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Theorem stoweidlem21 37881
Description: Once the Stone Weierstrass theorem has been proven for approximating nonnegative functions, then this lemma is used to extend the result to functions with (possibly) negative values. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem21.1  |-  F/_ t G
stoweidlem21.2  |-  F/_ t H
stoweidlem21.3  |-  F/_ t S
stoweidlem21.4  |-  F/ t
ph
stoweidlem21.5  |-  G  =  ( t  e.  T  |->  ( ( H `  t )  +  S
) )
stoweidlem21.6  |-  ( ph  ->  F : T --> RR )
stoweidlem21.7  |-  ( ph  ->  S  e.  RR )
stoweidlem21.8  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem21.9  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
stoweidlem21.10  |-  ( ph  ->  A. f  e.  A  f : T --> RR )
stoweidlem21.11  |-  ( ph  ->  H  e.  A )
stoweidlem21.12  |-  ( ph  ->  A. t  e.  T  ( abs `  ( ( H `  t )  -  ( ( F `
 t )  -  S ) ) )  <  E )
Assertion
Ref Expression
stoweidlem21  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
Distinct variable groups:    f, g,
t, T    A, f,
g    f, E, g    f, F, g    f, G, g   
f, H, g    ph, f,
g    S, g    x, t, T    x, A    x, S    ph, x
Allowed substitution hints:    ph( t)    A( t)    S( t, f)    E( x, t)    F( x, t)    G( x, t)    H( x, t)

Proof of Theorem stoweidlem21
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 stoweidlem21.5 . . . 4  |-  G  =  ( t  e.  T  |->  ( ( H `  t )  +  S
) )
2 stoweidlem21.4 . . . . 5  |-  F/ t
ph
3 stoweidlem21.7 . . . . . . . 8  |-  ( ph  ->  S  e.  RR )
4 fvconst2g 6118 . . . . . . . 8  |-  ( ( S  e.  RR  /\  t  e.  T )  ->  ( ( T  X.  { S } ) `  t )  =  S )
53, 4sylan 474 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( T  X.  { S } ) `  t
)  =  S )
65eqcomd 2457 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  S  =  ( ( T  X.  { S }
) `  t )
)
76oveq2d 6306 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  =  ( ( H `
 t )  +  ( ( T  X.  { S } ) `  t ) ) )
82, 7mpteq2da 4488 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  S
) )  =  ( t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) ) )
91, 8syl5eq 2497 . . 3  |-  ( ph  ->  G  =  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S }
) `  t )
) ) )
10 stoweidlem21.11 . . . 4  |-  ( ph  ->  H  e.  A )
11 fconstmpt 4878 . . . . . 6  |-  ( T  X.  { S }
)  =  ( s  e.  T  |->  S )
12 stoweidlem21.3 . . . . . . 7  |-  F/_ t S
13 nfcv 2592 . . . . . . 7  |-  F/_ s S
14 eqidd 2452 . . . . . . 7  |-  ( s  =  t  ->  S  =  S )
1512, 13, 14cbvmpt 4494 . . . . . 6  |-  ( s  e.  T  |->  S )  =  ( t  e.  T  |->  S )
1611, 15eqtri 2473 . . . . 5  |-  ( T  X.  { S }
)  =  ( t  e.  T  |->  S )
1712nfeq2 2607 . . . . . . . . . 10  |-  F/ t  x  =  S
18 simpl 459 . . . . . . . . . 10  |-  ( ( x  =  S  /\  t  e.  T )  ->  x  =  S )
1917, 18mpteq2da 4488 . . . . . . . . 9  |-  ( x  =  S  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  S ) )
2019eleq1d 2513 . . . . . . . 8  |-  ( x  =  S  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  S )  e.  A ) )
2120imbi2d 318 . . . . . . 7  |-  ( x  =  S  ->  (
( ph  ->  ( t  e.  T  |->  x )  e.  A )  <->  ( ph  ->  ( t  e.  T  |->  S )  e.  A
) ) )
22 stoweidlem21.9 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
2322expcom 437 . . . . . . 7  |-  ( x  e.  RR  ->  ( ph  ->  ( t  e.  T  |->  x )  e.  A ) )
2421, 23vtoclga 3113 . . . . . 6  |-  ( S  e.  RR  ->  ( ph  ->  ( t  e.  T  |->  S )  e.  A ) )
253, 24mpcom 37 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  S )  e.  A
)
2616, 25syl5eqel 2533 . . . 4  |-  ( ph  ->  ( T  X.  { S } )  e.  A
)
27 stoweidlem21.8 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
28 stoweidlem21.2 . . . . 5  |-  F/_ t H
29 nfcv 2592 . . . . . 6  |-  F/_ t T
3012nfsn 4029 . . . . . 6  |-  F/_ t { S }
3129, 30nfxp 4861 . . . . 5  |-  F/_ t
( T  X.  { S } )
3227, 28, 31stoweidlem8 37868 . . . 4  |-  ( (
ph  /\  H  e.  A  /\  ( T  X.  { S } )  e.  A )  ->  (
t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
3310, 26, 32mpd3an23 1366 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
349, 33eqeltrd 2529 . 2  |-  ( ph  ->  G  e.  A )
35 simpr 463 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
36 stoweidlem21.10 . . . . . . . . . . . 12  |-  ( ph  ->  A. f  e.  A  f : T --> RR )
37 feq1 5710 . . . . . . . . . . . . 13  |-  ( f  =  H  ->  (
f : T --> RR  <->  H : T
--> RR ) )
3837rspccva 3149 . . . . . . . . . . . 12  |-  ( ( A. f  e.  A  f : T --> RR  /\  H  e.  A )  ->  H : T --> RR )
3936, 10, 38syl2anc 667 . . . . . . . . . . 11  |-  ( ph  ->  H : T --> RR )
4039fnvinran 37335 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
413adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  RR )
4240, 41readdcld 9670 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  e.  RR )
431fvmpt2 5957 . . . . . . . . 9  |-  ( ( t  e.  T  /\  ( ( H `  t )  +  S
)  e.  RR )  ->  ( G `  t )  =  ( ( H `  t
)  +  S ) )
4435, 42, 43syl2anc 667 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  t )  =  ( ( H `
 t )  +  S ) )
4544oveq1d 6305 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
4640recnd 9669 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  CC )
47 stoweidlem21.6 . . . . . . . . . 10  |-  ( ph  ->  F : T --> RR )
4847fnvinran 37335 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
4948recnd 9669 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
503recnd 9669 . . . . . . . . 9  |-  ( ph  ->  S  e.  CC )
5150adantr 467 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  CC )
5246, 49, 51subsub3d 10016 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  -  ( ( F `  t )  -  S ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
5345, 52eqtr4d 2488 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )
5453fveq2d 5869 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  =  ( abs `  ( ( H `  t )  -  ( ( F `
 t )  -  S ) ) ) )
55 stoweidlem21.12 . . . . . 6  |-  ( ph  ->  A. t  e.  T  ( abs `  ( ( H `  t )  -  ( ( F `
 t )  -  S ) ) )  <  E )
5655r19.21bi 2757 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )  < 
E )
5754, 56eqbrtrd 4423 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  <  E
)
5857ex 436 . . 3  |-  ( ph  ->  ( t  e.  T  ->  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) )  <  E ) )
592, 58ralrimi 2788 . 2  |-  ( ph  ->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t ) ) )  <  E )
60 stoweidlem21.1 . . . . 5  |-  F/_ t G
6160nfeq2 2607 . . . 4  |-  F/ t  f  =  G
62 fveq1 5864 . . . . . . 7  |-  ( f  =  G  ->  (
f `  t )  =  ( G `  t ) )
6362oveq1d 6305 . . . . . 6  |-  ( f  =  G  ->  (
( f `  t
)  -  ( F `
 t ) )  =  ( ( G `
 t )  -  ( F `  t ) ) )
6463fveq2d 5869 . . . . 5  |-  ( f  =  G  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  =  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )
6564breq1d 4412 . . . 4  |-  ( f  =  G  ->  (
( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  E  <->  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6661, 65ralbid 2822 . . 3  |-  ( f  =  G  ->  ( A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6766rspcev 3150 . 2  |-  ( ( G  e.  A  /\  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t ) ) )  <  E )  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
6834, 59, 67syl2anc 667 1  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444   F/wnf 1667    e. wcel 1887   F/_wnfc 2579   A.wral 2737   E.wrex 2738   {csn 3968   class class class wbr 4402    |-> cmpt 4461    X. cxp 4832   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538    + caddc 9542    < 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-8 1889  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-pow 4581  ax-pr 4639  ax-un 6583  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  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-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  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-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-ltxr 9680  df-sub 9862
This theorem is referenced by:  stoweidlem62  37923  stoweidlem62OLD  37924
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