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Theorem stoweidlem21 29759
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 5924 . . . . . . . 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 2442 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  S  =  ( ( T  X.  { S }
) `  t )
)
76oveq2d 6102 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  =  ( ( H `
 t )  +  ( ( T  X.  { S } ) `  t ) ) )
82, 7mpteq2da 4370 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  S
) )  =  ( t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) ) )
91, 8syl5eq 2481 . . 3  |-  ( ph  ->  G  =  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S }
) `  t )
) ) )
10 stoweidlem21.11 . . . 4  |-  ( ph  ->  H  e.  A )
11 fconstmpt 4874 . . . . . 6  |-  ( T  X.  { S }
)  =  ( s  e.  T  |->  S )
12 stoweidlem21.3 . . . . . . 7  |-  F/_ t S
13 nfcv 2573 . . . . . . 7  |-  F/_ s S
14 eqidd 2438 . . . . . . 7  |-  ( s  =  t  ->  S  =  S )
1512, 13, 14cbvmpt 4375 . . . . . 6  |-  ( s  e.  T  |->  S )  =  ( t  e.  T  |->  S )
1611, 15eqtri 2457 . . . . 5  |-  ( T  X.  { S }
)  =  ( t  e.  T  |->  S )
1712nfeq2 2584 . . . . . . . . . 10  |-  F/ t  x  =  S
18 simpl 457 . . . . . . . . . 10  |-  ( ( x  =  S  /\  t  e.  T )  ->  x  =  S )
1917, 18mpteq2da 4370 . . . . . . . . 9  |-  ( x  =  S  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  S ) )
2019eleq1d 2503 . . . . . . . 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 3029 . . . . . 6  |-  ( S  e.  RR  ->  ( ph  ->  ( t  e.  T  |->  S )  e.  A ) )
253, 24mpcom 36 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  S )  e.  A
)
2616, 25syl5eqel 2521 . . . 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 2573 . . . . . 6  |-  F/_ t T
3012nfsn 3927 . . . . . 6  |-  F/_ t { S }
3129, 30nfxp 4858 . . . . 5  |-  F/_ t
( T  X.  { S } )
3227, 28, 31stoweidlem8 29746 . . . 4  |-  ( (
ph  /\  H  e.  A  /\  ( T  X.  { S } )  e.  A )  ->  (
t  e.  T  |->  ( ( H `  t
)  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
3310, 26, 32mpd3an23 1316 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( H `  t )  +  ( ( T  X.  { S } ) `  t
) ) )  e.  A )
349, 33eqeltrd 2511 . 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 5535 . . . . . . . . . . . . 13  |-  ( f  =  H  ->  (
f : T --> RR  <->  H : T
--> RR ) )
3837rspccva 3065 . . . . . . . . . . . 12  |-  ( ( A. f  e.  A  f : T --> RR  /\  H  e.  A )  ->  H : T --> RR )
3936, 10, 38syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  H : T --> RR )
4039fnvinran 29679 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
413adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  RR )
4240, 41readdcld 9405 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  +  S )  e.  RR )
431fvmpt2 5774 . . . . . . . . 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 6101 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
4640recnd 9404 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  CC )
47 stoweidlem21.6 . . . . . . . . . 10  |-  ( ph  ->  F : T --> RR )
4847fnvinran 29679 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
4948recnd 9404 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
503recnd 9404 . . . . . . . . 9  |-  ( ph  ->  S  e.  CC )
5150adantr 465 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  S  e.  CC )
5246, 49, 51subsub3d 9741 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  -  ( ( F `  t )  -  S ) )  =  ( ( ( H `  t )  +  S )  -  ( F `  t ) ) )
5345, 52eqtr4d 2472 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( G `  t
)  -  ( F `
 t ) )  =  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )
5453fveq2d 5688 . . . . 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 2808 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( abs `  ( ( H `
 t )  -  ( ( F `  t )  -  S
) ) )  < 
E )
5754, 56eqbrtrd 4305 . . . 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 2791 . 2  |-  ( ph  ->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t ) ) )  <  E )
60 stoweidlem21.1 . . . . 5  |-  F/_ t G
6160nfeq2 2584 . . . 4  |-  F/ t  f  =  G
62 fveq1 5683 . . . . . . 7  |-  ( f  =  G  ->  (
f `  t )  =  ( G `  t ) )
6362oveq1d 6101 . . . . . 6  |-  ( f  =  G  ->  (
( f `  t
)  -  ( F `
 t ) )  =  ( ( G `
 t )  -  ( F `  t ) ) )
6463fveq2d 5688 . . . . 5  |-  ( f  =  G  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  =  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) )
6564breq1d 4295 . . . 4  |-  ( f  =  G  ->  (
( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  E  <->  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6661, 65ralbid 2727 . . 3  |-  ( f  =  G  ->  ( A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E  <->  A. t  e.  T  ( abs `  ( ( G `  t )  -  ( F `  t )
) )  <  E
) )
6766rspcev 3066 . 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 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2560   A.wral 2709   E.wrex 2710   {csn 3870   class class class wbr 4285    e. cmpt 4343    X. cxp 4830   -->wf 5407   ` cfv 5411  (class class class)co 6086   CCcc 9272   RRcr 9273    + caddc 9277    < clt 9410    - cmin 9587   abscabs 12715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-br 4286  df-opab 4344  df-mpt 4345  df-id 4628  df-po 4633  df-so 4634  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-ltxr 9415  df-sub 9589
This theorem is referenced by:  stoweidlem62  29800
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