Theorem stoweid 27679

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

Step	Hyp	Ref	Expression
1		simpr 448	. . . 4 |- ( ( ph /\ T = (/) ) -> T = (/) )
2		stoweid.10	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
3	2	ralrimiva 2749	. . . . . 6 |- ( ph -> A. x e. RR ( t e. T |-> x ) e. A )
4		1re 9046	. . . . . 6 |- 1 e. RR
5		id 20	. . . . . . . . 9 |- ( x = 1 -> x = 1 )
6	5	mpteq2dv 4256	. . . . . . . 8 |- ( x = 1 -> ( t e. T |-> x ) = ( t e. T |-> 1 ) )
7	6	eleq1d 2470	. . . . . . 7 |- ( x = 1 -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> 1 ) e. A ) )
8	7	rspccv 3009	. . . . . 6 |- ( A. x e. RR ( t e. T |-> x ) e. A -> ( 1 e. RR -> ( t e. T |-> 1 ) e. A ) )
9	3, 4, 8	ee10 1382	. . . . 5 |- ( ph -> ( t e. T |-> 1 ) e. A )
10	9	adantr 452	. . . 4 |- ( ( ph /\ T = (/) ) -> ( t e. T |-> 1 ) e. A )
11	1, 10	stoweidlem9 27625	. . 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 = (/)
15	13, 14	nfan 1842	. . . 4 |- F/ f ( ph /\ -. T = (/) )
16		stoweid.2	. . . . 5 |- F/ t ph
17		nfv 1626	. . . . 5 |- F/ t -. T = (/)
18	16, 17	nfan 1842	. . . 4 |- F/ t ( ph /\ -. T = (/) )
19		eqid 2404	. . . 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 )
23	22	adantr 452	. . . 4 |- ( ( ph /\ -. T = (/) ) -> J e. Comp )
24		stoweid.6	. . . 4 |- C = ( J Cn K )
25		stoweid.7	. . . . 5 |- ( ph -> A C_ C )
26	25	adantr 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 )
28	27	3adant1r 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 )
30	29	3adant1r 1177	. . . 4 |- ( ( ( ph /\ -. T = (/) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
31	2	adantlr 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 ) )
33	32	adantlr 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 )
35	34	adantr 452	. . . 4 |- ( ( ph /\ -. T = (/) ) -> F e. C )
36		stoweid.13	. . . . . 6 |- ( ph -> E e. RR+ )
37		4re 10029	. . . . . . . . 9 |- 4 e. RR
38		4pos 10042	. . . . . . . . 9 |- 0 < 4
39	37, 38	elrpii 10571	. . . . . . . 8 |- 4 e. RR+
40	39	a1i 11	. . . . . . 7 |- ( ph -> 4 e. RR+ )
41	40	rpreccld 10614	. . . . . 6 |- ( ph -> ( 1 / 4 ) e. RR+ )
42		ifcl 3735	. . . . . 6 |- ( ( E e. RR+ /\ ( 1 / 4 ) e. RR+ ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
43	36, 41, 42	syl2anc 643	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
44	43	adantr 452	. . . 4 |- ( ( ph /\ -. T = (/) ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
45		df-ne 2569	. . . . . 6 |- ( T =/= (/) <-> -. T = (/) )
46	45	biimpri 198	. . . . 5 |- ( -. T = (/) -> T =/= (/) )
47	46	adantl 453	. . . 4 |- ( ( ph /\ -. T = (/) ) -> T =/= (/) )
48	36	rpred 10604	. . . . . . 7 |- ( ph -> E e. RR )
49		0re 9047	. . . . . . . . . 10 |- 0 e. RR
50	49, 38	gtneii 9141	. . . . . . . . 9 |- 4 =/= 0
51	37, 50	rereccli 9735	. . . . . . . 8 |- ( 1 / 4 ) e. RR
52	51	a1i 11	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR )
53		ifcl 3735	. . . . . . 7 |- ( ( E e. RR /\ ( 1 / 4 ) e. RR ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
54	48, 52, 53	syl2anc 643	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
55		3re 10027	. . . . . . . 8 |- 3 e. RR
56		3ne0 10041	. . . . . . . 8 |- 3 =/= 0
57	55, 56	rereccli 9735	. . . . . . 7 |- ( 1 / 3 ) e. RR
58	57	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 3 ) e. RR )
59	36	rpxrd 10605	. . . . . . 7 |- ( ph -> E e. RR* )
60	41	rpxrd 10605	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR* )
61		xrmin2 10722	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
62	59, 60, 61	syl2anc 643	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
63		3lt4 10101	. . . . . . . 8 |- 3 < 4
64		3pos 10040	. . . . . . . . 9 |- 0 < 3
65	55, 37, 64, 38	ltrecii 9883	. . . . . . . 8 |- ( 3 < 4 <-> ( 1 / 4 ) < ( 1 / 3 ) )
66	63, 65	mpbi 200	. . . . . . 7 |- ( 1 / 4 ) < ( 1 / 3 )
67	66	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 4 ) < ( 1 / 3 ) )
68	54, 52, 58, 62, 67	lelttrd 9184	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) < ( 1 / 3 ) )
69	68	adantr 452	. . . 4 |- ( ( ph /\ -. T = (/) ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) < ( 1 / 3 ) )
70	12, 15, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 33, 35, 44, 47, 69	stoweidlem62 27678	. . 3 |- ( ( ph /\ -. T = (/) ) -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) )
71	11, 70	pm2.61dan 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
73	16, 72	nfan 1842	. . . 4 |- F/ t ( ph /\ f e. A )
74		xrmin1 10721	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
75	59, 60, 74	syl2anc 643	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
76	75	ad2antrr 707	. . . . 5 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
77	25	ad2antrr 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 )
79	77, 78	sseldd 3309	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f e. C )
80	20, 21, 24, 79	fcnre 27563	. . . . . . . . . 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 )
82	80, 81	jca 519	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( f : T --> RR /\ t e. T ) )
83		ffvelrn 5827	. . . . . . . . 9 |- ( ( f : T --> RR /\ t e. T ) -> ( f ` t ) e. RR )
84		recn 9036	. . . . . . . . 9 |- ( ( f ` t ) e. RR -> ( f ` t ) e. CC )
85	82, 83, 84	3syl 19	. . . . . . . 8 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( f ` t ) e. CC )
86	34	ad2antrr 707	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F e. C )
87	20, 21, 24, 86	fcnre 27563	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F : T --> RR )
88	87, 81	jca 519	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( F : T --> RR /\ t e. T ) )
89		ffvelrn 5827	. . . . . . . . 9 |- ( ( F : T --> RR /\ t e. T ) -> ( F ` t ) e. RR )
90		recn 9036	. . . . . . . . 9 |- ( ( F ` t ) e. RR -> ( F ` t ) e. CC )
91	88, 89, 90	3syl 19	. . . . . . . 8 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( F ` t ) e. CC )
92	85, 91	subcld 9367	. . . . . . 7 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( ( f ` t ) - ( F ` t ) ) e. CC )
93	92	abscld 12193	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( abs ` ( ( f ` t ) - ( F ` t ) ) ) e. RR )
94	37, 38	gt0ne0ii 9519	. . . . . . . . . 10 |- 4 =/= 0
95	4, 37, 94	3pm3.2i 1132	. . . . . . . . 9 |- ( 1 e. RR /\ 4 e. RR /\ 4 =/= 0 )
96		redivcl 9689	. . . . . . . . 9 |- ( ( 1 e. RR /\ 4 e. RR /\ 4 =/= 0 ) -> ( 1 / 4 ) e. RR )
97	95, 96	mp1i 12	. . . . . . . 8 |- ( ph -> ( 1 / 4 ) e. RR )
98	48, 97, 53	syl2anc 643	. . . . . . 7 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
99	98	ad2antrr 707	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
100	48	ad2antrr 707	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> E e. RR )
101		ltletr 9122	. . . . . 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 ) )
102	93, 99, 100, 101	syl3anc 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 ) )
103	76, 102	mpan2d 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 ) )
104	73, 103	ralimdaa 2743	. . 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 ) )
105	104	reximdva 2778	. 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 ) )
106	71, 105	mpd 15	1 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   F/wnf 1550   = wceq 1649   e. wcel 1721   F/_wnfc 2527   =/= wne 2567   A.wral 2666   E.wrex 2667   C_ wss 3280   (/)c0 3588   ifcif 3699   U.cuni 3975   class class class wbr 4172   e. cmpt 4226   `'ccnv 4836   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   + caddc 8949   x. cmul 8951   RR*cxr 9075   < clt 9076   <_ cle 9077   - cmin 9247   / cdiv 9633   3c3 10006   4c4 10007   RR+crp 10568   (,)cioo 10872   abscabs 11994   topGenctg 13620   Cn ccn 17242   Compccmp 17403

This theorem is referenced by:  stowei  27680

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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-cn 17245  df-cnp 17246  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305