6556, 64syl 17 . . . . 5  |-  ( ph  ->  ( E  /  2
)  <  E )
66 stoweidlem62.17 . . . . 5  |-  ( ph  ->  E  <  ( 1  /  3 ) )
6759, 58, 63, 65, 66lttrd 9779 . . . 4  |-  ( ph  ->  ( E  /  2
)  <  ( 1  /  3 ) )
683, 4, 5, 6, 7, 8, 9, 10, 19, 25, 26, 27, 41, 55, 57, 67stoweidlem61 37224 . . 3  |-  ( ph  ->  E. h  e.  A  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  ( 2  x.  ( E  /  2
) ) )
69 nfra1 2787 . . . . . . 7  |-  F/ t A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  ( 2  x.  ( E  /  2
) )
704, 69nfan 1958 . . . . . 6  |-  F/ t ( ph  /\  A. t  e.  T  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  (
2  x.  ( E  /  2 ) ) )
71 rsp 2772 . . . . . . 7  |-  ( A. t  e.  T  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  (
2  x.  ( E  /  2 ) )  ->  ( t  e.  T  ->  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  (
2  x.  ( E  /  2 ) ) ) )
7256rpcnd 11308 . . . . . . . . . 10  |-  ( ph  ->  E  e.  CC )
73 2cnd 10651 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
74 2ne0 10671 . . . . . . . . . . 11  |-  2  =/=  0
7574a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  =/=  0 )
7672, 73, 75divcan2d 10365 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( E  /  2 ) )  =  E )
7776breq2d 4409 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  ( 2  x.  ( E  / 
2 ) )  <->  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )
7877biimpd 209 . . . . . . 7  |-  ( ph  ->  ( ( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  ( 2  x.  ( E  / 
2 ) )  -> 
( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  E ) )
7971, 78sylan9r 658 . . . . . 6  |-  ( (
ph  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  (
2  x.  ( E  /  2 ) ) )  ->  ( t  e.  T  ->  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )
8070, 79ralrimi 2806 . . . . 5  |-  ( (
ph  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  (
2  x.  ( E  /  2 ) ) )  ->  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
)
8180ex 434 . . . 4  |-  ( ph  ->  ( A. t  e.  T  ( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  ( 2  x.  ( E  / 
2 ) )  ->  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )
8281reximdv 2880 . . 3  |-  ( ph  ->  ( E. h  e.  A  A. t  e.  T  ( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  ( 2  x.  ( E  / 
2 ) )  ->  E. h  e.  A  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )
8368, 82mpd 15 . 2  |-  ( ph  ->  E. h  e.  A  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E )
84 nfmpt1 4486 . . 3  |-  F/_ t
( t  e.  T  |->  ( ( h `  t )  +  sup ( ran  F ,  RR ,  `'  <  ) ) )
85 nfcv 2566 . . 3  |-  F/_ t
h
86 nfv 1730 . . . . 5  |-  F/ t  h  e.  A
87 nfra1 2787 . . . . 5  |-  F/ t A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E
8886, 87nfan 1958 . . . 4  |-  F/ t ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E )
894, 88nfan 1958 . . 3  |-  F/ t ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )
90 eqid 2404 . . 3  |-  ( t  e.  T  |->  ( ( h `  t )  +  sup ( ran 
F ,  RR ,  `'  <  ) ) )  =  ( t  e.  T  |->  ( ( h `
 t )  +  sup ( ran  F ,  RR ,  `'  <  ) ) )
9144adantr 465 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  F : T --> RR )
9239adantr 465 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
93183adant1r 1225 . . 3  |-  ( ( ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
9426adantlr 715 . . 3  |-  ( ( ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
95 stoweidlem62.2 . . . . 5  |-  F/ f
ph
9610sseld 3443 . . . . . . . 8  |-  ( ph  ->  ( f  e.  A  ->  f  e.  C ) )
979eleq2i 2482 . . . . . . . 8  |-  ( f  e.  C  <->  f  e.  ( J  Cn  K
) )
9896, 97syl6ib 228 . . . . . . 7  |-  ( ph  ->  ( f  e.  A  ->  f  e.  ( J  Cn  K ) ) )
99 eqid 2404 . . . . . . . 8  |-  U. J  =  U. J
100 uniretop 21563 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
1015unieqi 4202 . . . . . . . . 9  |-  U. K  =  U. ( topGen `  ran  (,) )
102100, 101eqtr4i 2436 . . . . . . . 8  |-  RR  =  U. K
10399, 102cnf 20042 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  f : U. J --> RR )
10498, 103syl6 33 . . . . . 6  |-  ( ph  ->  ( f  e.  A  ->  f : U. J --> RR ) )
105 feq2 5699 . . . . . . 7  |-  ( T  =  U. J  -> 
( f : T --> RR 
<->  f : U. J --> RR ) )
1067, 105mp1i 13 . . . . . 6  |-  ( ph  ->  ( f : T --> RR 
<->  f : U. J --> RR ) )
107104, 106sylibrd 236 . . . . 5  |-  ( ph  ->  ( f  e.  A  ->  f : T --> RR ) )
10895, 107ralrimi 2806 . . . 4  |-  ( ph  ->  A. f  e.  A  f : T --> RR )
109108adantr 465 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  A. f  e.  A  f : T
--> RR )
110 simprl 758 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  h  e.  A )
11152eqcomd 2412 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  -  sup ( ran  F ,  RR ,  `'  <  ) )  =  ( H `  t
) )
112111oveq2d 6296 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( h `  t
)  -  ( ( F `  t )  -  sup ( ran 
F ,  RR ,  `'  <  ) ) )  =  ( ( h `
 t )  -  ( H `  t ) ) )
113112fveq2d 5855 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( h `
 t )  -  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) ) )  =  ( abs `  ( ( h `  t )  -  ( H `  t ) ) ) )
114113adantlr 715 . . . . . 6  |-  ( ( ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )  /\  t  e.  T
)  ->  ( abs `  ( ( h `  t )  -  (
( F `  t
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) ) )  =  ( abs `  ( ( h `  t )  -  ( H `  t )
) ) )
115 simplrr 765 . . . . . . 7  |-  ( ( ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )  /\  t  e.  T
)  ->  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
)
116 rspa 2773 . . . . . . 7  |-  ( ( A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E  /\  t  e.  T )  ->  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  E
)
117115, 116sylancom 667 . . . . . 6  |-  ( ( ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )  /\  t  e.  T
)  ->  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
)
118114, 117eqbrtrd 4417 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )  /\  t  e.  T
)  ->  ( abs `  ( ( h `  t )  -  (
( F `  t
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) ) )  <  E )
119118ex 434 . . . 4  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  (
t  e.  T  -> 
( abs `  (
( h `  t
)  -  ( ( F `  t )  -  sup ( ran 
F ,  RR ,  `'  <  ) ) ) )  <  E ) )
12089, 119ralrimi 2806 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  A. t  e.  T  ( abs `  ( ( h `  t )  -  (
( F `  t
)  -  sup ( ran  F ,  RR ,  `'  <  ) ) ) )  <  E )
12184, 85, 32, 89, 90, 91, 92, 93, 94, 109, 110, 120stoweidlem21 37184 . 2  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  E
)
12283, 121rexlimddv 2902 1  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407   F/wnf 1639    e. wcel 1844   F/_wnfc 2552    =/= wne 2600   A.wral 2756   E.wrex 2757    C_ wss 3416   (/)c0 3740   {csn 3974   U.cuni 4193   class class class wbr 4397    |-> cmpt 4455    X. cxp 4823   `'ccnv 4824   ran crn 4826   -->wf 5567   ` cfv 5571  (class class class)co 6280   supcsup 7936   RRcr 9523   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    < clt 9660    <_ cle 9661    - cmin 9843   -ucneg 9844    / cdiv 10249   2c2 10628   3c3 10629   RR+crp 11267   (,)cioo 11584   abscabs 13218   topGenctg 15054   Topctop 19688    Cn ccn 20020   Compccmp 20181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-fi 7907  df-sup 7937  df-oi 7971  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-q 11230  df-rp 11268  df-xneg 11373  df-xadd 11374  df-xmul 11375  df-ioo 11588  df-ioc 11589  df-ico 11590  df-icc 11591  df-fz 11729  df-fzo 11857  df-fl 11968  df-seq 12154  df-exp 12213  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-clim 13462  df-rlim 13463  df-sum 13660  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-sca 14927  df-vsca 14928  df-ip 14929  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-hom 14935  df-cco 14936  df-rest 15039  df-topn 15040  df-0g 15058  df-gsum 15059  df-topgen 15060  df-pt 15061  df-prds 15064  df-xrs 15118  df-qtop 15123  df-imas 15124  df-xps 15126  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-mulg 16386  df-cntz 16681  df-cmn 17126  df-psmet 18733  df-xmet 18734  df-met 18735  df-bl 18736  df-mopn 18737  df-cnfld 18743  df-top 19693  df-bases 19695  df-topon 19696  df-topsp 19697  df-cld 19814  df-cn 20023  df-cnp 20024  df-cmp 20182  df-tx 20357  df-hmeo 20550  df-xms 21117  df-ms 21118  df-tms 21119
This theorem is referenced by:  stoweid  37226
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