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Theorem stoweidlem9 37869
Description: Lemma for stoweid 37925: 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 4483 . . . . 5  |-  ( T  =  (/)  ->  ( t  e.  T  |->  1 )  =  ( t  e.  (/)  |->  1 ) )
3 mpt0 5705 . . . . 5  |-  ( t  e.  (/)  |->  1 )  =  (/)
42, 3syl6eq 2501 . . . 4  |-  ( T  =  (/)  ->  ( t  e.  T  |->  1 )  =  (/) )
51, 4syl 17 . . 3  |-  ( ph  ->  ( t  e.  T  |->  1 )  =  (/) )
6 stoweidlem9.2 . . 3  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
75, 6eqeltrrd 2530 . 2  |-  ( ph  -> 
(/)  e.  A )
8 rzal 3871 . . 3  |-  ( T  =  (/)  ->  A. t  e.  T  ( abs `  ( ( (/) `  t
)  -  ( F `
 t ) ) )  <  E )
91, 8syl 17 . 2  |-  ( ph  ->  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E )
10 fveq1 5864 . . . . . . 7  |-  ( g  =  (/)  ->  ( g `
 t )  =  ( (/) `  t ) )
1110oveq1d 6305 . . . . . 6  |-  ( g  =  (/)  ->  ( ( g `  t )  -  ( F `  t ) )  =  ( ( (/) `  t
)  -  ( F `
 t ) ) )
1211fveq2d 5869 . . . . 5  |-  ( g  =  (/)  ->  ( abs `  ( ( g `  t )  -  ( F `  t )
) )  =  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) ) )
1312breq1d 4412 . . . 4  |-  ( g  =  (/)  ->  ( ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  E  <->  ( abs `  ( ( (/) `  t
)  -  ( F `
 t ) ) )  <  E ) )
1413ralbidv 2827 . . 3  |-  ( g  =  (/)  ->  ( A. t  e.  T  ( abs `  ( ( g `
 t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( (
(/) `  t )  -  ( F `  t ) ) )  <  E ) )
1514rspcev 3150 . 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 667 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 1444    e. wcel 1887   A.wral 2737   E.wrex 2738   (/)c0 3731   class class class wbr 4402    |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   1c1 9540    < clt 9675    - cmin 9860   abscabs 13297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590  df-ov 6293
This theorem is referenced by:  stoweid  37925
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