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Theorem stoweidlem21 31964
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 6126 . . . . . . . 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 2465 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  S  =  ( ( T  X.  { S }
) `  t )
)
76oveq2d 6312 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  =  ( ( H `
 t )  +  ( ( T  X.  { S } ) `  t ) ) )
82, 7mpteq2da 4542 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  S
) )  =  ( t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) ) )
91, 8syl5eq 2510 . . 3  |-  ( ph  ->  G  =  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S }
) `  t )
) ) )
10 stoweidlem21.11 . . . 4  |-  ( ph  ->  H  e.  A )
11 fconstmpt 5052 . . . . . 6  |-  ( T  X.  { S }
)  =  ( s  e.  T  |->  S )
12 stoweidlem21.3 . . . . . . 7  |-  F/_ t S
13 nfcv 2619 . . . . . . 7  |-  F/_ s S
14 eqidd 2458 . . . . . . 7  |-  ( s  =  t  ->  S  =  S )
1512, 13, 14cbvmpt 4547 . . . . . 6  |-  ( s  e.  T  |->  S )  =  ( t  e.  T  |->  S )
1611, 15eqtri 2486 . . . . 5  |-  ( T  X.  { S }
)  =  ( t  e.  T  |->  S )
1712nfeq2 2636 . . . . . . . . . 10  |-  F/ t  x  =  S
18 simpl 457 . . . . . . . . . 10  |-  ( ( x  =  S  /\  t  e.  T )  ->  x  =  S )
1917, 18mpteq2da 4542 . . . . . . . . 9  |-  ( x  =  S  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  S ) )
2019eleq1d 2526 . . . . . . . 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 3173 . . . . . 6  |-  ( S  e.  RR  ->  ( ph  ->  ( t  e.  T  |->  S )  e.  A ) )
253, 24mpcom 36 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  S )  e.  A
)
2616, 25syl5eqel 2549 . . . 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 2619 . . . . . 6  |-  F/_ t T
3012nfsn 4089 . . . . . 6  |-  F/_ t { S }
3129, 30nfxp 5035 . . . . 5  |-  F/_ t
( T  X.  { S } )
3227, 28, 31stoweidlem8 31951 . . . 4  |-  ( (
ph  /\  H  e.  A  /\  ( T  X.  { S } )  e.  A )  ->  (
t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
3310, 26, 32mpd3an23 1326 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
349, 33eqeltrd 2545 . 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 5719 . . . . . . . . . . . . 13  |-  ( f  =  H  ->  (
f : T --> RR  <->  H : T
--> RR ) )
3837rspccva 3209 . . . . . . . . . . . 12  |-  ( ( A. f  e.  A  f : T --> RR  /\  H  e.  A )  ->  H : T --> RR )
3936, 10, 38syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  H : T --> RR )
4039fnvinran 31550 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
413adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  RR )
4240, 41readdcld 9640 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  e.  RR )
431fvmpt2 5964 . . . . . . . . 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 6311 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
4640recnd 9639 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  CC )
47 stoweidlem21.6 . . . . . . . . . 10  |-  ( ph  ->  F : T --> RR )
4847fnvinran 31550 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
4948recnd 9639 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
503recnd 9639 . . . . . . . . 9  |-  ( ph  ->  S  e.  CC )
5150adantr 465 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  CC )
5246, 49, 51subsub3d 9980 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  -  ( ( F `  t )  -  S ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
5345, 52eqtr4d 2501 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )
5453fveq2d 5876 . . . . 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 2826 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )  < 
E )
5754, 56eqbrtrd 4476 . . . 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 2857 . 2  |-  ( ph  ->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t ) ) )  <  E )
60 stoweidlem21.1 . . . . 5  |-  F/_ t G
6160nfeq2 2636 . . . 4  |-  F/ t  f  =  G
62 fveq1 5871 . . . . . . 7  |-  ( f  =  G  ->  (
f `  t )  =  ( G `  t ) )
6362oveq1d 6311 . . . . . 6  |-  ( f  =  G  ->  (
( f `  t
)  -  ( F `
 t ) )  =  ( ( G `
 t )  -  ( F `  t ) ) )
6463fveq2d 5876 . . . . 5  |-  ( f  =  G  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  =  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )
6564breq1d 4466 . . . 4  |-  ( f  =  G  ->  (
( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  E  <->  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6661, 65ralbid 2891 . . 3  |-  ( f  =  G  ->  ( A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6766rspcev 3210 . 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 973    = wceq 1395   F/wnf 1617    e. wcel 1819   F/_wnfc 2605   A.wral 2807   E.wrex 2808   {csn 4032   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508    + caddc 9512    < clt 9645    - cmin 9824   abscabs 13078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-sub 9826
This theorem is referenced by:  stoweidlem62  32005
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