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Theorem stoweidlem21 29954
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 6030 . . . . . . . 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 2459 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  S  =  ( ( T  X.  { S }
) `  t )
)
76oveq2d 6206 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  =  ( ( H `
 t )  +  ( ( T  X.  { S } ) `  t ) ) )
82, 7mpteq2da 4475 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  S
) )  =  ( t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) ) )
91, 8syl5eq 2504 . . 3  |-  ( ph  ->  G  =  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S }
) `  t )
) ) )
10 stoweidlem21.11 . . . 4  |-  ( ph  ->  H  e.  A )
11 fconstmpt 4980 . . . . . 6  |-  ( T  X.  { S }
)  =  ( s  e.  T  |->  S )
12 stoweidlem21.3 . . . . . . 7  |-  F/_ t S
13 nfcv 2613 . . . . . . 7  |-  F/_ s S
14 eqidd 2452 . . . . . . 7  |-  ( s  =  t  ->  S  =  S )
1512, 13, 14cbvmpt 4480 . . . . . 6  |-  ( s  e.  T  |->  S )  =  ( t  e.  T  |->  S )
1611, 15eqtri 2480 . . . . 5  |-  ( T  X.  { S }
)  =  ( t  e.  T  |->  S )
1712nfeq2 2629 . . . . . . . . . 10  |-  F/ t  x  =  S
18 simpl 457 . . . . . . . . . 10  |-  ( ( x  =  S  /\  t  e.  T )  ->  x  =  S )
1917, 18mpteq2da 4475 . . . . . . . . 9  |-  ( x  =  S  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  S ) )
2019eleq1d 2520 . . . . . . . 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 3132 . . . . . 6  |-  ( S  e.  RR  ->  ( ph  ->  ( t  e.  T  |->  S )  e.  A ) )
253, 24mpcom 36 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  S )  e.  A
)
2616, 25syl5eqel 2543 . . . 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 2613 . . . . . 6  |-  F/_ t T
3012nfsn 4032 . . . . . 6  |-  F/_ t { S }
3129, 30nfxp 4964 . . . . 5  |-  F/_ t
( T  X.  { S } )
3227, 28, 31stoweidlem8 29941 . . . 4  |-  ( (
ph  /\  H  e.  A  /\  ( T  X.  { S } )  e.  A )  ->  (
t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
3310, 26, 32mpd3an23 1317 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
349, 33eqeltrd 2539 . 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 5640 . . . . . . . . . . . . 13  |-  ( f  =  H  ->  (
f : T --> RR  <->  H : T
--> RR ) )
3837rspccva 3168 . . . . . . . . . . . 12  |-  ( ( A. f  e.  A  f : T --> RR  /\  H  e.  A )  ->  H : T --> RR )
3936, 10, 38syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  H : T --> RR )
4039fnvinran 29874 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
413adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  RR )
4240, 41readdcld 9514 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  e.  RR )
431fvmpt2 5880 . . . . . . . . 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 6205 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
4640recnd 9513 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  CC )
47 stoweidlem21.6 . . . . . . . . . 10  |-  ( ph  ->  F : T --> RR )
4847fnvinran 29874 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
4948recnd 9513 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
503recnd 9513 . . . . . . . . 9  |-  ( ph  ->  S  e.  CC )
5150adantr 465 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  CC )
5246, 49, 51subsub3d 9850 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  -  ( ( F `  t )  -  S ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
5345, 52eqtr4d 2495 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )
5453fveq2d 5793 . . . . 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 2910 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )  < 
E )
5754, 56eqbrtrd 4410 . . . 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 2815 . 2  |-  ( ph  ->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t ) ) )  <  E )
60 stoweidlem21.1 . . . . 5  |-  F/_ t G
6160nfeq2 2629 . . . 4  |-  F/ t  f  =  G
62 fveq1 5788 . . . . . . 7  |-  ( f  =  G  ->  (
f `  t )  =  ( G `  t ) )
6362oveq1d 6205 . . . . . 6  |-  ( f  =  G  ->  (
( f `  t
)  -  ( F `
 t ) )  =  ( ( G `
 t )  -  ( F `  t ) ) )
6463fveq2d 5793 . . . . 5  |-  ( f  =  G  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  =  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )
6564breq1d 4400 . . . 4  |-  ( f  =  G  ->  (
( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  E  <->  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6661, 65ralbid 2836 . . 3  |-  ( f  =  G  ->  ( A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6766rspcev 3169 . 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 965    = wceq 1370   F/wnf 1590    e. wcel 1758   F/_wnfc 2599   A.wral 2795   E.wrex 2796   {csn 3975   class class class wbr 4390    |-> cmpt 4448    X. cxp 4936   -->wf 5512   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382    + caddc 9386    < clt 9519    - cmin 9696   abscabs 12825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-ltxr 9524  df-sub 9698
This theorem is referenced by:  stoweidlem62  29995
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