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Theorem stoweidlem9 27426
Description: Lemma for stoweid 27480: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem9.1  |-  ( ph  ->  T  =  (/) )
stoweidlem9.2  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
Assertion
Ref Expression
stoweidlem9  |-  ( ph  ->  E. g  e.  A  A. t  e.  T  ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  E )
Distinct variable groups:    t, g    A, g    g, E    g, F    T, g, t
Allowed substitution hints:    ph( t, g)    A( t)    E( t)    F( t)

Proof of Theorem stoweidlem9
StepHypRef Expression
1 stoweidlem9.1 . . . 4  |-  ( ph  ->  T  =  (/) )
2 mpteq1 4230 . . . . 5  |-  ( T  =  (/)  ->  ( t  e.  T  |->  1 )  =  ( t  e.  (/)  |->  1 ) )
3 mpt0 5512 . . . . 5  |-  ( t  e.  (/)  |->  1 )  =  (/)
42, 3syl6eq 2435 . . . 4  |-  ( T  =  (/)  ->  ( t  e.  T  |->  1 )  =  (/) )
51, 4syl 16 . . 3  |-  ( ph  ->  ( t  e.  T  |->  1 )  =  (/) )
6 stoweidlem9.2 . . 3  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
75, 6eqeltrrd 2462 . 2  |-  ( ph  -> 
(/)  e.  A )
8 rzal 3672 . . 3  |-  ( T  =  (/)  ->  A. t  e.  T  ( abs `  ( ( (/) `  t
)  -  ( F `
 t ) ) )  <  E )
91, 8syl 16 . 2  |-  ( ph  ->  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E )
10 fveq1 5667 . . . . . . 7  |-  ( g  =  (/)  ->  ( g `
 t )  =  ( (/) `  t ) )
1110oveq1d 6035 . . . . . 6  |-  ( g  =  (/)  ->  ( ( g `  t )  -  ( F `  t ) )  =  ( ( (/) `  t
)  -  ( F `
 t ) ) )
1211fveq2d 5672 . . . . 5  |-  ( g  =  (/)  ->  ( abs `  ( ( g `  t )  -  ( F `  t )
) )  =  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) ) )
1312breq1d 4163 . . . 4  |-  ( g  =  (/)  ->  ( ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  E  <->  ( abs `  ( ( (/) `  t
)  -  ( F `
 t ) ) )  <  E ) )
1413ralbidv 2669 . . 3  |-  ( g  =  (/)  ->  ( A. t  e.  T  ( abs `  ( ( g `
 t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E ) )
1514rspcev 2995 . 2  |-  ( (
(/)  e.  A  /\  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E )  ->  E. g  e.  A  A. t  e.  T  ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  E )
167, 9, 15syl2anc 643 1  |-  ( ph  ->  E. g  e.  A  A. t  e.  T  ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   (/)c0 3571   class class class wbr 4153    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   1c1 8924    < clt 9053    - cmin 9223   abscabs 11966
This theorem is referenced by:  stoweid  27480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402  df-ov 6023
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