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Theorem stoweidlem62OLD 37934
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.) Obsolete version of stoweidlem62 37933 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
stoweidlem62OLD.1  |-  F/_ t F
stoweidlem62OLD.2  |-  F/ f
ph
stoweidlem62OLD.3  |-  F/ t
ph
stoweidlem62OLD.4  |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
stoweidlem62OLD.5  |-  K  =  ( topGen `  ran  (,) )
stoweidlem62OLD.6  |-  T  = 
U. J
stoweidlem62OLD.7  |-  ( ph  ->  J  e.  Comp )
stoweidlem62OLD.8  |-  C  =  ( J  Cn  K
)
stoweidlem62OLD.9  |-  ( ph  ->  A  C_  C )
stoweidlem62OLD.10  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem62OLD.11  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem62OLD.12  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
stoweidlem62OLD.13  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
stoweidlem62OLD.14  |-  ( ph  ->  F  e.  C )
stoweidlem62OLD.15  |-  ( ph  ->  E  e.  RR+ )
stoweidlem62OLD.16  |-  ( ph  ->  T  =/=  (/) )
stoweidlem62OLD.17  |-  ( ph  ->  E  <  ( 1  /  3 ) )
Assertion
Ref Expression
stoweidlem62OLD  |-  ( 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 stoweidlem62OLD
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 stoweidlem62OLD.4 . . . . 5  |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
2 nfmpt1 4495 . . . . 5  |-  F/_ t
( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
31, 2nfcxfr 2592 . . . 4  |-  F/_ t H
4 stoweidlem62OLD.3 . . . 4  |-  F/ t
ph
5 stoweidlem62OLD.5 . . . 4  |-  K  =  ( topGen `  ran  (,) )
6 stoweidlem62OLD.7 . . . 4  |-  ( ph  ->  J  e.  Comp )
7 stoweidlem62OLD.6 . . . 4  |-  T  = 
U. J
8 stoweidlem62OLD.16 . . . 4  |-  ( ph  ->  T  =/=  (/) )
9 stoweidlem62OLD.8 . . . 4  |-  C  =  ( J  Cn  K
)
10 stoweidlem62OLD.9 . . . 4  |-  ( ph  ->  A  C_  C )
11 eleq1 2519 . . . . . . 7  |-  ( g  =  h  ->  (
g  e.  A  <->  h  e.  A ) )
12113anbi3d 1347 . . . . . 6  |-  ( g  =  h  ->  (
( ph  /\  f  e.  A  /\  g  e.  A )  <->  ( ph  /\  f  e.  A  /\  h  e.  A )
) )
13 fveq1 5869 . . . . . . . . 9  |-  ( g  =  h  ->  (
g `  t )  =  ( h `  t ) )
1413oveq2d 6311 . . . . . . . 8  |-  ( g  =  h  ->  (
( f `  t
)  +  ( g `
 t ) )  =  ( ( f `
 t )  +  ( h `  t
) ) )
1514mpteq2dv 4493 . . . . . . 7  |-  ( g  =  h  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( f `  t )  +  ( h `  t ) ) ) )
1615eleq1d 2515 . . . . . 6  |-  ( g  =  h  ->  (
( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A  <->  ( t  e.  T  |->  ( ( f `  t
)  +  ( h `
 t ) ) )  e.  A ) )
1712, 16imbi12d 322 . . . . 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 stoweidlem62OLD.10 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
1917, 18chvarv 2109 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  h  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( h `  t ) ) )  e.  A )
2013oveq2d 6311 . . . . . . . 8  |-  ( g  =  h  ->  (
( f `  t
)  x.  ( g `
 t ) )  =  ( ( f `
 t )  x.  ( h `  t
) ) )
2120mpteq2dv 4493 . . . . . . 7  |-  ( g  =  h  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( f `  t )  x.  ( h `  t ) ) ) )
2221eleq1d 2515 . . . . . 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 322 . . . . 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 stoweidlem62OLD.11 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
2523, 24chvarv 2109 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  h  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( h `  t ) ) )  e.  A )
26 stoweidlem62OLD.12 . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
27 stoweidlem62OLD.13 . . . 4  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
28 stoweidlem62OLD.1 . . . . . 6  |-  F/_ t F
2928nfrn 5080 . . . . . . 7  |-  F/_ t ran  F
30 nfcv 2594 . . . . . . 7  |-  F/_ t RR
31 nfcv 2594 . . . . . . 7  |-  F/_ t `'  <
3229, 30, 31nfsup 7970 . . . . . 6  |-  F/_ t sup ( ran  F ,  RR ,  `'  <  )
33 eqid 2453 . . . . . 6  |-  ( T  X.  { -u sup ( ran  F ,  RR ,  `'  <  ) } )  =  ( T  X.  { -u sup ( ran  F ,  RR ,  `'  <  ) } )
34 cmptop 20422 . . . . . . 7  |-  ( J  e.  Comp  ->  J  e. 
Top )
356, 34syl 17 . . . . . 6  |-  ( ph  ->  J  e.  Top )
36 stoweidlem62OLD.14 . . . . . 6  |-  ( ph  ->  F  e.  C )
3736, 9syl6eleq 2541 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3828, 4, 7, 5, 6, 37, 8stoweidlem29OLD 37900 . . . . . . 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 1022 . . . . . 6  |-  ( ph  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4028, 32, 4, 7, 33, 5, 35, 9, 36, 39stoweidlem47 37918 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )  e.  C )
411, 40syl5eqel 2535 . . . 4  |-  ( ph  ->  H  e.  C )
4238simp3d 1023 . . . . . . . . 9  |-  ( ph  ->  A. t  e.  T  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  t ) )
4342r19.21bi 2759 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  sup ( ran  F ,  RR ,  `'  <  )  <_ 
( F `  t
) )
445, 7, 9, 36fcnre 37356 . . . . . . . . . 10  |-  ( ph  ->  F : T --> RR )
4544fnvinran 37345 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
4639adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
4745, 46subge0d 10210 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
0  <_  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) )  <->  sup ( ran  F ,  RR ,  `'  <  )  <_  ( F `  t ) ) )
4843, 47mpbird 236 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  0  <_  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
49 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
5045, 46resubcld 10054 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )
511fvmpt2 5962 . . . . . . . 8  |-  ( ( t  e.  T  /\  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) )  e.  RR )  -> 
( H `  t
)  =  ( ( F `  t )  -  sup ( ran 
F ,  RR ,  `'  <  ) ) )
5249, 50, 51syl2anc 667 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  ( ( F `
 t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
5348, 52breqtrrd 4432 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  0  <_  ( H `  t
) )
5453ex 436 . . . . 5  |-  ( ph  ->  ( t  e.  T  ->  0  <_  ( H `  t ) ) )
554, 54ralrimi 2790 . . . 4  |-  ( ph  ->  A. t  e.  T 
0  <_  ( H `  t ) )
56 stoweidlem62OLD.15 . . . . 5  |-  ( ph  ->  E  e.  RR+ )
5756rphalfcld 11360 . . . 4  |-  ( ph  ->  ( E  /  2
)  e.  RR+ )
5856rpred 11348 . . . . . 6  |-  ( ph  ->  E  e.  RR )
5958rehalfcld 10866 . . . . 5  |-  ( ph  ->  ( E  /  2
)  e.  RR )
60 3re 10690 . . . . . . 7  |-  3  e.  RR
61 3ne0 10711 . . . . . . 7  |-  3  =/=  0
6260, 61rereccli 10379 . . . . . 6  |-  ( 1  /  3 )  e.  RR
6362a1i 11 . . . . 5  |-  ( ph  ->  ( 1  /  3
)  e.  RR )
64 rphalflt 11336 . . . . . 6  |-  ( E  e.  RR+  ->  ( E  /  2 )  < 
E )
6556, 64syl 17 . . . . 5  |-  ( ph  ->  ( E  /  2
)  <  E )
66 stoweidlem62OLD.17 . . . . 5  |-  ( ph  ->  E  <  ( 1  /  3 ) )
6759, 58, 63, 65, 66lttrd 9801 . . . 4  |-  ( ph  ->  ( E  /  2
)  <  ( 1  /  3 ) )
683, 4, 5, 6, 7, 8, 9, 10, 19, 25, 26, 27, 41, 55, 57, 67stoweidlem61 37932 . . 3  |-  ( ph  ->  E. h  e.  A  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  ( 2  x.  ( E  /  2
) ) )
69 nfra1 2771 . . . . . . 7  |-  F/ t A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  ( 2  x.  ( E  /  2
) )
704, 69nfan 2013 . . . . . 6  |-  F/ t ( ph  /\  A. t  e.  T  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  (
2  x.  ( E  /  2 ) ) )
71 rsp 2756 . . . . . . 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 11350 . . . . . . . . . 10  |-  ( ph  ->  E  e.  CC )
73 2cnd 10689 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
74 2ne0 10709 . . . . . . . . . . 11  |-  2  =/=  0
7574a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  =/=  0 )
7672, 73, 75divcan2d 10392 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( E  /  2 ) )  =  E )
7776breq2d 4417 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  ( 2  x.  ( E  / 
2 ) )  <->  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )
7877biimpd 211 . . . . . . 7  |-  ( ph  ->  ( ( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  ( 2  x.  ( E  / 
2 ) )  -> 
( abs `  (
( h `  t
)  -  ( H `
 t ) ) )  <  E ) )
7971, 78sylan9r 664 . . . . . 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 2790 . . . . 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 436 . . . 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 2863 . . 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 4495 . . 3  |-  F/_ t
( t  e.  T  |->  ( ( h `  t )  +  sup ( ran  F ,  RR ,  `'  <  ) ) )
85 nfcv 2594 . . 3  |-  F/_ t
h
86 nfv 1763 . . . . 5  |-  F/ t  h  e.  A
87 nfra1 2771 . . . . 5  |-  F/ t A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E
8886, 87nfan 2013 . . . 4  |-  F/ t ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E )
894, 88nfan 2013 . . 3  |-  F/ t ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )
90 eqid 2453 . . 3  |-  ( t  e.  T  |->  ( ( h `  t )  +  sup ( ran 
F ,  RR ,  `'  <  ) ) )  =  ( t  e.  T  |->  ( ( h `
 t )  +  sup ( ran  F ,  RR ,  `'  <  ) ) )
9144adantr 467 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  F : T --> RR )
9239adantr 467 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  sup ( ran  F ,  RR ,  `'  <  )  e.  RR )
93183adant1r 1262 . . 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 722 . . 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 stoweidlem62OLD.2 . . . . 5  |-  F/ f
ph
9610sseld 3433 . . . . . . . 8  |-  ( ph  ->  ( f  e.  A  ->  f  e.  C ) )
979eleq2i 2523 . . . . . . . 8  |-  ( f  e.  C  <->  f  e.  ( J  Cn  K
) )
9896, 97syl6ib 230 . . . . . . 7  |-  ( ph  ->  ( f  e.  A  ->  f  e.  ( J  Cn  K ) ) )
99 eqid 2453 . . . . . . . 8  |-  U. J  =  U. J
100 uniretop 21795 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
1015unieqi 4210 . . . . . . . . 9  |-  U. K  =  U. ( topGen `  ran  (,) )
102100, 101eqtr4i 2478 . . . . . . . 8  |-  RR  =  U. K
10399, 102cnf 20274 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  f : U. J --> RR )
10498, 103syl6 34 . . . . . 6  |-  ( ph  ->  ( f  e.  A  ->  f : U. J --> RR ) )
105 feq2 5716 . . . . . . 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 238 . . . . 5  |-  ( ph  ->  ( f  e.  A  ->  f : T --> RR ) )
10895, 107ralrimi 2790 . . . 4  |-  ( ph  ->  A. f  e.  A  f : T --> RR )
109108adantr 467 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  A. f  e.  A  f : T
--> RR )
110 simprl 765 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  h  e.  A )
11152eqcomd 2459 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  -  sup ( ran  F ,  RR ,  `'  <  ) )  =  ( H `  t
) )
112111oveq2d 6311 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( h `  t
)  -  ( ( F `  t )  -  sup ( ran 
F ,  RR ,  `'  <  ) ) )  =  ( ( h `
 t )  -  ( H `  t ) ) )
113112fveq2d 5874 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( h `
 t )  -  ( ( F `  t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) ) )  =  ( abs `  ( ( h `  t )  -  ( H `  t ) ) ) )
114113adantlr 722 . . . . . 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 772 . . . . . . 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 2757 . . . . . . 7  |-  ( ( A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E  /\  t  e.  T )  ->  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  E
)
117115, 116sylancom 674 . . . . . 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 4426 . . . . 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 436 . . . 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 2790 . . 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 37891 . 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 2885 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 188    /\ wa 371    /\ w3a 986    = wceq 1446   F/wnf 1669    e. wcel 1889   F/_wnfc 2581    =/= wne 2624   A.wral 2739   E.wrex 2740    C_ wss 3406   (/)c0 3733   {csn 3970   U.cuni 4201   class class class wbr 4405    |-> cmpt 4464    X. cxp 4835   `'ccnv 4836   ran crn 4838   -->wf 5581   ` cfv 5585  (class class class)co 6295   supcsup 7959   RRcr 9543   0cc0 9544   1c1 9545    + caddc 9547    x. cmul 9549    < clt 9680    <_ cle 9681    - cmin 9865   -ucneg 9866    / cdiv 10276   2c2 10666   3c3 10667   RR+crp 11309   (,)cioo 11642   abscabs 13309   topGenctg 15348   Topctop 19929    Cn ccn 20252   Compccmp 20413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-addf 9623  ax-mulf 9624
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ioc 11647  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-rlim 13565  df-sum 13765  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-rest 15333  df-topn 15334  df-0g 15352  df-gsum 15353  df-topgen 15354  df-pt 15355  df-prds 15358  df-xrs 15412  df-qtop 15418  df-imas 15419  df-xps 15422  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-mulg 16688  df-cntz 16983  df-cmn 17444  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-cn 20255  df-cnp 20256  df-cmp 20414  df-tx 20589  df-hmeo 20782  df-xms 21347  df-ms 21348  df-tms 21349
This theorem is referenced by: (None)
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