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Theorem stoweid 27915
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 2639 . . . . . 6  |-  ( ph  ->  A. x  e.  RR  ( t  e.  T  |->  x )  e.  A
)
4 1re 8853 . . . . . . 7  |-  1  e.  RR
54a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  RR )
6 id 19 . . . . . . . . 9  |-  ( x  =  1  ->  x  =  1 )
76mpteq2dv 4123 . . . . . . . 8  |-  ( x  =  1  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  1 ) )
87eleq1d 2362 . . . . . . 7  |-  ( x  =  1  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  1 )  e.  A ) )
98rspccv 2894 . . . . . 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 27861 . . 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 1609 . . . . 5  |-  F/ f
ph
15 nfv 1609 . . . . 5  |-  F/ f  -.  T  =  (/)
1614, 15nfan 1783 . . . 4  |-  F/ f ( ph  /\  -.  T  =  (/) )
17 stoweid.2 . . . . 5  |-  F/ t
ph
18 nfv 1609 . . . . 5  |-  F/ t  -.  T  =  (/)
1917, 18nfan 1783 . . . 4  |-  F/ t ( ph  /\  -.  T  =  (/) )
20 eqid 2296 . . . 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 10374 . . . . . . . . 9  |-  1  e.  RR+
4645a1i 10 . . . . . . . 8  |-  ( ph  ->  1  e.  RR+ )
47 4re 9835 . . . . . . . . . . 11  |-  4  e.  RR
48 4pos 9848 . . . . . . . . . . 11  |-  0  <  4
4947, 48pm3.2i 441 . . . . . . . . . 10  |-  ( 4  e.  RR  /\  0  <  4 )
50 elrp 10372 . . . . . . . . . 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 10423 . . . . . . 7  |-  ( ph  ->  ( 1  /  4
)  e.  RR+ )
5444, 53jca 518 . . . . . 6  |-  ( ph  ->  ( E  e.  RR+  /\  ( 1  /  4
)  e.  RR+ )
)
55 ifcl 3614 . . . . . 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 2461 . . . . . 6  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
5958biimpri 197 . . . . 5  |-  ( -.  T  =  (/)  ->  T  =/=  (/) )
6059adantl 452 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  T  =/=  (/) )
61 rpre 10376 . . . . . . . . . 10  |-  ( E  e.  RR+  ->  E  e.  RR )
6244, 61syl 15 . . . . . . . . 9  |-  ( ph  ->  E  e.  RR )
63 0re 8854 . . . . . . . . . . . . . 14  |-  0  e.  RR
6463, 48pm3.2i 441 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  /\  0  <  4 )
65 ltne 8933 . . . . . . . . . . . . 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 9495 . . . . . . . . . . 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 3614 . . . . . . . 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 9833 . . . . . . . . . 10  |-  3  e.  RR
75 3ne0 9847 . . . . . . . . . 10  |-  3  =/=  0
764, 74, 753pm3.2i 1130 . . . . . . . . 9  |-  ( 1  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )
77 redivcl 9495 . . . . . . . . 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 10377 . . . . . . . . . 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 10377 . . . . . . . . . 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 10523 . . . . . . . 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 9905 . . . . . . . . 9  |-  3  <  4
90 3pos 9846 . . . . . . . . . . 11  |-  0  <  3
9190, 48pm3.2i 441 . . . . . . . . . 10  |-  ( 0  <  3  /\  0  <  4 )
9274, 47ltreci 9683 . . . . . . . . . 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 8928 . . . . . 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 27914 . . 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 1609 . . . . . 6  |-  F/ t  f  e.  A
10317, 102nfan 1783 . . . . 5  |-  F/ t ( ph  /\  f  e.  A )
104 xrmin1 10522 . . . . . . . . 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 3188 . . . . . . . . . . . . . . . 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 27799 . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . 13  |-  ( ( f : T --> RR  /\  t  e.  T )  ->  ( f `  t
)  e.  RR )
116 recn 8843 . . . . . . . . . . . . 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 27799 . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . 13  |-  ( ( F : T --> RR  /\  t  e.  T )  ->  ( F `  t
)  e.  RR )
122 recn 8843 . . . . . . . . . . . . 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 9067 . . . . . . . . . . 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 11779 . . . . . . . . . 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 9325 . . . . . . . . . . . . . . 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 8929 . . . . . . . 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 2637 . . . 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 2627 . . . 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 2668 . 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 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   ifcif 3578   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   3c3 9812   4c4 9813   RR+crp 10370   (,)cioo 10672   abscabs 11735   topGenctg 13358    Cn ccn 16970   Compccmp 17129
This theorem is referenced by:  stowei  27916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cn 16973  df-cnp 16974  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903
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