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Theorem stoweid 27779
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 448 . . . 4  |-  ( (
ph  /\  T  =  (/) )  ->  T  =  (/) )
2 stoweid.10 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
32ralrimiva 2781 . . . . . 6  |-  ( ph  ->  A. x  e.  RR  ( t  e.  T  |->  x )  e.  A
)
4 1re 9082 . . . . . 6  |-  1  e.  RR
5 id 20 . . . . . . . . 9  |-  ( x  =  1  ->  x  =  1 )
65mpteq2dv 4288 . . . . . . . 8  |-  ( x  =  1  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  1 ) )
76eleq1d 2501 . . . . . . 7  |-  ( x  =  1  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  1 )  e.  A ) )
87rspccv 3041 . . . . . 6  |-  ( A. x  e.  RR  (
t  e.  T  |->  x )  e.  A  -> 
( 1  e.  RR  ->  ( t  e.  T  |->  1 )  e.  A
) )
93, 4, 8ee10 1385 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
109adantr 452 . . . 4  |-  ( (
ph  /\  T  =  (/) )  ->  ( t  e.  T  |->  1 )  e.  A )
111, 10stoweidlem9 27725 . . 3  |-  ( (
ph  /\  T  =  (/) )  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) ) )
12 stoweid.1 . . . 4  |-  F/_ t F
13 nfv 1629 . . . . 5  |-  F/ f
ph
14 nfv 1629 . . . . 5  |-  F/ f  -.  T  =  (/)
1513, 14nfan 1846 . . . 4  |-  F/ f ( ph  /\  -.  T  =  (/) )
16 stoweid.2 . . . . 5  |-  F/ t
ph
17 nfv 1629 . . . . 5  |-  F/ t  -.  T  =  (/)
1816, 17nfan 1846 . . . 4  |-  F/ t ( ph  /\  -.  T  =  (/) )
19 eqid 2435 . . . 4  |-  ( t  e.  T  |->  ( ( F `  t )  -  sup ( ran 
F ,  RR ,  `'  <  ) ) )  =  ( t  e.  T  |->  ( ( F `
 t )  -  sup ( ran  F ,  RR ,  `'  <  ) ) )
20 stoweid.3 . . . 4  |-  K  =  ( topGen `  ran  (,) )
21 stoweid.5 . . . 4  |-  T  = 
U. J
22 stoweid.4 . . . . 5  |-  ( ph  ->  J  e.  Comp )
2322adantr 452 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  J  e.  Comp )
24 stoweid.6 . . . 4  |-  C  =  ( J  Cn  K
)
25 stoweid.7 . . . . 5  |-  ( ph  ->  A  C_  C )
2625adantr 452 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  A  C_  C )
27 stoweid.8 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
28273adant1r 1177 . . . 4  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A
)
29 stoweid.9 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
30293adant1r 1177 . . . 4  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) )  e.  A
)
312adantlr 696 . . . 4  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A
)
32 stoweid.11 . . . . 5  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. h  e.  A  ( h `  r
)  =/=  ( h `
 t ) )
3332adantlr 696 . . . 4  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t
) )  ->  E. h  e.  A  ( h `  r )  =/=  (
h `  t )
)
34 stoweid.12 . . . . 5  |-  ( ph  ->  F  e.  C )
3534adantr 452 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  F  e.  C )
36 stoweid.13 . . . . . 6  |-  ( ph  ->  E  e.  RR+ )
37 4re 10065 . . . . . . . . 9  |-  4  e.  RR
38 4pos 10078 . . . . . . . . 9  |-  0  <  4
3937, 38elrpii 10607 . . . . . . . 8  |-  4  e.  RR+
4039a1i 11 . . . . . . 7  |-  ( ph  ->  4  e.  RR+ )
4140rpreccld 10650 . . . . . 6  |-  ( ph  ->  ( 1  /  4
)  e.  RR+ )
42 ifcl 3767 . . . . . 6  |-  ( ( E  e.  RR+  /\  (
1  /  4 )  e.  RR+ )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR+ )
4336, 41, 42syl2anc 643 . . . . 5  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR+ )
4443adantr 452 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR+ )
45 df-ne 2600 . . . . . 6  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
4645biimpri 198 . . . . 5  |-  ( -.  T  =  (/)  ->  T  =/=  (/) )
4746adantl 453 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  T  =/=  (/) )
4836rpred 10640 . . . . . . 7  |-  ( ph  ->  E  e.  RR )
49 0re 9083 . . . . . . . . . 10  |-  0  e.  RR
5049, 38gtneii 9177 . . . . . . . . 9  |-  4  =/=  0
5137, 50rereccli 9771 . . . . . . . 8  |-  ( 1  /  4 )  e.  RR
5251a1i 11 . . . . . . 7  |-  ( ph  ->  ( 1  /  4
)  e.  RR )
53 ifcl 3767 . . . . . . 7  |-  ( ( E  e.  RR  /\  ( 1  /  4
)  e.  RR )  ->  if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) )  e.  RR )
5448, 52, 53syl2anc 643 . . . . . 6  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR )
55 3re 10063 . . . . . . . 8  |-  3  e.  RR
56 3ne0 10077 . . . . . . . 8  |-  3  =/=  0
5755, 56rereccli 9771 . . . . . . 7  |-  ( 1  /  3 )  e.  RR
5857a1i 11 . . . . . 6  |-  ( ph  ->  ( 1  /  3
)  e.  RR )
5936rpxrd 10641 . . . . . . 7  |-  ( ph  ->  E  e.  RR* )
6041rpxrd 10641 . . . . . . 7  |-  ( ph  ->  ( 1  /  4
)  e.  RR* )
61 xrmin2 10758 . . . . . . 7  |-  ( ( E  e.  RR*  /\  (
1  /  4 )  e.  RR* )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  ( 1  / 
4 ) )
6259, 60, 61syl2anc 643 . . . . . 6  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  <_  (
1  /  4 ) )
63 3lt4 10137 . . . . . . . 8  |-  3  <  4
64 3pos 10076 . . . . . . . . 9  |-  0  <  3
6555, 37, 64, 38ltrecii 9919 . . . . . . . 8  |-  ( 3  <  4  <->  ( 1  /  4 )  < 
( 1  /  3
) )
6663, 65mpbi 200 . . . . . . 7  |-  ( 1  /  4 )  < 
( 1  /  3
)
6766a1i 11 . . . . . 6  |-  ( ph  ->  ( 1  /  4
)  <  ( 1  /  3 ) )
6854, 52, 58, 62, 67lelttrd 9220 . . . . 5  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  <  (
1  /  3 ) )
6968adantr 452 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <  ( 1  / 
3 ) )
7012, 15, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 33, 35, 44, 47, 69stoweidlem62 27778 . . 3  |-  ( (
ph  /\  -.  T  =  (/) )  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) ) )
7111, 70pm2.61dan 767 . 2  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) ) )
72 nfv 1629 . . . . 5  |-  F/ t  f  e.  A
7316, 72nfan 1846 . . . 4  |-  F/ t ( ph  /\  f  e.  A )
74 xrmin1 10757 . . . . . . 7  |-  ( ( E  e.  RR*  /\  (
1  /  4 )  e.  RR* )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  E )
7559, 60, 74syl2anc 643 . . . . . 6  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  <_  E
)
7675ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  E )
7725ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  A  C_  C )
78 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f  e.  A )
7977, 78sseldd 3341 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f  e.  C )
8020, 21, 24, 79fcnre 27663 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f : T --> RR )
81 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  t  e.  T )
8280, 81jca 519 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
f : T --> RR  /\  t  e.  T )
)
83 ffvelrn 5860 . . . . . . . . 9  |-  ( ( f : T --> RR  /\  t  e.  T )  ->  ( f `  t
)  e.  RR )
84 recn 9072 . . . . . . . . 9  |-  ( ( f `  t )  e.  RR  ->  (
f `  t )  e.  CC )
8582, 83, 843syl 19 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
f `  t )  e.  CC )
8634ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  F  e.  C )
8720, 21, 24, 86fcnre 27663 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  F : T --> RR )
8887, 81jca 519 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( F : T --> RR  /\  t  e.  T )
)
89 ffvelrn 5860 . . . . . . . . 9  |-  ( ( F : T --> RR  /\  t  e.  T )  ->  ( F `  t
)  e.  RR )
90 recn 9072 . . . . . . . . 9  |-  ( ( F `  t )  e.  RR  ->  ( F `  t )  e.  CC )
9188, 89, 903syl 19 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
9285, 91subcld 9403 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
( f `  t
)  -  ( F `
 t ) )  e.  CC )
9392abscld 12230 . . . . . 6  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  e.  RR )
9437, 38gt0ne0ii 9555 . . . . . . . . . 10  |-  4  =/=  0
954, 37, 943pm3.2i 1132 . . . . . . . . 9  |-  ( 1  e.  RR  /\  4  e.  RR  /\  4  =/=  0 )
96 redivcl 9725 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  4  e.  RR  /\  4  =/=  0 )  ->  (
1  /  4 )  e.  RR )
9795, 96mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( 1  /  4
)  e.  RR )
9848, 97, 53syl2anc 643 . . . . . . 7  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR )
9998ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR )
10048ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  E  e.  RR )
101 ltletr 9158 . . . . . 6  |-  ( ( ( 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
) )
10293, 99, 100, 101syl3anc 1184 . . . . 5  |-  ( ( ( 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
) )
10376, 102mpan2d 656 . . . 4  |-  ( ( ( 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 ) )
10473, 103ralimdaa 2775 . . 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 ) )
105104reximdva 2810 . 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 ) )
10671, 105mpd 15 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    /\ wa 359    /\ w3a 936   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558    =/= wne 2598   A.wral 2697   E.wrex 2698    C_ wss 3312   (/)c0 3620   ifcif 3731   U.cuni 4007   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987   RR*cxr 9111    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   3c3 10042   4c4 10043   RR+crp 10604   (,)cioo 10908   abscabs 12031   topGenctg 13657    Cn ccn 17280   Compccmp 17441
This theorem is referenced by:  stowei  27780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-cn 17283  df-cnp 17284  df-cmp 17442  df-tx 17586  df-hmeo 17779  df-xms 18342  df-ms 18343  df-tms 18344
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