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Theorem stoweid 29829
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 461 . . . 4  |-  ( (
ph  /\  T  =  (/) )  ->  T  =  (/) )
2 stoweid.10 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
32ralrimiva 2794 . . . . . 6  |-  ( ph  ->  A. x  e.  RR  ( t  e.  T  |->  x )  e.  A
)
4 1re 9377 . . . . . 6  |-  1  e.  RR
5 id 22 . . . . . . . . 9  |-  ( x  =  1  ->  x  =  1 )
65mpteq2dv 4374 . . . . . . . 8  |-  ( x  =  1  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  1 ) )
76eleq1d 2504 . . . . . . 7  |-  ( x  =  1  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  1 )  e.  A ) )
87rspccv 3065 . . . . . 6  |-  ( A. x  e.  RR  (
t  e.  T  |->  x )  e.  A  -> 
( 1  e.  RR  ->  ( t  e.  T  |->  1 )  e.  A
) )
93, 4, 8mpisyl 18 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
109adantr 465 . . . 4  |-  ( (
ph  /\  T  =  (/) )  ->  ( t  e.  T  |->  1 )  e.  A )
111, 10stoweidlem9 29775 . . 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 1673 . . . . 5  |-  F/ f
ph
14 nfv 1673 . . . . 5  |-  F/ f  -.  T  =  (/)
1513, 14nfan 1861 . . . 4  |-  F/ f ( ph  /\  -.  T  =  (/) )
16 stoweid.2 . . . . 5  |-  F/ t
ph
17 nfv 1673 . . . . 5  |-  F/ t  -.  T  =  (/)
1816, 17nfan 1861 . . . 4  |-  F/ t ( ph  /\  -.  T  =  (/) )
19 eqid 2438 . . . 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 465 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  J  e.  Comp )
24 stoweid.6 . . . 4  |-  C  =  ( J  Cn  K
)
25 stoweid.7 . . . . 5  |-  ( ph  ->  A  C_  C )
2625adantr 465 . . . 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 1211 . . . 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 1211 . . . 4  |-  ( ( ( ph  /\  -.  T  =  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) )  e.  A
)
312adantlr 714 . . . 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 714 . . . 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 465 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  F  e.  C )
36 stoweid.13 . . . . . 6  |-  ( ph  ->  E  e.  RR+ )
37 4re 10390 . . . . . . . . 9  |-  4  e.  RR
38 4pos 10409 . . . . . . . . 9  |-  0  <  4
3937, 38elrpii 10986 . . . . . . . 8  |-  4  e.  RR+
4039a1i 11 . . . . . . 7  |-  ( ph  ->  4  e.  RR+ )
4140rpreccld 11029 . . . . . 6  |-  ( ph  ->  ( 1  /  4
)  e.  RR+ )
4236, 41ifcld 3827 . . . . 5  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR+ )
4342adantr 465 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR+ )
44 df-ne 2603 . . . . . 6  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
4544biimpri 206 . . . . 5  |-  ( -.  T  =  (/)  ->  T  =/=  (/) )
4645adantl 466 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  T  =/=  (/) )
4736rpred 11019 . . . . . . 7  |-  ( ph  ->  E  e.  RR )
48 4ne0 10410 . . . . . . . . 9  |-  4  =/=  0
4937, 48rereccli 10088 . . . . . . . 8  |-  ( 1  /  4 )  e.  RR
5049a1i 11 . . . . . . 7  |-  ( ph  ->  ( 1  /  4
)  e.  RR )
5147, 50ifcld 3827 . . . . . 6  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR )
52 3re 10387 . . . . . . . 8  |-  3  e.  RR
53 3ne0 10408 . . . . . . . 8  |-  3  =/=  0
5452, 53rereccli 10088 . . . . . . 7  |-  ( 1  /  3 )  e.  RR
5554a1i 11 . . . . . 6  |-  ( ph  ->  ( 1  /  3
)  e.  RR )
5636rpxrd 11020 . . . . . . 7  |-  ( ph  ->  E  e.  RR* )
5741rpxrd 11020 . . . . . . 7  |-  ( ph  ->  ( 1  /  4
)  e.  RR* )
58 xrmin2 11142 . . . . . . 7  |-  ( ( E  e.  RR*  /\  (
1  /  4 )  e.  RR* )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  ( 1  / 
4 ) )
5956, 57, 58syl2anc 661 . . . . . 6  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  <_  (
1  /  4 ) )
60 3lt4 10483 . . . . . . . 8  |-  3  <  4
61 3pos 10407 . . . . . . . . 9  |-  0  <  3
6252, 37, 61, 38ltrecii 10241 . . . . . . . 8  |-  ( 3  <  4  <->  ( 1  /  4 )  < 
( 1  /  3
) )
6360, 62mpbi 208 . . . . . . 7  |-  ( 1  /  4 )  < 
( 1  /  3
)
6463a1i 11 . . . . . 6  |-  ( ph  ->  ( 1  /  4
)  <  ( 1  /  3 ) )
6551, 50, 55, 59, 64lelttrd 9521 . . . . 5  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  <  (
1  /  3 ) )
6665adantr 465 . . . 4  |-  ( (
ph  /\  -.  T  =  (/) )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <  ( 1  / 
3 ) )
6712, 15, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 33, 35, 43, 46, 66stoweidlem62 29828 . . 3  |-  ( (
ph  /\  -.  T  =  (/) )  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) ) )
6811, 67pm2.61dan 789 . 2  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  if ( E  <_  ( 1  / 
4 ) ,  E ,  ( 1  / 
4 ) ) )
69 nfv 1673 . . . . 5  |-  F/ t  f  e.  A
7016, 69nfan 1861 . . . 4  |-  F/ t ( ph  /\  f  e.  A )
71 xrmin1 11141 . . . . . . 7  |-  ( ( E  e.  RR*  /\  (
1  /  4 )  e.  RR* )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  E )
7256, 57, 71syl2anc 661 . . . . . 6  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  <_  E
)
7372ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  <_  E )
7425ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  A  C_  C )
75 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f  e.  A )
7674, 75sseldd 3352 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f  e.  C )
7720, 21, 24, 76fcnre 29718 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  f : T --> RR )
78 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  t  e.  T )
7977, 78jca 532 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
f : T --> RR  /\  t  e.  T )
)
80 ffvelrn 5836 . . . . . . . . 9  |-  ( ( f : T --> RR  /\  t  e.  T )  ->  ( f `  t
)  e.  RR )
81 recn 9364 . . . . . . . . 9  |-  ( ( f `  t )  e.  RR  ->  (
f `  t )  e.  CC )
8279, 80, 813syl 20 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
f `  t )  e.  CC )
8334ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  F  e.  C )
8420, 21, 24, 83fcnre 29718 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  F : T --> RR )
8584, 78jca 532 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( F : T --> RR  /\  t  e.  T )
)
86 ffvelrn 5836 . . . . . . . . 9  |-  ( ( F : T --> RR  /\  t  e.  T )  ->  ( F `  t
)  e.  RR )
87 recn 9364 . . . . . . . . 9  |-  ( ( F `  t )  e.  RR  ->  ( F `  t )  e.  CC )
8885, 86, 873syl 20 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
8982, 88subcld 9711 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  (
( f `  t
)  -  ( F `
 t ) )  e.  CC )
9089abscld 12914 . . . . . 6  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  ( abs `  ( ( f `
 t )  -  ( F `  t ) ) )  e.  RR )
914, 37, 483pm3.2i 1166 . . . . . . . . 9  |-  ( 1  e.  RR  /\  4  e.  RR  /\  4  =/=  0 )
92 redivcl 10042 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  4  e.  RR  /\  4  =/=  0 )  ->  (
1  /  4 )  e.  RR )
9391, 92mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( 1  /  4
)  e.  RR )
9447, 93ifcld 3827 . . . . . . 7  |-  ( ph  ->  if ( E  <_ 
( 1  /  4
) ,  E , 
( 1  /  4
) )  e.  RR )
9594ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  if ( E  <_  ( 1  /  4 ) ,  E ,  ( 1  /  4 ) )  e.  RR )
9647ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  f  e.  A )  /\  t  e.  T )  ->  E  e.  RR )
97 ltletr 9458 . . . . . 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
) )
9890, 95, 96, 97syl3anc 1218 . . . . 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
) )
9973, 98mpan2d 674 . . . 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 ) )
10070, 99ralimdaa 2788 . . 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 ) )
101100reximdva 2823 . 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 ) )
10268, 101mpd 15 1  |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2561    =/= wne 2601   A.wral 2710   E.wrex 2711    C_ wss 3323   (/)c0 3632   ifcif 3786   U.cuni 4086   class class class wbr 4287    e. cmpt 4345   `'ccnv 4834   ran crn 4836   -->wf 5409   ` cfv 5413  (class class class)co 6086   supcsup 7682   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587    / cdiv 9985   3c3 10364   4c4 10365   RR+crp 10983   (,)cioo 11292   abscabs 12715   topGenctg 14368    Cn ccn 18808   Compccmp 18969
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  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  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-cn 18811  df-cnp 18812  df-cmp 18970  df-tx 19115  df-hmeo 19308  df-xms 19875  df-ms 19876  df-tms 19877
This theorem is referenced by:  stowei  29830
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