Theorem sitgclg 30404

Description: Closure of the Bochner integral on simple functions, generic version. See sitgclbn 30405 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.) (Proof shortened by AV, 12-Dec-2019.)

Hypotheses

Ref	Expression
sitgval.b	|- B = ( Base ` W )
sitgval.j	|- J = ( TopOpen ` W )
sitgval.s	|- S = (sigaGen ` J )
sitgval.0	|- .0. = ( 0g ` W )
sitgval.x	|- .x. = ( .s ` W )
sitgval.h	|- H = (RRHom ` (Scalar ` W ) )
sitgval.1	|- ( ph -> W e. V )
sitgval.2	|- ( ph -> M e. U. ran measures )
sibfmbl.1	|- ( ph -> F e. dom ( Wsitg M ) )
sitgclg.g	|- G = (Scalar ` W )
sitgclg.d	|- D = ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) )
sitgclg.1	|- ( ph -> W e. TopSp )
sitgclg.2	|- ( ph -> W e. CMnd )
sitgclg.3	|- ( ph -> (Scalar ` W ) e. ℝExt )
sitgclg.4	|- ( ( ph /\ m e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> ( m .x. x ) e. B )

Assertion

Ref	Expression
sitgclg	|- ( ph -> ( ( Wsitg M ) ` F ) e. B )

Distinct variable groups:   B, m   x, F   m, H   x, m, M   S, m   m, W, x   .0. , m, x   .x. , m   ph, x   x, B   m, F   m, G   ph, m

Allowed substitution hints:   D( x, m)   S( x)   .x. ( x)   G( x)   H( x)   J( x, m)   V( x, m)

Proof of Theorem sitgclg

Step	Hyp	Ref	Expression
1		sitgval.b	. . 3 |- B = ( Base ` W )
2		sitgval.j	. . 3 |- J = ( TopOpen ` W )
3		sitgval.s	. . 3 |- S = (sigaGen ` J )
4		sitgval.0	. . 3 |- .0. = ( 0g ` W )
5		sitgval.x	. . 3 |- .x. = ( .s ` W )
6		sitgval.h	. . 3 |- H = (RRHom ` (Scalar ` W ) )
7		sitgval.1	. . 3 |- ( ph -> W e. V )
8		sitgval.2	. . 3 |- ( ph -> M e. U. ran measures )
9		sibfmbl.1	. . 3 |- ( ph -> F e. dom ( Wsitg M ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	sitgfval 30403	. 2 |- ( ph -> ( ( Wsitg M ) ` F ) = ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) )
11		sitgclg.2	. . 3 |- ( ph -> W e. CMnd )
12		rnexg 7098	. . . 4 |- ( F e. dom ( Wsitg M ) -> ran F e. _V )
13		difexg 4808	. . . 4 |- ( ran F e. _V -> ( ran F \ { .0. } ) e. _V )
14	9, 12, 13	3syl 18	. . 3 |- ( ph -> ( ran F \ { .0. } ) e. _V )
15		simpl 473	. . . . 5 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> ph )
16	1, 2, 3, 4, 5, 6, 7, 8, 9	sibfima 30400	. . . . . 6 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
17		sitgclg.d	. . . . . . . . . . 11 |- D = ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) )
18		sitgclg.g	. . . . . . . . . . . . 13 |- G = (Scalar ` W )
19	18	fveq2i 6194	. . . . . . . . . . . 12 |- ( dist ` G ) = ( dist ` (Scalar ` W ) )
20	18	fveq2i 6194	. . . . . . . . . . . . 13 |- ( Base ` G ) = ( Base ` (Scalar ` W ) )
21	20, 20	xpeq12i 5137	. . . . . . . . . . . 12 |- ( ( Base ` G ) X. ( Base ` G ) ) = ( ( Base ` (Scalar ` W ) ) X. ( Base ` (Scalar ` W ) ) )
22	19, 21	reseq12i 5394	. . . . . . . . . . 11 |- ( ( dist ` G ) |` ( ( Base ` G ) X. ( Base ` G ) ) ) = ( ( dist ` (Scalar ` W ) ) |` ( ( Base ` (Scalar ` W ) ) X. ( Base ` (Scalar ` W ) ) ) )
23	17, 22	eqtri 2644	. . . . . . . . . 10 |- D = ( ( dist ` (Scalar ` W ) ) |` ( ( Base ` (Scalar ` W ) ) X. ( Base ` (Scalar ` W ) ) ) )
24		eqid 2622	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
25		eqid 2622	. . . . . . . . . 10 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
26	18	fveq2i 6194	. . . . . . . . . 10 |- ( TopOpen ` G ) = ( TopOpen ` (Scalar ` W ) )
27	18	fveq2i 6194	. . . . . . . . . 10 |- ( ZMod ` G ) = ( ZMod ` (Scalar ` W ) )
28		sitgclg.3	. . . . . . . . . . . . 13 |- ( ph -> (Scalar ` W ) e. ℝExt )
29	18, 28	syl5eqel 2705	. . . . . . . . . . . 12 |- ( ph -> G e. ℝExt )
30		rrextdrg 30046	. . . . . . . . . . . 12 |- ( G e. ℝExt -> G e. DivRing )
31	29, 30	syl 17	. . . . . . . . . . 11 |- ( ph -> G e. DivRing )
32	18, 31	syl5eqelr 2706	. . . . . . . . . 10 |- ( ph -> (Scalar ` W ) e. DivRing )
33		rrextnrg 30045	. . . . . . . . . . . 12 |- ( G e. ℝExt -> G e. NrmRing )
34	29, 33	syl 17	. . . . . . . . . . 11 |- ( ph -> G e. NrmRing )
35	18, 34	syl5eqelr 2706	. . . . . . . . . 10 |- ( ph -> (Scalar ` W ) e. NrmRing )
36		eqid 2622	. . . . . . . . . . . 12 |- ( ZMod ` G ) = ( ZMod ` G )
37	36	rrextnlm 30047	. . . . . . . . . . 11 |- ( G e. ℝExt -> ( ZMod ` G ) e. NrmMod )
38	29, 37	syl 17	. . . . . . . . . 10 |- ( ph -> ( ZMod ` G ) e. NrmMod )
39	18	fveq2i 6194	. . . . . . . . . . 11 |- (chr ` G ) = (chr ` (Scalar ` W ) )
40		rrextchr 30048	. . . . . . . . . . . 12 |- ( G e. ℝExt -> (chr ` G ) = 0 )
41	29, 40	syl 17	. . . . . . . . . . 11 |- ( ph -> (chr ` G ) = 0 )
42	39, 41	syl5eqr 2670	. . . . . . . . . 10 |- ( ph -> (chr ` (Scalar ` W ) ) = 0 )
43		rrextcusp 30049	. . . . . . . . . . . 12 |- ( G e. ℝExt -> G e. CUnifSp )
44	29, 43	syl 17	. . . . . . . . . . 11 |- ( ph -> G e. CUnifSp )
45	18, 44	syl5eqelr 2706	. . . . . . . . . 10 |- ( ph -> (Scalar ` W ) e. CUnifSp )
46	18	fveq2i 6194	. . . . . . . . . . 11 |- (UnifSt ` G ) = (UnifSt ` (Scalar ` W ) )
47		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` G ) = ( Base ` G )
48	47, 17	rrextust 30052	. . . . . . . . . . . 12 |- ( G e. ℝExt -> (UnifSt ` G ) = (metUnif ` D ) )
49	29, 48	syl 17	. . . . . . . . . . 11 |- ( ph -> (UnifSt ` G ) = (metUnif ` D ) )
50	46, 49	syl5eqr 2670	. . . . . . . . . 10 |- ( ph -> (UnifSt ` (Scalar ` W ) ) = (metUnif ` D ) )
51	23, 24, 25, 26, 27, 32, 35, 38, 42, 45, 50	rrhf 30042	. . . . . . . . 9 |- ( ph -> (RRHom ` (Scalar ` W ) ) : RR --> ( Base ` (Scalar ` W ) ) )
52	6	feq1i 6036	. . . . . . . . 9 |- ( H : RR --> ( Base ` (Scalar ` W ) ) <-> (RRHom ` (Scalar ` W ) ) : RR --> ( Base ` (Scalar ` W ) ) )
53	51, 52	sylibr 224	. . . . . . . 8 |- ( ph -> H : RR --> ( Base ` (Scalar ` W ) ) )
54		ffun 6048	. . . . . . . 8 |- ( H : RR --> ( Base ` (Scalar ` W ) ) -> Fun H )
55	53, 54	syl 17	. . . . . . 7 |- ( ph -> Fun H )
56		rge0ssre 12280	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
57		fdm 6051	. . . . . . . . 9 |- ( H : RR --> ( Base ` (Scalar ` W ) ) -> dom H = RR )
58	53, 57	syl 17	. . . . . . . 8 |- ( ph -> dom H = RR )
59	56, 58	syl5sseqr 3654	. . . . . . 7 |- ( ph -> ( 0 [,) +oo ) C_ dom H )
60		funfvima2 6493	. . . . . . 7 |- ( ( Fun H /\ ( 0 [,) +oo ) C_ dom H ) -> ( ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) -> ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) ) )
61	55, 59, 60	syl2anc 693	. . . . . 6 |- ( ph -> ( ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) -> ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) ) )
62	15, 16, 61	sylc 65	. . . . 5 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) )
63		dmmeas 30264	. . . . . . . . . . . 12 |- ( M e. U. ran measures -> dom M e. U. ran sigAlgebra )
64	8, 63	syl 17	. . . . . . . . . . 11 |- ( ph -> dom M e. U. ran sigAlgebra )
65		fvex 6201	. . . . . . . . . . . . . . 15 |- ( TopOpen ` W ) e. _V
66	2, 65	eqeltri 2697	. . . . . . . . . . . . . 14 |- J e. _V
67	66	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> J e. _V )
68	67	sgsiga 30205	. . . . . . . . . . . 12 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
69	3, 68	syl5eqel 2705	. . . . . . . . . . 11 |- ( ph -> S e. U. ran sigAlgebra )
70	1, 2, 3, 4, 5, 6, 7, 8, 9	sibfmbl 30397	. . . . . . . . . . 11 |- ( ph -> F e. ( dom MMblFnM S ) )
71	64, 69, 70	mbfmf 30317	. . . . . . . . . 10 |- ( ph -> F : U. dom M --> U. S )
72		frn 6053	. . . . . . . . . 10 |- ( F : U. dom M --> U. S -> ran F C_ U. S )
73	71, 72	syl 17	. . . . . . . . 9 |- ( ph -> ran F C_ U. S )
74	3	unieqi 4445	. . . . . . . . . . 11 |- U. S = U. (sigaGen ` J )
75		unisg 30206	. . . . . . . . . . . 12 |- ( J e. _V -> U. (sigaGen ` J ) = U. J )
76	66, 75	mp1i 13	. . . . . . . . . . 11 |- ( ph -> U. (sigaGen ` J ) = U. J )
77	74, 76	syl5eq 2668	. . . . . . . . . 10 |- ( ph -> U. S = U. J )
78		sitgclg.1	. . . . . . . . . . 11 |- ( ph -> W e. TopSp )
79	1, 2	tpsuni 20740	. . . . . . . . . . 11 |- ( W e. TopSp -> B = U. J )
80	78, 79	syl 17	. . . . . . . . . 10 |- ( ph -> B = U. J )
81	77, 80	eqtr4d 2659	. . . . . . . . 9 |- ( ph -> U. S = B )
82	73, 81	sseqtrd 3641	. . . . . . . 8 |- ( ph -> ran F C_ B )
83	82	ssdifd 3746	. . . . . . 7 |- ( ph -> ( ran F \ { .0. } ) C_ ( B \ { .0. } ) )
84	83	sselda 3603	. . . . . 6 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> x e. ( B \ { .0. } ) )
85	84	eldifad 3586	. . . . 5 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> x e. B )
86		simp2 1062	. . . . . 6 |- ( ( ph /\ ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) )
87		eleq1 2689	. . . . . . . . 9 |- ( m = ( H ` ( M ` ( `' F " { x } ) ) ) -> ( m e. ( H " ( 0 [,) +oo ) ) <-> ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) ) )
88	87	3anbi2d 1404	. . . . . . . 8 |- ( m = ( H ` ( M ` ( `' F " { x } ) ) ) -> ( ( ph /\ m e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) <-> ( ph /\ ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) ) )
89		oveq1 6657	. . . . . . . . 9 |- ( m = ( H ` ( M ` ( `' F " { x } ) ) ) -> ( m .x. x ) = ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) )
90	89	eleq1d 2686	. . . . . . . 8 |- ( m = ( H ` ( M ` ( `' F " { x } ) ) ) -> ( ( m .x. x ) e. B <-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) e. B ) )
91	88, 90	imbi12d 334	. . . . . . 7 |- ( m = ( H ` ( M ` ( `' F " { x } ) ) ) -> ( ( ( ph /\ m e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> ( m .x. x ) e. B ) <-> ( ( ph /\ ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) e. B ) ) )
92		sitgclg.4	. . . . . . 7 |- ( ( ph /\ m e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> ( m .x. x ) e. B )
93	91, 92	vtoclg 3266	. . . . . 6 |- ( ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) -> ( ( ph /\ ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) e. B ) )
94	86, 93	mpcom 38	. . . . 5 |- ( ( ph /\ ( H ` ( M ` ( `' F " { x } ) ) ) e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) e. B )
95	15, 62, 85, 94	syl3anc 1326	. . . 4 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) e. B )
96		eqid 2622	. . . 4 |- ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) = ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) )
97	95, 96	fmptd 6385	. . 3 |- ( ph -> ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) : ( ran F \ { .0. } ) --> B )
98		mptexg 6484	. . . . . 6 |- ( ( ran F \ { .0. } ) e. _V -> ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) e. _V )
99	14, 98	syl 17	. . . . 5 |- ( ph -> ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) e. _V )
100		fvex 6201	. . . . . 6 |- ( 0g ` W ) e. _V
101	4, 100	eqeltri 2697	. . . . 5 |- .0. e. _V
102		suppimacnv 7306	. . . . 5 |- ( ( ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) e. _V /\ .0. e. _V ) -> ( ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) supp .0. ) = ( `' ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) " ( _V \ { .0. } ) ) )
103	99, 101, 102	sylancl 694	. . . 4 |- ( ph -> ( ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) supp .0. ) = ( `' ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) " ( _V \ { .0. } ) ) )
104	1, 2, 3, 4, 5, 6, 7, 8, 9	sibfrn 30399	. . . . 5 |- ( ph -> ran F e. Fin )
105		cnvimass 5485	. . . . . . 7 |- ( `' ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) " ( _V \ { .0. } ) ) C_ dom ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) )
106	96	dmmptss 5631	. . . . . . 7 |- dom ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) C_ ( ran F \ { .0. } )
107	105, 106	sstri 3612	. . . . . 6 |- ( `' ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) " ( _V \ { .0. } ) ) C_ ( ran F \ { .0. } )
108		difss 3737	. . . . . 6 |- ( ran F \ { .0. } ) C_ ran F
109	107, 108	sstri 3612	. . . . 5 |- ( `' ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) " ( _V \ { .0. } ) ) C_ ran F
110		ssfi 8180	. . . . 5 |- ( ( ran F e. Fin /\ ( `' ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) " ( _V \ { .0. } ) ) C_ ran F ) -> ( `' ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) " ( _V \ { .0. } ) ) e. Fin )
111	104, 109, 110	sylancl 694	. . . 4 |- ( ph -> ( `' ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) " ( _V \ { .0. } ) ) e. Fin )
112	103, 111	eqeltrd 2701	. . 3 |- ( ph -> ( ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) supp .0. ) e. Fin )
113	1, 4, 11, 14, 97, 112	gsumcl2 18315	. 2 |- ( ph -> ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) e. B )
114	10, 113	eqeltrd 2701	1 |- ( ph -> ( ( Wsitg M ) ` F ) e. B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   RRcr 9935   0cc0 9936   +oocpnf 10071   (,)cioo 12175   [,)cico 12177   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   distcds 15950   TopOpenctopn 16082   topGenctg 16098   0gc0g 16100   gsum_g cgsu 16101  CMndccmn 18193   DivRingcdr 18747  metUnifcmetu 19737   ZModczlm 19849  chrcchr 19850   TopSpctps 20736  UnifStcuss 22057  CUnifSpccusp 22101  NrmRingcnrg 22384  NrmModcnlm 22385  RRHomcrrh 30037   ℝExt crrext 30038  sigAlgebracsiga 30170  sigaGencsigagen 30201  measurescmeas 30258  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-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

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-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-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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  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-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  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-subrg 18778  df-abv 18817  df-lmod 18865  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-cn 21031  df-cnp 21032  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-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-cncf 22681  df-cfil 23053  df-cmet 23055  df-cms 23132  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

This theorem is referenced by:  sitgclbn  30405  sitmcl  30413