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Theorem stoweidlem21 31278
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 6107 . . . . . . . 8  |-  ( ( S  e.  RR  /\  t  e.  T )  ->  ( ( T  X.  { S } ) `  t )  =  S )
53, 4sylan 471 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( T  X.  { S } ) `  t
)  =  S )
65eqcomd 2470 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  S  =  ( ( T  X.  { S }
) `  t )
)
76oveq2d 6293 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  =  ( ( H `
 t )  +  ( ( T  X.  { S } ) `  t ) ) )
82, 7mpteq2da 4527 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  S
) )  =  ( t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) ) )
91, 8syl5eq 2515 . . 3  |-  ( ph  ->  G  =  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S }
) `  t )
) ) )
10 stoweidlem21.11 . . . 4  |-  ( ph  ->  H  e.  A )
11 fconstmpt 5037 . . . . . 6  |-  ( T  X.  { S }
)  =  ( s  e.  T  |->  S )
12 stoweidlem21.3 . . . . . . 7  |-  F/_ t S
13 nfcv 2624 . . . . . . 7  |-  F/_ s S
14 eqidd 2463 . . . . . . 7  |-  ( s  =  t  ->  S  =  S )
1512, 13, 14cbvmpt 4532 . . . . . 6  |-  ( s  e.  T  |->  S )  =  ( t  e.  T  |->  S )
1611, 15eqtri 2491 . . . . 5  |-  ( T  X.  { S }
)  =  ( t  e.  T  |->  S )
1712nfeq2 2641 . . . . . . . . . 10  |-  F/ t  x  =  S
18 simpl 457 . . . . . . . . . 10  |-  ( ( x  =  S  /\  t  e.  T )  ->  x  =  S )
1917, 18mpteq2da 4527 . . . . . . . . 9  |-  ( x  =  S  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  S ) )
2019eleq1d 2531 . . . . . . . 8  |-  ( x  =  S  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  S )  e.  A ) )
2120imbi2d 316 . . . . . . 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 435 . . . . . . 7  |-  ( x  e.  RR  ->  ( ph  ->  ( t  e.  T  |->  x )  e.  A ) )
2421, 23vtoclga 3172 . . . . . 6  |-  ( S  e.  RR  ->  ( ph  ->  ( t  e.  T  |->  S )  e.  A ) )
253, 24mpcom 36 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  S )  e.  A
)
2616, 25syl5eqel 2554 . . . 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 2624 . . . . . 6  |-  F/_ t T
3012nfsn 4080 . . . . . 6  |-  F/_ t { S }
3129, 30nfxp 5020 . . . . 5  |-  F/_ t
( T  X.  { S } )
3227, 28, 31stoweidlem8 31265 . . . 4  |-  ( (
ph  /\  H  e.  A  /\  ( T  X.  { S } )  e.  A )  ->  (
t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
3310, 26, 32mpd3an23 1321 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
349, 33eqeltrd 2550 . 2  |-  ( ph  ->  G  e.  A )
35 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
36 stoweidlem21.10 . . . . . . . . . . . 12  |-  ( ph  ->  A. f  e.  A  f : T --> RR )
37 feq1 5706 . . . . . . . . . . . . 13  |-  ( f  =  H  ->  (
f : T --> RR  <->  H : T
--> RR ) )
3837rspccva 3208 . . . . . . . . . . . 12  |-  ( ( A. f  e.  A  f : T --> RR  /\  H  e.  A )  ->  H : T --> RR )
3936, 10, 38syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  H : T --> RR )
4039fnvinran 30924 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
413adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  RR )
4240, 41readdcld 9614 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  e.  RR )
431fvmpt2 5950 . . . . . . . . 9  |-  ( ( t  e.  T  /\  ( ( H `  t )  +  S
)  e.  RR )  ->  ( G `  t )  =  ( ( H `  t
)  +  S ) )
4435, 42, 43syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( G `  t )  =  ( ( H `
 t )  +  S ) )
4544oveq1d 6292 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
4640recnd 9613 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  CC )
47 stoweidlem21.6 . . . . . . . . . 10  |-  ( ph  ->  F : T --> RR )
4847fnvinran 30924 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
4948recnd 9613 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
503recnd 9613 . . . . . . . . 9  |-  ( ph  ->  S  e.  CC )
5150adantr 465 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  CC )
5246, 49, 51subsub3d 9951 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  -  ( ( F `  t )  -  S ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
5345, 52eqtr4d 2506 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )
5453fveq2d 5863 . . . . 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 2828 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )  < 
E )
5754, 56eqbrtrd 4462 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( G `
 t )  -  ( F `  t ) ) )  <  E
)
5857ex 434 . . 3  |-  ( ph  ->  ( t  e.  T  ->  ( abs `  (
( G `  t
)  -  ( F `
 t ) ) )  <  E ) )
592, 58ralrimi 2859 . 2  |-  ( ph  ->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t ) ) )  <  E )
60 stoweidlem21.1 . . . . 5  |-  F/_ t G
6160nfeq2 2641 . . . 4  |-  F/ t  f  =  G
62 fveq1 5858 . . . . . . 7  |-  ( f  =  G  ->  (
f `  t )  =  ( G `  t ) )
6362oveq1d 6292 . . . . . 6  |-  ( f  =  G  ->  (
( f `  t
)  -  ( F `
 t ) )  =  ( ( G `
 t )  -  ( F `  t ) ) )
6463fveq2d 5863 . . . . 5  |-  ( f  =  G  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  =  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )
6564breq1d 4452 . . . 4  |-  ( f  =  G  ->  (
( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  E  <->  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6661, 65ralbid 2893 . . 3  |-  ( f  =  G  ->  ( A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6766rspcev 3209 . 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 661 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 369    /\ w3a 968    = wceq 1374   F/wnf 1594    e. wcel 1762   F/_wnfc 2610   A.wral 2809   E.wrex 2810   {csn 4022   class class class wbr 4442    |-> cmpt 4500    X. cxp 4992   -->wf 5577   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482    + caddc 9486    < clt 9619    - cmin 9796   abscabs 13019
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 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-ltxr 9624  df-sub 9798
This theorem is referenced by:  stoweidlem62  31319
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