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

Theorem stoweidlem62 37917
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.) (Revised by AV, 13-Sep-2020.)
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 )  - inf ( 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 )  - inf ( ran  F ,  RR ,  <  ) ) )
2 nfmpt1 4491 . . . . 5  |-  F/_ t
( t  e.  T  |->  ( ( F `  t )  - inf ( ran  F ,  RR ,  <  ) ) )
31, 2nfcxfr 2589 . . . 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 2516 . . . . . . 7  |-  ( g  =  h  ->  (
g  e.  A  <->  h  e.  A ) )
12113anbi3d 1344 . . . . . 6  |-  ( g  =  h  ->  (
( ph  /\  f  e.  A  /\  g  e.  A )  <->  ( ph  /\  f  e.  A  /\  h  e.  A )
) )
13 fveq1 5862 . . . . . . . . 9  |-  ( g  =  h  ->  (
g `  t )  =  ( h `  t ) )
1413oveq2d 6304 . . . . . . . 8  |-  ( g  =  h  ->  (
( f `  t
)  +  ( g `
 t ) )  =  ( ( f `
 t )  +  ( h `  t
) ) )
1514mpteq2dv 4489 . . . . . . 7  |-  ( g  =  h  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( f `  t )  +  ( h `  t ) ) ) )
1615eleq1d 2512 . . . . . 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 stoweidlem62.10 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
1917, 18chvarv 2106 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  h  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( h `  t ) ) )  e.  A )
2013oveq2d 6304 . . . . . . . 8  |-  ( g  =  h  ->  (
( f `  t
)  x.  ( g `
 t ) )  =  ( ( f `
 t )  x.  ( h `  t
) ) )
2120mpteq2dv 4489 . . . . . . 7  |-  ( g  =  h  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  =  ( t  e.  T  |->  ( ( f `  t )  x.  ( h `  t ) ) ) )
2221eleq1d 2512 . . . . . 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 stoweidlem62.11 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
2523, 24chvarv 2106 . . . 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 5076 . . . . . . 7  |-  F/_ t ran  F
30 nfcv 2591 . . . . . . 7  |-  F/_ t RR
31 nfcv 2591 . . . . . . 7  |-  F/_ t  <
3229, 30, 31nfinf 7995 . . . . . 6  |-  F/_ tinf ( ran  F ,  RR ,  <  )
33 eqid 2450 . . . . . 6  |-  ( T  X.  { -uinf ( ran 
F ,  RR ,  <  ) } )  =  ( T  X.  { -uinf ( ran  F ,  RR ,  <  ) } )
34 cmptop 20403 . . . . . . 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 2538 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3828, 4, 7, 5, 6, 37, 8stoweidlem29 37883 . . . . . . 7  |-  ( ph  ->  (inf ( ran  F ,  RR ,  <  )  e.  ran  F  /\ inf ( ran  F ,  RR ,  <  )  e.  RR  /\  A. t  e.  T inf ( ran  F ,  RR ,  <  )  <_  ( F `  t ) ) )
3938simp2d 1020 . . . . . 6  |-  ( ph  -> inf ( ran  F ,  RR ,  <  )  e.  RR )
4028, 32, 4, 7, 33, 5, 35, 9, 36, 39stoweidlem47 37902 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t )  - inf ( ran  F ,  RR ,  <  ) ) )  e.  C )
411, 40syl5eqel 2532 . . . 4  |-  ( ph  ->  H  e.  C )
4238simp3d 1021 . . . . . . . . 9  |-  ( ph  ->  A. t  e.  T inf ( ran  F ,  RR ,  <  )  <_  ( F `  t )
)
4342r19.21bi 2756 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  -> inf ( ran 
F ,  RR ,  <  )  <_  ( F `  t ) )
445, 7, 9, 36fcnre 37340 . . . . . . . . . 10  |-  ( ph  ->  F : T --> RR )
4544fnvinran 37329 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
4639adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  -> inf ( ran 
F ,  RR ,  <  )  e.  RR )
4745, 46subge0d 10200 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
0  <_  ( ( F `  t )  - inf ( ran  F ,  RR ,  <  ) )  <-> inf ( ran  F ,  RR ,  <  )  <_  ( F `  t )
) )
4843, 47mpbird 236 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  0  <_  ( ( F `  t )  - inf ( ran  F ,  RR ,  <  ) ) )
49 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
5045, 46resubcld 10044 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  - inf ( ran  F ,  RR ,  <  ) )  e.  RR )
511fvmpt2 5955 . . . . . . . 8  |-  ( ( t  e.  T  /\  ( ( F `  t )  - inf ( ran  F ,  RR ,  <  ) )  e.  RR )  ->  ( H `  t )  =  ( ( F `  t
)  - inf ( ran  F ,  RR ,  <  ) ) )
5249, 50, 51syl2anc 666 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  ( ( F `
 t )  - inf ( ran  F ,  RR ,  <  ) ) )
5348, 52breqtrrd 4428 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  0  <_  ( H `  t
) )
5453ex 436 . . . . 5  |-  ( ph  ->  ( t  e.  T  ->  0  <_  ( H `  t ) ) )
554, 54ralrimi 2787 . . . 4  |-  ( ph  ->  A. t  e.  T 
0  <_  ( H `  t ) )
56 stoweidlem62.15 . . . . 5  |-  ( ph  ->  E  e.  RR+ )
5756rphalfcld 11350 . . . 4  |-  ( ph  ->  ( E  /  2
)  e.  RR+ )
5856rpred 11338 . . . . . 6  |-  ( ph  ->  E  e.  RR )
5958rehalfcld 10856 . . . . 5  |-  ( ph  ->  ( E  /  2
)  e.  RR )
60 3re 10680 . . . . . . 7  |-  3  e.  RR
61 3ne0 10701 . . . . . . 7  |-  3  =/=  0
6260, 61rereccli 10369 . . . . . 6  |-  ( 1  /  3 )  e.  RR
6362a1i 11 . . . . 5  |-  ( ph  ->  ( 1  /  3
)  e.  RR )
64 rphalflt 11326 . . . . . 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 9793 . . . 4  |-  ( ph  ->  ( E  /  2
)  <  ( 1  /  3 ) )
683, 4, 5, 6, 7, 8, 9, 10, 19, 25, 26, 27, 41, 55, 57, 67stoweidlem61 37916 . . 3  |-  ( ph  ->  E. h  e.  A  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  ( 2  x.  ( E  /  2
) ) )
69 nfra1 2768 . . . . . . 7  |-  F/ t A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  ( 2  x.  ( E  /  2
) )
704, 69nfan 2010 . . . . . 6  |-  F/ t ( ph  /\  A. t  e.  T  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  (
2  x.  ( E  /  2 ) ) )
71 rsp 2753 . . . . . . 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 11340 . . . . . . . . . 10  |-  ( ph  ->  E  e.  CC )
73 2cnd 10679 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
74 2ne0 10699 . . . . . . . . . . 11  |-  2  =/=  0
7574a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  =/=  0 )
7672, 73, 75divcan2d 10382 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( E  /  2 ) )  =  E )
7776breq2d 4413 . . . . . . . 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 663 . . . . . 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 2787 . . . . 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 2860 . . 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 4491 . . 3  |-  F/_ t
( t  e.  T  |->  ( ( h `  t )  + inf ( ran  F ,  RR ,  <  ) ) )
85 nfcv 2591 . . 3  |-  F/_ t
h
86 nfv 1760 . . . . 5  |-  F/ t  h  e.  A
87 nfra1 2768 . . . . 5  |-  F/ t A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E
8886, 87nfan 2010 . . . 4  |-  F/ t ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E )
894, 88nfan 2010 . . 3  |-  F/ t ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )
90 eqid 2450 . . 3  |-  ( t  e.  T  |->  ( ( h `  t )  + inf ( ran  F ,  RR ,  <  )
) )  =  ( t  e.  T  |->  ( ( h `  t
)  + inf ( 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
) )  -> inf ( ran 
F ,  RR ,  <  )  e.  RR )
93183adant1r 1260 . . 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 720 . . 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 3430 . . . . . . . 8  |-  ( ph  ->  ( f  e.  A  ->  f  e.  C ) )
979eleq2i 2520 . . . . . . . 8  |-  ( f  e.  C  <->  f  e.  ( J  Cn  K
) )
9896, 97syl6ib 230 . . . . . . 7  |-  ( ph  ->  ( f  e.  A  ->  f  e.  ( J  Cn  K ) ) )
99 eqid 2450 . . . . . . . 8  |-  U. J  =  U. J
100 uniretop 21776 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
1015unieqi 4206 . . . . . . . . 9  |-  U. K  =  U. ( topGen `  ran  (,) )
102100, 101eqtr4i 2475 . . . . . . . 8  |-  RR  =  U. K
10399, 102cnf 20255 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  f : U. J --> RR )
10498, 103syl6 34 . . . . . 6  |-  ( ph  ->  ( f  e.  A  ->  f : U. J --> RR ) )
105 feq2 5709 . . . . . . 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 2787 . . . 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 763 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  h  e.  A )
11152eqcomd 2456 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  - inf ( ran  F ,  RR ,  <  ) )  =  ( H `
 t ) )
112111oveq2d 6304 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( h `  t
)  -  ( ( F `  t )  - inf ( ran  F ,  RR ,  <  )
) )  =  ( ( h `  t
)  -  ( H `
 t ) ) )
113112fveq2d 5867 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( h `
 t )  -  ( ( F `  t )  - inf ( ran  F ,  RR ,  <  ) ) ) )  =  ( abs `  (
( h `  t
)  -  ( H `
 t ) ) ) )
114113adantlr 720 . . . . . 6  |-  ( ( ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )  /\  t  e.  T
)  ->  ( abs `  ( ( h `  t )  -  (
( F `  t
)  - inf ( ran  F ,  RR ,  <  ) ) ) )  =  ( abs `  (
( h `  t
)  -  ( H `
 t ) ) ) )
115 simplrr 770 . . . . . . 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 2754 . . . . . . 7  |-  ( ( A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E  /\  t  e.  T )  ->  ( abs `  ( ( h `
 t )  -  ( H `  t ) ) )  <  E
)
117115, 116sylancom 672 . . . . . 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 4422 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t ) ) )  <  E ) )  /\  t  e.  T
)  ->  ( abs `  ( ( h `  t )  -  (
( F `  t
)  - inf ( 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 )  - inf ( ran  F ,  RR ,  <  )
) ) )  < 
E ) )
12089, 119ralrimi 2787 . . 3  |-  ( (
ph  /\  ( h  e.  A  /\  A. t  e.  T  ( abs `  ( ( h `  t )  -  ( H `  t )
) )  <  E
) )  ->  A. t  e.  T  ( abs `  ( ( h `  t )  -  (
( F `  t
)  - inf ( ran  F ,  RR ,  <  ) ) ) )  < 
E )
12184, 85, 32, 89, 90, 91, 92, 93, 94, 109, 110, 120stoweidlem21 37875 . 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 2882 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 984    = wceq 1443   F/wnf 1666    e. wcel 1886   F/_wnfc 2578    =/= wne 2621   A.wral 2736   E.wrex 2737    C_ wss 3403   (/)c0 3730   {csn 3967   U.cuni 4197   class class class wbr 4401    |-> cmpt 4460    X. cxp 4831   ran crn 4834   -->wf 5577   ` cfv 5581  (class class class)co 6288  infcinf 7952   RRcr 9535   0cc0 9536   1c1 9537    + caddc 9539    x. cmul 9541    < clt 9672    <_ cle 9673    - cmin 9857   -ucneg 9858    / cdiv 10266   2c2 10656   3c3 10657   RR+crp 11299   (,)cioo 11632   abscabs 13290   topGenctg 15329   Topctop 19910    Cn ccn 20233   Compccmp 20394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-rlim 13546  df-sum 13746  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-cn 20236  df-cnp 20237  df-cmp 20395  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330
This theorem is referenced by:  stoweid  37919
  Copyright terms: Public domain W3C validator