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Theorem stoweidlem9 27861
Description: Lemma for stoweid 27915: 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 . . . . 5  |-  ( ph  ->  T  =  (/) )
2 mpteq1 4116 . . . . . 6  |-  ( T  =  (/)  ->  ( t  e.  T  |->  1 )  =  ( t  e.  (/)  |->  1 ) )
3 mpt0 5387 . . . . . 6  |-  ( t  e.  (/)  |->  1 )  =  (/)
42, 3syl6eq 2344 . . . . 5  |-  ( T  =  (/)  ->  ( t  e.  T  |->  1 )  =  (/) )
51, 4syl 15 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  1 )  =  (/) )
6 stoweidlem9.2 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
75, 6eqeltrrd 2371 . . 3  |-  ( ph  -> 
(/)  e.  A )
8 rzal 3568 . . . 4  |-  ( T  =  (/)  ->  A. t  e.  T  ( abs `  ( ( (/) `  t
)  -  ( F `
 t ) ) )  <  E )
91, 8syl 15 . . 3  |-  ( ph  ->  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E )
107, 9jca 518 . 2  |-  ( ph  ->  ( (/)  e.  A  /\  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E ) )
11 fveq1 5540 . . . . . . 7  |-  ( g  =  (/)  ->  ( g `
 t )  =  ( (/) `  t ) )
1211oveq1d 5889 . . . . . 6  |-  ( g  =  (/)  ->  ( ( g `  t )  -  ( F `  t ) )  =  ( ( (/) `  t
)  -  ( F `
 t ) ) )
1312fveq2d 5545 . . . . 5  |-  ( g  =  (/)  ->  ( abs `  ( ( g `  t )  -  ( F `  t )
) )  =  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) ) )
1413breq1d 4049 . . . 4  |-  ( g  =  (/)  ->  ( ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  E  <->  ( abs `  ( ( (/) `  t
)  -  ( F `
 t ) ) )  <  E ) )
1514ralbidv 2576 . . 3  |-  ( g  =  (/)  ->  ( A. t  e.  T  ( abs `  ( ( g `
 t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E ) )
1615rspcev 2897 . 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 )
1710, 16syl 15 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    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   (/)c0 3468   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   1c1 8754    < clt 8883    - cmin 9053   abscabs 11735
This theorem is referenced by:  stoweid  27915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877
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