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Theorem stoweidlem9 27725
Description: Lemma for stoweid 27779: 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 4281 . . . . 5  |-  ( T  =  (/)  ->  ( t  e.  T  |->  1 )  =  ( t  e.  (/)  |->  1 ) )
3 mpt0 5564 . . . . 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 3721 . . 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 5719 . . . . . . 7  |-  ( g  =  (/)  ->  ( g `
 t )  =  ( (/) `  t ) )
1110oveq1d 6088 . . . . . 6  |-  ( g  =  (/)  ->  ( ( g `  t )  -  ( F `  t ) )  =  ( ( (/) `  t
)  -  ( F `
 t ) ) )
1211fveq2d 5724 . . . . 5  |-  ( g  =  (/)  ->  ( abs `  ( ( g `  t )  -  ( F `  t )
) )  =  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) ) )
1312breq1d 4214 . . . 4  |-  ( g  =  (/)  ->  ( ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  E  <->  ( abs `  ( ( (/) `  t
)  -  ( F `
 t ) ) )  <  E ) )
1413ralbidv 2717 . . 3  |-  ( g  =  (/)  ->  ( A. t  e.  T  ( abs `  ( ( g `
 t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E ) )
1514rspcev 3044 . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   (/)c0 3620   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   1c1 8983    < clt 9112    - cmin 9283   abscabs 12031
This theorem is referenced by:  stoweid  27779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-ov 6076
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