Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem62 Structured version   Unicode version

Theorem stoweidlem62 31378
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 4536 . . . . 5  |-  F/_ t
( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
31, 2nfcxfr 2627 . . . 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 2539 . . . . . . 7  |-  ( g  =  h  ->  (
g  e.  A  <->  h  e.  A ) )
12113anbi3d 1305 . . . . . 6  |-  ( g  =  h  ->  (
( ph  /\  f  e.  A  /\  g  e.  A )  <->  ( ph  /\  f  e.  A  /\  h  e.  A )
) )
13 fveq1 5864 . . . . . . . . 9  |-  ( g  =  h  ->  (
g `  t )  =  ( h `  t ) )
1413oveq2d 6299 . . . . . . . 8  |-  ( g  =  h  ->  (
( f `  t
)  +  ( g `
 t ) )  =  ( ( f `
 t )  +  ( h `  t
) ) )
1514mpteq2dv 4534 . . . . . . 7  |-  ( g  =  h  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( f `  t )  +  ( h `  t ) ) ) )
1615eleq1d 2536 . . . . . 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 1983 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  h  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( h `  t ) ) )  e.  A )
2013oveq2d 6299 . . . . . . . 8  |-  ( g  =  h  ->  (
( f `  t
)  x.  ( g `
 t ) )  =  ( ( f `
 t )  x.  ( h `  t
) ) )
2120mpteq2dv 4534 . . . . . . 7  |-  ( g  =  h  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( f `  t )  x.  ( h `  t ) ) ) )
2221eleq1d 2536 . . . . . 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 1983 . . . 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 5244 . . . . . . 7  |-  F/_ t ran  F
30 nfcv 2629 . . . . . . 7  |-  F/_ t RR
31 nfcv 2629 . . . . . . 7  |-  F/_ t `'  <
3229, 30, 31nfsup 7910 . . . . . 6  |-  F/_ t sup ( ran  F ,  RR ,  `'  <  )
33 eqid 2467 . . . . . 6  |-  ( T  X.  { -u sup ( ran  F ,  RR ,  `'  <  ) } )  =  ( T  X.  { -u sup ( ran  F ,  RR ,  `'  <  ) } )
34 cmptop 19677 . . . . . . 7  |-  ( J  e.  Comp  ->  J  e. 
Top )
356, 34syl 16 . . . . . 6  |-  ( ph  ->  J  e.  Top )
36 stoweidlem62.14 . . . . . 6  |-  ( ph  ->  F  e.  C )
3736, 9syl6eleq 2565 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3828, 4, 7, 5, 6, 37, 8stoweidlem29 31345 . . . . . . 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 1009 . . . . . 6  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4028, 32, 4, 7, 33, 5, 35, 9, 36, 39stoweidlem47 31363 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  e.  C )
411, 40syl5eqel 2559 . . . 4  |-  ( ph  ->  H  e.  C )
4238simp3d 1010 . . . . . . . . 9  |-  ( ph  ->  A. t  e.  T  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  t ) )
4342r19.21bi 2833 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  sup ( ran  F ,  RR ,  `'  <  )  <_ 
( F `  t
) )
445, 7, 9, 36fcnre 30994 . . . . . . . . . 10  |-  ( ph  ->  F : T --> RR )
4544fnvinran 30983 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
4639adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4745, 46subge0d 10141 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
0  <_  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <->  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  t ) ) )
4843, 47mpbird 232 . . . . . . 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 9986 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
511fvmpt2 5956 . . . . . . . 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 4473 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  0  <_  ( H `  t
) )
5453ex 434 . . . . 5  |-  ( ph  ->  ( t  e.  T  ->  0  <_  ( H `  t ) ) )
554, 54ralrimi 2864 . . . 4  |-  ( ph  ->  A. t  e.  T 
0  <_  ( H `  t ) )
56 stoweidlem62.15 . . . . 5  |-  ( ph  ->  E  e.  RR+ )
5756rphalfcld 11267 . . . 4  |-  ( ph  ->  ( E  /  2
)  e.  RR+ )
5856rpred 11255 . . . . . 6  |-  ( ph  ->  E  e.  RR )
5958rehalfcld 10784 . . . . 5  |-  ( ph  ->  ( E  /  2
)  e.  RR )
60 3re 10608 . . . . . . 7  |-  3  e.  RR
61 3ne0 10629 . . . . . . 7  |-  3  =/=  0
6260, 61rereccli 10308 . . . . . 6  |-  ( 1  /  3 )  e.  RR
6362a1i 11 . . . . 5  |-  ( ph  ->  ( 1  /  3
)  e.  RR )
64 rphalflt 11245 . . . . . 6  |-  ( E  e.  RR+  ->  ( E  /  2 )  < 
E )
6556, 64syl 16 . . . . 5  |-  ( ph  ->  ( E  /  2
)  <  E )
66 stoweidlem62.17 . . . . 5  |-  ( ph  ->  E  <  ( 1  /  3 ) )
6759, 58, 63, 65, 66lttrd 9741 . . . 4  |-  ( ph  ->  ( E  /  2
)  <  ( 1  /  3 ) )
683, 4, 5, 6, 7, 8, 9, 10, 19, 25, 26, 27, 41, 55, 57, 67stoweidlem61 31377 . . 3  |-  ( ph  ->  E. h  e.  A  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  ( 2  x.  ( E  /  2
) ) )
69 nfra1 2845 . . . . . . 7  |-  F/ t A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  ( 2  x.  ( E  /  2
) )
704, 69nfan 1875 . . . . . 6  |-  F/ t ( ph  /\  A. t  e.  T  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  (
2  x.  ( E  /  2 ) ) )
71 rsp 2830 . . . . . . 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 11257 . . . . . . . . . 10  |-  ( ph  ->  E  e.  CC )
73 2cnd 10607 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
74 2ne0 10627 . . . . . . . . . . 11  |-  2  =/=  0
7574a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  =/=  0 )
7672, 73, 75divcan2d 10321 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( E  /  2 ) )  =  E )
7776breq2d 4459 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  ( 2  x.  ( E  / 
2 ) )  <->  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )
7877biimpd 207 . . . . . . 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 2864 . . . . 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 2937 . . 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 4536 . . 3  |-  F/_ t
( t  e.  T  |->  ( ( h `  t )  +  sup ( ran  F ,  RR ,  `'  <  ) ) )
85 nfcv 2629 . . 3  |-  F/_ t
h
86 nfv 1683 . . . . 5  |-  F/ t  h  e.  A
87 nfra1 2845 . . . . 5  |-  F/ t A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E
8886, 87nfan 1875 . . . 4  |-  F/ t ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E )
894, 88nfan 1875 . . 3  |-  F/ t ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )
90 eqid 2467 . . 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 1221 . . 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 714 . . 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 3503 . . . . . . . 8  |-  ( ph  ->  ( f  e.  A  ->  f  e.  C ) )
979eleq2i 2545 . . . . . . . 8  |-  ( f  e.  C  <->  f  e.  ( J  Cn  K
) )
9896, 97syl6ib 226 . . . . . . 7  |-  ( ph  ->  ( f  e.  A  ->  f  e.  ( J  Cn  K ) ) )
99 eqid 2467 . . . . . . . 8  |-  U. J  =  U. J
100 uniretop 21020 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
1015unieqi 4254 . . . . . . . . 9  |-  U. K  =  U. ( topGen `  ran  (,) )
102100, 101eqtr4i 2499 . . . . . . . 8  |-  RR  =  U. K
10399, 102cnf 19529 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  f : U. J --> RR )
10498, 103syl6 33 . . . . . 6  |-  ( ph  ->  ( f  e.  A  ->  f : U. J --> RR ) )
105 feq2 5713 . . . . . . 7  |-  ( T  =  U. J  -> 
( f : T --> RR 
<->  f : U. J --> RR ) )
1067, 105mp1i 12 . . . . . 6  |-  ( ph  ->  ( f : T --> RR 
<->  f : U. J --> RR ) )
107104, 106sylibrd 234 . . . . 5  |-  ( ph  ->  ( f  e.  A  ->  f : T --> RR ) )
10895, 107ralrimi 2864 . . . 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 755 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  h  e.  A )
11152eqcomd 2475 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  -  sup ( ran  F ,  RR ,  `'  <  ) )  =  ( H `  t
) )
112111oveq2d 6299 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( h `  t
)  -  ( ( F `  t )  -  sup ( ran 
F ,  RR ,  `'  <  ) ) )  =  ( ( h `
 t )  -  ( H `  t ) ) )
113112fveq2d 5869 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( h `
 t )  -  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) ) )  =  ( abs `  ( ( h `  t )  -  ( H `  t ) ) ) )
114113adantlr 714 . . . . . 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 760 . . . . . . 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 rsp 2830 . . . . . . . 8  |-  ( A. t  e.  T  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  E  ->  ( t  e.  T  ->  ( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  E ) )
117116imp 429 . . . . . . 7  |-  ( ( A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E  /\  t  e.  T )  ->  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  E
)
118115, 117sylancom 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
)
119114, 118eqbrtrd 4467 . . . . 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 )
120119ex 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 ) )
12189, 120ralrimi 2864 . . 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 )
12284, 85, 32, 89, 90, 91, 92, 93, 94, 109, 110, 121stoweidlem21 31337 . 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
)
12383, 122rexlimddv 2959 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 184    /\ wa 369    /\ w3a 973    = wceq 1379   F/wnf 1599    e. wcel 1767   F/_wnfc 2615    =/= wne 2662   A.wral 2814   E.wrex 2815    C_ wss 3476   (/)c0 3785   {csn 4027   U.cuni 4245   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   ran crn 5000   -->wf 5583   ` cfv 5587  (class class class)co 6283   supcsup 7899   RRcr 9490   0cc0 9491   1c1 9492    + caddc 9494    x. cmul 9496    < clt 9627    <_ cle 9628    - cmin 9804   -ucneg 9805    / cdiv 10205   2c2 10584   3c3 10585   RR+crp 11219   (,)cioo 11528   abscabs 13029   topGenctg 14692   Topctop 19177    Cn ccn 19507   Compccmp 19668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ioc 11533  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-rlim 13274  df-sum 13471  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-cn 19510  df-cnp 19511  df-cmp 19669  df-tx 19814  df-hmeo 20007  df-xms 20574  df-ms 20575  df-tms 20576
This theorem is referenced by:  stoweid  31379
  Copyright terms: Public domain W3C validator