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Theorem stoweid 27482
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 2734 . . . . . 6  |-  ( ph  ->  A. x  e.  RR  ( t  e.  T  |->  x )  e.  A
)
4 1re 9025 . . . . . 6  |-  1  e.  RR
5 id 20 . . . . . . . . 9  |-  ( x  =  1  ->  x  =  1 )
65mpteq2dv 4239 . . . . . . . 8  |-  ( x  =  1  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  1 ) )
76eleq1d 2455 . . . . . . 7  |-  ( x  =  1  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  1 )  e.  A ) )
87rspccv 2994 . . . . . 6  |-  ( A. x  e.  RR  (
t  e.  T  |->  x )  e.  A  -> 
( 1  e.  RR  ->  ( t  e.  T  |->  1 )  e.  A
) )
93, 4, 8ee10 1382 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
109adantr 452 . . . 4  |-  ( (
ph  /\  T  =  (/) )  ->  ( t  e.  T  |->  1 )  e.  A )
111, 10stoweidlem9 27428 . . 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 1626 . . . . 5  |-  F/ f
ph
14 nfv 1626 . . . . 5  |-  F/ f  -.  T  =  (/)
1513, 14nfan 1836 . . . 4  |-  F/ f ( ph  /\  -.  T  =  (/) )
16 stoweid.2 . . . . 5  |-  F/ t
ph
17 nfv 1626 . . . . 5  |-  F/ t  -.  T  =  (/)
1816, 17nfan 1836 . . . 4  |-  F/ t ( ph  /\  -.  T  =  (/) )
19 eqid 2389 . . . 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 10007 . . . . . . . . 9  |-  4  e.  RR
38 4pos 10020 . . . . . . . . 9  |-  0  <  4
3937, 38elrpii 10549 . . . . . . . 8  |-  4  e.  RR+
4039a1i 11 . . . . . . 7  |-  ( ph  ->  4  e.  RR+ )
4140rpreccld 10592 . . . . . 6  |-  ( ph  ->  ( 1  /  4
)  e.  RR+ )
42 ifcl 3720 . . . . . 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 2554 . . . . . 6  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
4645biimpri 198 . . . . 5  |-  ( -.  T  =  (/)  ->  T  =/=  (/) )
4746adantl 453 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  T  =/=  (/) )
4836rpred 10582 . . . . . . 7  |-  ( ph  ->  E  e.  RR )
49 0re 9026 . . . . . . . . . 10  |-  0  e.  RR
5049, 38gtneii 9118 . . . . . . . . 9  |-  4  =/=  0
5137, 50rereccli 9713 . . . . . . . 8  |-  ( 1  /  4 )  e.  RR
5251a1i 11 . . . . . . 7  |-  ( ph  ->  ( 1  /  4
)  e.  RR )
53 ifcl 3720 . . . . . . 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 10005 . . . . . . . 8  |-  3  e.  RR
56 3ne0 10019 . . . . . . . 8  |-  3  =/=  0
5755, 56rereccli 9713 . . . . . . 7  |-  ( 1  /  3 )  e.  RR
5857a1i 11 . . . . . 6  |-  ( ph  ->  ( 1  /  3
)  e.  RR )
5936rpxrd 10583 . . . . . . 7  |-  ( ph  ->  E  e.  RR* )
6041rpxrd 10583 . . . . . . 7  |-  ( ph  ->  ( 1  /  4
)  e.  RR* )
61 xrmin2 10700 . . . . . . 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 10079 . . . . . . . 8  |-  3  <  4
64 3pos 10018 . . . . . . . . 9  |-  0  <  3
6555, 37, 64, 38ltrecii 9861 . . . . . . . 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 9162 . . . . 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 27481 . . 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 1626 . . . . 5  |-  F/ t  f  e.  A
7316, 72nfan 1836 . . . 4  |-  F/ t ( ph  /\  f  e.  A )
74 xrmin1 10699 . . . . . . 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 3294 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f  e.  C )
8020, 21, 24, 79fcnre 27366 . . . . . . . . . 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 5809 . . . . . . . . 9  |-  ( ( f : T --> RR  /\  t  e.  T )  ->  ( f `  t
)  e.  RR )
84 recn 9015 . . . . . . . . 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 27366 . . . . . . . . . 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 5809 . . . . . . . . 9  |-  ( ( F : T --> RR  /\  t  e.  T )  ->  ( F `  t
)  e.  RR )
90 recn 9015 . . . . . . . . 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 9345 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
( f `  t
)  -  ( F `
 t ) )  e.  CC )
9392abscld 12167 . . . . . 6  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  e.  RR )
9437, 38gt0ne0ii 9497 . . . . . . . . . 10  |-  4  =/=  0
954, 37, 943pm3.2i 1132 . . . . . . . . 9  |-  ( 1  e.  RR  /\  4  e.  RR  /\  4  =/=  0 )
96 redivcl 9667 . . . . . . . . 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 9101 . . . . . 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 2728 . . 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 2763 . 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 1550    = wceq 1649    e. wcel 1717   F/_wnfc 2512    =/= wne 2552   A.wral 2651   E.wrex 2652    C_ wss 3265   (/)c0 3573   ifcif 3684   U.cuni 3959   class class class wbr 4155    e. cmpt 4209   `'ccnv 4819   ran crn 4821   -->wf 5392   ` cfv 5396  (class class class)co 6022   supcsup 7382   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930   RR*cxr 9054    < clt 9055    <_ cle 9056    - cmin 9225    / cdiv 9611   3c3 9984   4c4 9985   RR+crp 10546   (,)cioo 10850   abscabs 11968   topGenctg 13594    Cn ccn 17212   Compccmp 17373
This theorem is referenced by:  stowei  27483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004  ax-mulf 9005
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-oi 7414  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-ioo 10854  df-ioc 10855  df-ico 10856  df-icc 10857  df-fz 10978  df-fzo 11068  df-fl 11131  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-rlim 12212  df-sum 12409  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-hom 13482  df-cco 13483  df-rest 13579  df-topn 13580  df-topgen 13596  df-pt 13597  df-prds 13600  df-xrs 13655  df-0g 13656  df-gsum 13657  df-qtop 13662  df-imas 13663  df-xps 13665  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-submnd 14668  df-mulg 14744  df-cntz 15045  df-cmn 15343  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cld 17008  df-cn 17215  df-cnp 17216  df-cmp 17374  df-tx 17517  df-hmeo 17710  df-xms 18261  df-ms 18262  df-tms 18263
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