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Theorem stoweidlem9 29652
Description: Lemma for stoweid 29706: 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 4362 . . . . 5  |-  ( T  =  (/)  ->  ( t  e.  T  |->  1 )  =  ( t  e.  (/)  |->  1 ) )
3 mpt0 5528 . . . . 5  |-  ( t  e.  (/)  |->  1 )  =  (/)
42, 3syl6eq 2483 . . . 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 2510 . 2  |-  ( ph  -> 
(/)  e.  A )
8 rzal 3771 . . 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 5680 . . . . . . 7  |-  ( g  =  (/)  ->  ( g `
 t )  =  ( (/) `  t ) )
1110oveq1d 6097 . . . . . 6  |-  ( g  =  (/)  ->  ( ( g `  t )  -  ( F `  t ) )  =  ( ( (/) `  t
)  -  ( F `
 t ) ) )
1211fveq2d 5685 . . . . 5  |-  ( g  =  (/)  ->  ( abs `  ( ( g `  t )  -  ( F `  t )
) )  =  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) ) )
1312breq1d 4292 . . . 4  |-  ( g  =  (/)  ->  ( ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  E  <->  ( abs `  ( ( (/) `  t
)  -  ( F `
 t ) ) )  <  E ) )
1413ralbidv 2727 . . 3  |-  ( g  =  (/)  ->  ( A. t  e.  T  ( abs `  ( ( g `
 t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E ) )
1514rspcev 3064 . 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 656 1  |-  ( ph  ->  E. g  e.  A  A. t  e.  T  ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1757   A.wral 2707   E.wrex 2708   (/)c0 3627   class class class wbr 4282    e. cmpt 4340   ` cfv 5408  (class class class)co 6082   1c1 9273    < clt 9408    - cmin 9585   abscabs 12709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pr 4521
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2966  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-br 4283  df-opab 4341  df-mpt 4342  df-id 4625  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-iota 5371  df-fun 5410  df-fn 5411  df-fv 5416  df-ov 6085
This theorem is referenced by:  stoweid  29706
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