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Theorem stoweid 27812
Description: This theorem proves the Stone-Weierstrass theorem for real valued functions: let  J be a compact topology on  T, and  C be the set of real continuous functions on  T. Assume that  A is a subalgebra of  C (closed under addition and multiplication of functions) containing constant functions and discriminating points (if  r and  t are distinct points in  T, then there exists a function  h in  A such that h(r) is distinct from h(t) ). Then, for any continuous function 
F and for any positive real  E, there exists a function  f in the subalgebra  A, such that  f approximates  F up to  E ( E represents the usual ε value). As a classical example, given any a,b reals, the closed interval  T  =  [
a ,  b ] could be taken, along with the subalgebra  A of real polynomials on  T, and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on  [ a ,  b ]. The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweid.1  |-  F/_ t F
stoweid.2  |-  F/ t
ph
stoweid.3  |-  K  =  ( topGen `  ran  (,) )
stoweid.4  |-  ( ph  ->  J  e.  Comp )
stoweid.5  |-  T  = 
U. J
stoweid.6  |-  C  =  ( J  Cn  K
)
stoweid.7  |-  ( ph  ->  A  C_  C )
stoweid.8  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweid.9  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweid.10  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
stoweid.11  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. h  e.  A  ( h `  r
)  =/=  ( h `
 t ) )
stoweid.12  |-  ( ph  ->  F  e.  C )
stoweid.13  |-  ( ph  ->  E  e.  RR+ )
Assertion
Ref Expression
stoweid  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
Distinct variable groups:    f, g,
t, A    f, h, r, x, t, A    f, E, g, t    f, F, g    f, J, r, t    T, f, g, t    ph, f, g    h, E, r, x    h, F, r, x    T, h, r, x    ph, h, r, x    t, K
Allowed substitution hints:    ph( t)    C( x, t, f, g, h, r)    F( t)    J( x, g, h)    K( x, f, g, h, r)

Proof of Theorem stoweid
StepHypRef Expression
1 simpr 447 . . . 4  |-  ( (
ph  /\  T  =  (/) )  ->  T  =  (/) )
2 stoweid.10 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
32ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. x  e.  RR  ( t  e.  T  |->  x )  e.  A
)
4 1re 8837 . . . . . . 7  |-  1  e.  RR
54a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  RR )
6 id 19 . . . . . . . . 9  |-  ( x  =  1  ->  x  =  1 )
76mpteq2dv 4107 . . . . . . . 8  |-  ( x  =  1  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  1 ) )
87eleq1d 2349 . . . . . . 7  |-  ( x  =  1  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  1 )  e.  A ) )
98rspccv 2881 . . . . . 6  |-  ( A. x  e.  RR  (
t  e.  T  |->  x )  e.  A  -> 
( 1  e.  RR  ->  ( t  e.  T  |->  1 )  e.  A
) )
103, 5, 9sylc 56 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
1110adantr 451 . . . 4  |-  ( (
ph  /\  T  =  (/) )  ->  ( t  e.  T  |->  1 )  e.  A )
121, 11stoweidlem9 27758 . . 3  |-  ( (
ph  /\  T  =  (/) )  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) ) )
13 stoweid.1 . . . 4  |-  F/_ t F
14 nfv 1605 . . . . 5  |-  F/ f
ph
15 nfv 1605 . . . . 5  |-  F/ f  -.  T  =  (/)
1614, 15nfan 1771 . . . 4  |-  F/ f ( ph  /\  -.  T  =  (/) )
17 stoweid.2 . . . . 5  |-  F/ t
ph
18 nfv 1605 . . . . 5  |-  F/ t  -.  T  =  (/)
1917, 18nfan 1771 . . . 4  |-  F/ t ( ph  /\  -.  T  =  (/) )
20 eqid 2283 . . . 4  |-  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran 
F ,  RR ,  `'  <  ) ) )  =  ( t  e.  T  |->  ( ( F `
 t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
21 stoweid.3 . . . 4  |-  K  =  ( topGen `  ran  (,) )
22 stoweid.5 . . . 4  |-  T  = 
U. J
23 stoweid.4 . . . . 5  |-  ( ph  ->  J  e.  Comp )
2423adantr 451 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  J  e.  Comp )
25 stoweid.6 . . . 4  |-  C  =  ( J  Cn  K
)
26 stoweid.7 . . . . 5  |-  ( ph  ->  A  C_  C )
2726adantr 451 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  A  C_  C )
28 simp1l 979 . . . . . 6  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  ph )
29 simp2 956 . . . . . 6  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  f  e.  A )
30 simp3 957 . . . . . 6  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  g  e.  A )
3128, 29, 303jca 1132 . . . . 5  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  ( ph  /\  f  e.  A  /\  g  e.  A ) )
32 stoweid.8 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
3331, 32syl 15 . . . 4  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A
)
34 stoweid.9 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
3531, 34syl 15 . . . 4  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) )  e.  A
)
362adantlr 695 . . . 4  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A
)
37 simpll 730 . . . . . 6  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t
) )  ->  ph )
38 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t
) )  ->  (
r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )
3937, 38jca 518 . . . . 5  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t
) )  ->  ( ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) ) )
40 stoweid.11 . . . . 5  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. h  e.  A  ( h `  r
)  =/=  ( h `
 t ) )
4139, 40syl 15 . . . 4  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t
) )  ->  E. h  e.  A  ( h `  r )  =/=  (
h `  t )
)
42 stoweid.12 . . . . 5  |-  ( ph  ->  F  e.  C )
4342adantr 451 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  F  e.  C )
44 stoweid.13 . . . . . . 7  |-  ( ph  ->  E  e.  RR+ )
45 1rp 10358 . . . . . . . . 9  |-  1  e.  RR+
4645a1i 10 . . . . . . . 8  |-  ( ph  ->  1  e.  RR+ )
47 4re 9819 . . . . . . . . . . 11  |-  4  e.  RR
48 4pos 9832 . . . . . . . . . . 11  |-  0  <  4
4947, 48pm3.2i 441 . . . . . . . . . 10  |-  ( 4  e.  RR  /\  0  <  4 )
50 elrp 10356 . . . . . . . . . 10  |-  ( 4  e.  RR+  <->  ( 4  e.  RR  /\  0  <  4 ) )
5149, 50mpbir 200 . . . . . . . . 9  |-  4  e.  RR+
5251a1i 10 . . . . . . . 8  |-  ( ph  ->  4  e.  RR+ )
5346, 52rpdivcld 10407 . . . . . . 7  |-  ( ph  ->  ( 1  /  4
)  e.  RR+ )
5444, 53jca 518 . . . . . 6  |-  ( ph  ->  ( E  e.  RR+  /\  ( 1  /  4
)  e.  RR+ )
)
55 ifcl 3601 . . . . . 6  |-  ( ( E  e.  RR+  /\  (
1  /  4 )  e.  RR+ )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR+ )
5654, 55syl 15 . . . . 5  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR+ )
5756adantr 451 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR+ )
58 df-ne 2448 . . . . . 6  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
5958biimpri 197 . . . . 5  |-  ( -.  T  =  (/)  ->  T  =/=  (/) )
6059adantl 452 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  T  =/=  (/) )
61 rpre 10360 . . . . . . . . . 10  |-  ( E  e.  RR+  ->  E  e.  RR )
6244, 61syl 15 . . . . . . . . 9  |-  ( ph  ->  E  e.  RR )
63 0re 8838 . . . . . . . . . . . . . 14  |-  0  e.  RR
6463, 48pm3.2i 441 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  /\  0  <  4 )
65 ltne 8917 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  0  <  4 )  -> 
4  =/=  0 )
6664, 65ax-mp 8 . . . . . . . . . . . 12  |-  4  =/=  0
674, 47, 663pm3.2i 1130 . . . . . . . . . . 11  |-  ( 1  e.  RR  /\  4  e.  RR  /\  4  =/=  0 )
68 redivcl 9479 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  4  e.  RR  /\  4  =/=  0 )  ->  (
1  /  4 )  e.  RR )
6967, 68ax-mp 8 . . . . . . . . . 10  |-  ( 1  /  4 )  e.  RR
7069a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( 1  /  4
)  e.  RR )
7162, 70jca 518 . . . . . . . 8  |-  ( ph  ->  ( E  e.  RR  /\  ( 1  /  4
)  e.  RR ) )
72 ifcl 3601 . . . . . . . 8  |-  ( ( E  e.  RR  /\  ( 1  /  4
)  e.  RR )  ->  if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) )  e.  RR )
7371, 72syl 15 . . . . . . 7  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR )
74 3re 9817 . . . . . . . . . 10  |-  3  e.  RR
75 3ne0 9831 . . . . . . . . . 10  |-  3  =/=  0
764, 74, 753pm3.2i 1130 . . . . . . . . 9  |-  ( 1  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )
77 redivcl 9479 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )  ->  (
1  /  3 )  e.  RR )
7876, 77ax-mp 8 . . . . . . . 8  |-  ( 1  /  3 )  e.  RR
7978a1i 10 . . . . . . 7  |-  ( ph  ->  ( 1  /  3
)  e.  RR )
8073, 70, 793jca 1132 . . . . . 6  |-  ( ph  ->  ( if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) )  e.  RR  /\  ( 1  /  4 )  e.  RR  /\  ( 1  /  3 )  e.  RR ) )
81 rpxr 10361 . . . . . . . . . 10  |-  ( E  e.  RR+  ->  E  e. 
RR* )
8244, 81syl 15 . . . . . . . . 9  |-  ( ph  ->  E  e.  RR* )
8353idi 2 . . . . . . . . . 10  |-  ( ph  ->  ( 1  /  4
)  e.  RR+ )
84 rpxr 10361 . . . . . . . . . 10  |-  ( ( 1  /  4 )  e.  RR+  ->  ( 1  /  4 )  e. 
RR* )
8583, 84syl 15 . . . . . . . . 9  |-  ( ph  ->  ( 1  /  4
)  e.  RR* )
8682, 85jca 518 . . . . . . . 8  |-  ( ph  ->  ( E  e.  RR*  /\  ( 1  /  4
)  e.  RR* )
)
87 xrmin2 10507 . . . . . . . 8  |-  ( ( E  e.  RR*  /\  (
1  /  4 )  e.  RR* )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  ( 1  / 
4 ) )
8886, 87syl 15 . . . . . . 7  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  <_  (
1  /  4 ) )
89 3lt4 9889 . . . . . . . . 9  |-  3  <  4
90 3pos 9830 . . . . . . . . . . 11  |-  0  <  3
9190, 48pm3.2i 441 . . . . . . . . . 10  |-  ( 0  <  3  /\  0  <  4 )
9274, 47ltreci 9667 . . . . . . . . . 10  |-  ( ( 0  <  3  /\  0  <  4 )  ->  ( 3  <  4  <->  ( 1  / 
4 )  <  (
1  /  3 ) ) )
9391, 92ax-mp 8 . . . . . . . . 9  |-  ( 3  <  4  <->  ( 1  /  4 )  < 
( 1  /  3
) )
9489, 93mpbi 199 . . . . . . . 8  |-  ( 1  /  4 )  < 
( 1  /  3
)
9594a1i 10 . . . . . . 7  |-  ( ph  ->  ( 1  /  4
)  <  ( 1  /  3 ) )
9688, 95jca 518 . . . . . 6  |-  ( ph  ->  ( if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) )  <_ 
( 1  /  4
)  /\  ( 1  /  4 )  < 
( 1  /  3
) ) )
97 lelttr 8912 . . . . . 6  |-  ( ( if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR  /\  ( 1  /  4
)  e.  RR  /\  ( 1  /  3
)  e.  RR )  ->  ( ( if ( E  <_  (
1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  ( 1  /  4 )  /\  ( 1  /  4
)  <  ( 1  /  3 ) )  ->  if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) )  < 
( 1  /  3
) ) )
9880, 96, 97sylc 56 . . . . 5  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  <  (
1  /  3 ) )
9998adantr 451 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <  ( 1  / 
3 ) )
10013, 16, 19, 20, 21, 22, 24, 25, 27, 33, 35, 36, 41, 43, 57, 60, 99stoweidlem62 27811 . . 3  |-  ( (
ph  /\  -.  T  =  (/) )  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) ) )
10112, 100pm2.61dan 766 . 2  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) ) )
102 nfv 1605 . . . . . 6  |-  F/ t  f  e.  A
10317, 102nfan 1771 . . . . 5  |-  F/ t ( ph  /\  f  e.  A )
104 xrmin1 10506 . . . . . . . . 9  |-  ( ( E  e.  RR*  /\  (
1  /  4 )  e.  RR* )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  E )
10586, 104syl 15 . . . . . . . 8  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  <_  E
)
106105ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  E )
10726ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  A  C_  C )
108 simplr 731 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f  e.  A )
109107, 108jca 518 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( A  C_  C  /\  f  e.  A ) )
110 ssel2 3175 . . . . . . . . . . . . . . . 16  |-  ( ( A  C_  C  /\  f  e.  A )  ->  f  e.  C )
111109, 110syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f  e.  C )
11221, 22, 25, 111fcnre 27696 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f : T --> RR )
113 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  t  e.  T )
114112, 113jca 518 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
f : T --> RR  /\  t  e.  T )
)
115 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( f : T --> RR  /\  t  e.  T )  ->  ( f `  t
)  e.  RR )
116 recn 8827 . . . . . . . . . . . . 13  |-  ( ( f `  t )  e.  RR  ->  (
f `  t )  e.  CC )
117114, 115, 1163syl 18 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
f `  t )  e.  CC )
11842ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  F  e.  C )
11921, 22, 25, 118fcnre 27696 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  F : T --> RR )
120119, 113jca 518 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( F : T --> RR  /\  t  e.  T )
)
121 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( F : T --> RR  /\  t  e.  T )  ->  ( F `  t
)  e.  RR )
122 recn 8827 . . . . . . . . . . . . 13  |-  ( ( F `  t )  e.  RR  ->  ( F `  t )  e.  CC )
123120, 121, 1223syl 18 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
124117, 123jca 518 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
( f `  t
)  e.  CC  /\  ( F `  t )  e.  CC ) )
125 subcl 9051 . . . . . . . . . . 11  |-  ( ( ( f `  t
)  e.  CC  /\  ( F `  t )  e.  CC )  -> 
( ( f `  t )  -  ( F `  t )
)  e.  CC )
126124, 125syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
( f `  t
)  -  ( F `
 t ) )  e.  CC )
127 abscl 11763 . . . . . . . . . 10  |-  ( ( ( f `  t
)  -  ( F `
 t ) )  e.  CC  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  e.  RR )
128126, 127syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  e.  RR )
12947, 48gt0ne0ii 9309 . . . . . . . . . . . . . . 15  |-  4  =/=  0
1304, 47, 1293pm3.2i 1130 . . . . . . . . . . . . . 14  |-  ( 1  e.  RR  /\  4  e.  RR  /\  4  =/=  0 )
131130a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1  e.  RR  /\  4  e.  RR  /\  4  =/=  0 ) )
132131, 68syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  /  4
)  e.  RR )
13362, 132jca 518 . . . . . . . . . . 11  |-  ( ph  ->  ( E  e.  RR  /\  ( 1  /  4
)  e.  RR ) )
134133, 72syl 15 . . . . . . . . . 10  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR )
135134ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR )
13662ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  E  e.  RR )
137128, 135, 1363jca 1132 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  e.  RR  /\  if ( E  <_  (
1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR  /\  E  e.  RR )
)
138 ltletr 8913 . . . . . . . 8  |-  ( ( ( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  e.  RR  /\  if ( E  <_  (
1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR  /\  E  e.  RR )  ->  ( ( ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  /\  if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) )  <_  E )  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  <  E
) )
139137, 138syl 15 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
( ( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  /\  if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) )  <_  E )  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  <  E
) )
140106, 139mpan2d 655 . . . . . 6  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  ->  ( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  E ) )
141140ex 423 . . . . 5  |-  ( (
ph  /\  f  e.  A )  ->  (
t  e.  T  -> 
( ( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  ->  ( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  E ) ) )
142103, 141ralrimi 2624 . . . 4  |-  ( (
ph  /\  f  e.  A )  ->  A. t  e.  T  ( ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  ->  ( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  E ) )
143 ralim 2614 . . . 4  |-  ( A. t  e.  T  (
( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  ->  ( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  E )  ->  ( A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  ->  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E ) )
144142, 143syl 15 . . 3  |-  ( (
ph  /\  f  e.  A )  ->  ( A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) )  ->  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E ) )
145144reximdva 2655 . 2  |-  ( ph  ->  ( E. f  e.  A  A. t  e.  T  ( abs `  (
( f `  t
)  -  ( F `
 t ) ) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E ) )
146101, 145mpd 14 1  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   ifcif 3565   U.cuni 3827   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   3c3 9796   4c4 9797   RR+crp 10354   (,)cioo 10656   abscabs 11719   topGenctg 13342    Cn ccn 16954   Compccmp 17113
This theorem is referenced by:  stowei  27813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-cn 16957  df-cnp 16958  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887
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