Theorem sitmcl 30413

Description: Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)

Hypotheses

Ref	Expression
sitmcl.0	|- ( ph -> W e. Mnd )
sitmcl.1	|- ( ph -> W e. *MetSp )
sitmcl.2	|- ( ph -> M e. U. ran measures )
sitmcl.3	|- ( ph -> F e. dom ( Wsitg M ) )
sitmcl.4	|- ( ph -> G e. dom ( Wsitg M ) )

Assertion

Ref	Expression
sitmcl	|- ( ph -> ( F ( Wsitm M ) G ) e. ( 0 [,] +oo ) )

Proof of Theorem sitmcl

Dummy variables x m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( dist ` W ) = ( dist ` W )
2		sitmcl.1	. . 3 |- ( ph -> W e. *MetSp )
3		sitmcl.2	. . 3 |- ( ph -> M e. U. ran measures )
4		sitmcl.3	. . 3 |- ( ph -> F e. dom ( Wsitg M ) )
5		sitmcl.4	. . 3 |- ( ph -> G e. dom ( Wsitg M ) )
6	1, 2, 3, 4, 5	sitmfval 30412	. 2 |- ( ph -> ( F ( Wsitm M ) G ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( F oF ( dist ` W ) G ) ) )
7		xrge0base 29685	. . 3 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
8		xrge0topn 29989	. . . 4 |- ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )
9	8	eqcomi 2631	. . 3 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) )
10		eqid 2622	. . 3 |- (sigaGen ` ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) = (sigaGen ` ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) )
11		xrge00 29686	. . 3 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
12		ovex 6678	. . . 4 |- ( 0 [,] +oo ) e. _V
13		eqid 2622	. . . . 5 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
14		ax-xrsvsca 29674	. . . . 5 |- xe = ( .s ` RR*s )
15	13, 14	ressvsca 16032	. . . 4 |- ( ( 0 [,] +oo ) e. _V -> xe = ( .s ` ( RR*s ↾s ( 0 [,] +oo ) ) ) )
16	12, 15	ax-mp 5	. . 3 |- xe = ( .s ` ( RR*s ↾s ( 0 [,] +oo ) ) )
17		ax-xrssca 29673	. . . . . 6 |- RRfld = (Scalar ` RR*s )
18	13, 17	resssca 16031	. . . . 5 |- ( ( 0 [,] +oo ) e. _V -> RRfld = (Scalar ` ( RR*s ↾s ( 0 [,] +oo ) ) ) )
19	12, 18	ax-mp 5	. . . 4 |- RRfld = (Scalar ` ( RR*s ↾s ( 0 [,] +oo ) ) )
20	19	fveq2i 6194	. . 3 |- (RRHom ` RRfld ) = (RRHom ` (Scalar ` ( RR*s ↾s ( 0 [,] +oo ) ) ) )
21		ovexd 6680	. . 3 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. _V )
22		eqid 2622	. . . . . . 7 |- ( Base ` W ) = ( Base ` W )
23		eqid 2622	. . . . . . 7 |- ( TopOpen ` W ) = ( TopOpen ` W )
24		eqid 2622	. . . . . . 7 |- (sigaGen ` ( TopOpen ` W ) ) = (sigaGen ` ( TopOpen ` W ) )
25		eqid 2622	. . . . . . 7 |- ( 0g ` W ) = ( 0g ` W )
26		eqid 2622	. . . . . . 7 |- ( .s ` W ) = ( .s ` W )
27		eqid 2622	. . . . . . 7 |- (RRHom ` (Scalar ` W ) ) = (RRHom ` (Scalar ` W ) )
28	22, 23, 24, 25, 26, 27, 2, 3, 4	sibff 30398	. . . . . 6 |- ( ph -> F : U. dom M --> U. ( TopOpen ` W ) )
29		xmstps 22258	. . . . . . . 8 |- ( W e. *MetSp -> W e. TopSp )
30	22, 23	tpsuni 20740	. . . . . . . 8 |- ( W e. TopSp -> ( Base ` W ) = U. ( TopOpen ` W ) )
31	2, 29, 30	3syl 18	. . . . . . 7 |- ( ph -> ( Base ` W ) = U. ( TopOpen ` W ) )
32		feq3 6028	. . . . . . 7 |- ( ( Base ` W ) = U. ( TopOpen ` W ) -> ( F : U. dom M --> ( Base ` W ) <-> F : U. dom M --> U. ( TopOpen ` W ) ) )
33	31, 32	syl 17	. . . . . 6 |- ( ph -> ( F : U. dom M --> ( Base ` W ) <-> F : U. dom M --> U. ( TopOpen ` W ) ) )
34	28, 33	mpbird 247	. . . . 5 |- ( ph -> F : U. dom M --> ( Base ` W ) )
35	22, 23, 24, 25, 26, 27, 2, 3, 5	sibff 30398	. . . . . 6 |- ( ph -> G : U. dom M --> U. ( TopOpen ` W ) )
36		feq3 6028	. . . . . . 7 |- ( ( Base ` W ) = U. ( TopOpen ` W ) -> ( G : U. dom M --> ( Base ` W ) <-> G : U. dom M --> U. ( TopOpen ` W ) ) )
37	31, 36	syl 17	. . . . . 6 |- ( ph -> ( G : U. dom M --> ( Base ` W ) <-> G : U. dom M --> U. ( TopOpen ` W ) ) )
38	35, 37	mpbird 247	. . . . 5 |- ( ph -> G : U. dom M --> ( Base ` W ) )
39		dmexg 7097	. . . . . 6 |- ( M e. U. ran measures -> dom M e. _V )
40		uniexg 6955	. . . . . 6 |- ( dom M e. _V -> U. dom M e. _V )
41	3, 39, 40	3syl 18	. . . . 5 |- ( ph -> U. dom M e. _V )
42	34, 38, 41	ofresid 29444	. . . 4 |- ( ph -> ( F oF ( dist ` W ) G ) = ( F oF ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) G ) )
43	2, 29	syl 17	. . . . 5 |- ( ph -> W e. TopSp )
44		eqid 2622	. . . . . . . 8 |- ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) = ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) )
45	22, 44	xmsxmet 22261	. . . . . . 7 |- ( W e. *MetSp -> ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) e. ( *Met ` ( Base ` W ) ) )
46		xmetpsmet 22153	. . . . . . 7 |- ( ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) e. ( *Met ` ( Base ` W ) ) -> ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) e. (PsMet ` ( Base ` W ) ) )
47	2, 45, 46	3syl 18	. . . . . 6 |- ( ph -> ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) e. (PsMet ` ( Base ` W ) ) )
48		psmetxrge0 22118	. . . . . 6 |- ( ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) e. (PsMet ` ( Base ` W ) ) -> ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) : ( ( Base ` W ) X. ( Base ` W ) ) --> ( 0 [,] +oo ) )
49	47, 48	syl 17	. . . . 5 |- ( ph -> ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) : ( ( Base ` W ) X. ( Base ` W ) ) --> ( 0 [,] +oo ) )
50		xrge0tps 29988	. . . . . 6 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. TopSp
51	50	a1i 11	. . . . 5 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. TopSp )
52	23, 22, 44	xmstopn 22256	. . . . . . . 8 |- ( W e. *MetSp -> ( TopOpen ` W ) = ( MetOpen ` ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) ) )
53	2, 52	syl 17	. . . . . . 7 |- ( ph -> ( TopOpen ` W ) = ( MetOpen ` ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) ) )
54		eqid 2622	. . . . . . . . 9 |- ( MetOpen ` ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) ) = ( MetOpen ` ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) )
55	54	methaus 22325	. . . . . . . 8 |- ( ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) e. ( *Met ` ( Base ` W ) ) -> ( MetOpen ` ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) ) e. Haus )
56	2, 45, 55	3syl 18	. . . . . . 7 |- ( ph -> ( MetOpen ` ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) ) e. Haus )
57	53, 56	eqeltrd 2701	. . . . . 6 |- ( ph -> ( TopOpen ` W ) e. Haus )
58		haust1 21156	. . . . . 6 |- ( ( TopOpen ` W ) e. Haus -> ( TopOpen ` W ) e. Fre )
59	57, 58	syl 17	. . . . 5 |- ( ph -> ( TopOpen ` W ) e. Fre )
60	2, 45	syl 17	. . . . . . 7 |- ( ph -> ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) e. ( *Met ` ( Base ` W ) ) )
61		sitmcl.0	. . . . . . . 8 |- ( ph -> W e. Mnd )
62	22, 25	mndidcl 17308	. . . . . . . 8 |- ( W e. Mnd -> ( 0g ` W ) e. ( Base ` W ) )
63	61, 62	syl 17	. . . . . . 7 |- ( ph -> ( 0g ` W ) e. ( Base ` W ) )
64		xmet0 22147	. . . . . . 7 |- ( ( ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) e. ( *Met ` ( Base ` W ) ) /\ ( 0g ` W ) e. ( Base ` W ) ) -> ( ( 0g ` W ) ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) ( 0g ` W ) ) = 0 )
65	60, 63, 64	syl2anc 693	. . . . . 6 |- ( ph -> ( ( 0g ` W ) ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) ( 0g ` W ) ) = 0 )
66	65, 11	syl6eq 2672	. . . . 5 |- ( ph -> ( ( 0g ` W ) ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) ( 0g ` W ) ) = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) ) )
67	22, 23, 24, 25, 26, 27, 2, 3, 4, 7, 43, 49, 5, 51, 59, 66	sibfof 30402	. . . 4 |- ( ph -> ( F oF ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) G ) e. dom ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) )
68	42, 67	eqeltrd 2701	. . 3 |- ( ph -> ( F oF ( dist ` W ) G ) e. dom ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) )
69		rebase 19952	. . . . 5 |- RR = ( Base ` RRfld )
70	69, 69	xpeq12i 5137	. . . 4 |- ( RR X. RR ) = ( ( Base ` RRfld ) X. ( Base ` RRfld ) )
71	70	reseq2i 5393	. . 3 |- ( ( dist ` RRfld ) |` ( RR X. RR ) ) = ( ( dist ` RRfld ) |` ( ( Base ` RRfld ) X. ( Base ` RRfld ) ) )
72		xrge0cmn 19788	. . . 4 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
73	72	a1i 11	. . 3 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
74		rerrext 30053	. . . . 5 |- RRfld e. ℝExt
75	19, 74	eqeltrri 2698	. . . 4 |- (Scalar ` ( RR*s ↾s ( 0 [,] +oo ) ) ) e. ℝExt
76	75	a1i 11	. . 3 |- ( ph -> (Scalar ` ( RR*s ↾s ( 0 [,] +oo ) ) ) e. ℝExt )
77		rrhre 30065	. . . . . . . . 9 |- (RRHom ` RRfld ) = ( _I |` RR )
78	77	imaeq1i 5463	. . . . . . . 8 |- ( (RRHom ` RRfld ) " ( 0 [,) +oo ) ) = ( ( _I |` RR ) " ( 0 [,) +oo ) )
79		0re 10040	. . . . . . . . . 10 |- 0 e. RR
80		pnfxr 10092	. . . . . . . . . 10 |- +oo e. RR*
81		icossre 12254	. . . . . . . . . 10 |- ( ( 0 e. RR /\ +oo e. RR* ) -> ( 0 [,) +oo ) C_ RR )
82	79, 80, 81	mp2an 708	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ RR
83		resiima 5480	. . . . . . . . 9 |- ( ( 0 [,) +oo ) C_ RR -> ( ( _I |` RR ) " ( 0 [,) +oo ) ) = ( 0 [,) +oo ) )
84	82, 83	ax-mp 5	. . . . . . . 8 |- ( ( _I |` RR ) " ( 0 [,) +oo ) ) = ( 0 [,) +oo )
85	78, 84	eqtri 2644	. . . . . . 7 |- ( (RRHom ` RRfld ) " ( 0 [,) +oo ) ) = ( 0 [,) +oo )
86		icossicc 12260	. . . . . . 7 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
87	85, 86	eqsstri 3635	. . . . . 6 |- ( (RRHom ` RRfld ) " ( 0 [,) +oo ) ) C_ ( 0 [,] +oo )
88	87	sseli 3599	. . . . 5 |- ( m e. ( (RRHom ` RRfld ) " ( 0 [,) +oo ) ) -> m e. ( 0 [,] +oo ) )
89	88	3ad2ant2 1083	. . . 4 |- ( ( ph /\ m e. ( (RRHom ` RRfld ) " ( 0 [,) +oo ) ) /\ x e. ( 0 [,] +oo ) ) -> m e. ( 0 [,] +oo ) )
90		simp3 1063	. . . 4 |- ( ( ph /\ m e. ( (RRHom ` RRfld ) " ( 0 [,) +oo ) ) /\ x e. ( 0 [,] +oo ) ) -> x e. ( 0 [,] +oo ) )
91		ge0xmulcl 12287	. . . 4 |- ( ( m e. ( 0 [,] +oo ) /\ x e. ( 0 [,] +oo ) ) -> ( m xe x ) e. ( 0 [,] +oo ) )
92	89, 90, 91	syl2anc 693	. . 3 |- ( ( ph /\ m e. ( (RRHom ` RRfld ) " ( 0 [,) +oo ) ) /\ x e. ( 0 [,] +oo ) ) -> ( m xe x ) e. ( 0 [,] +oo ) )
93	7, 9, 10, 11, 16, 20, 21, 3, 68, 19, 71, 51, 73, 76, 92	sitgclg 30404	. 2 |- ( ph -> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( F oF ( dist ` W ) G ) ) e. ( 0 [,] +oo ) )
94	6, 93	eqeltrd 2701	1 |- ( ph -> ( F ( Wsitm M ) G ) e. ( 0 [,] +oo ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   U.cuni 4436   _I cid 5023   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   xecxmu 11945   [,)cico 12177   [,]cicc 12178   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   .scvsca 15945   distcds 15950   ↾_t crest 16081   TopOpenctopn 16082   0gc0g 16100  ordTopcordt 16159   RR*scxrs 16160   Mndcmnd 17294  CMndccmn 18193  PsMetcpsmet 19730   *Metcxmt 19731   MetOpencmopn 19736  RRfldcrefld 19950   TopSpctps 20736   Frect1 21111   Hauscha 21112   *MetSpcxme 22122  RRHomcrrh 30037   ℝExt crrext 30038  sigaGencsigagen 30201  measurescmeas 30258  sitmcsitm 30390  sitgcsitg 30391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016  ax-xrssca 29673  ax-xrsvsca 29674

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444  df-gz 15634  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-ordt 16161  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-preset 16928  df-poset 16946  df-plt 16958  df-toset 17034  df-ps 17200  df-tsr 17201  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-od 17948  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-field 18750  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-nzr 19258  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-metu 19745  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zlm 19853  df-chr 19854  df-refld 19951  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-reg 21120  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-fcls 21745  df-cnext 21864  df-tmd 21876  df-tgp 21877  df-tsms 21930  df-trg 21963  df-ust 22004  df-utop 22035  df-uss 22060  df-usp 22061  df-ucn 22080  df-cfilu 22091  df-cusp 22102  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-ii 22680  df-cncf 22681  df-cfil 23053  df-cmet 23055  df-cms 23132  df-limc 23630  df-dv 23631  df-log 24303  df-omnd 29699  df-ogrp 29700  df-orng 29797  df-ofld 29798  df-qqh 30017  df-rrh 30039  df-rrext 30043  df-esum 30090  df-siga 30171  df-sigagen 30202  df-meas 30259  df-mbfm 30313  df-sitg 30392  df-sitm 30393

This theorem is referenced by:  sitmf  30414