Theorem sibfof 28788

Description: Applying function operations on simple functions results in simple functions with regard to the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.)

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 ) )
sibfof.c	|- C = ( Base ` K )
sibfof.0	|- ( ph -> W e. TopSp )
sibfof.1	|- ( ph -> .+ : ( B X. B ) --> C )
sibfof.2	|- ( ph -> G e. dom ( Wsitg M ) )
sibfof.3	|- ( ph -> K e. TopSp )
sibfof.4	|- ( ph -> J e. Fre )
sibfof.5	|- ( ph -> ( .0. .+ .0. ) = ( 0g ` K ) )

Assertion

Ref	Expression
sibfof	|- ( ph -> ( F oF .+ G ) e. dom ( Ksitg M ) )

Proof of Theorem sibfof

Dummy variables x y z b p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sibfof.1	. . . . . . . 8 |- ( ph -> .+ : ( B X. B ) --> C )
2		sibfof.0	. . . . . . . . . . 11 |- ( ph -> W e. TopSp )
3		sitgval.b	. . . . . . . . . . . 12 |- B = ( Base ` W )
4		sitgval.j	. . . . . . . . . . . 12 |- J = ( TopOpen ` W )
5	3, 4	tpsuni 19731	. . . . . . . . . . 11 |- ( W e. TopSp -> B = U. J )
6	2, 5	syl 17	. . . . . . . . . 10 |- ( ph -> B = U. J )
7	6	sqxpeqd 4849	. . . . . . . . 9 |- ( ph -> ( B X. B ) = ( U. J X. U. J ) )
8	7	feq2d 5701	. . . . . . . 8 |- ( ph -> ( .+ : ( B X. B ) --> C <-> .+ : ( U. J X. U. J ) --> C ) )
9	1, 8	mpbid 210	. . . . . . 7 |- ( ph -> .+ : ( U. J X. U. J ) --> C )
10	9	fovrnda 6427	. . . . . 6 |- ( ( ph /\ ( z e. U. J /\ x e. U. J ) ) -> ( z .+ x ) e. C )
11		sitgval.s	. . . . . . 7 |- S = (sigaGen ` J )
12		sitgval.0	. . . . . . 7 |- .0. = ( 0g ` W )
13		sitgval.x	. . . . . . 7 |- .x. = ( .s ` W )
14		sitgval.h	. . . . . . 7 |- H = (RRHom ` (Scalar ` W ) )
15		sitgval.1	. . . . . . 7 |- ( ph -> W e. V )
16		sitgval.2	. . . . . . 7 |- ( ph -> M e. U. ran measures )
17		sibfmbl.1	. . . . . . 7 |- ( ph -> F e. dom ( Wsitg M ) )
18	3, 4, 11, 12, 13, 14, 15, 16, 17	sibff 28784	. . . . . 6 |- ( ph -> F : U. dom M --> U. J )
19		sibfof.2	. . . . . . 7 |- ( ph -> G e. dom ( Wsitg M ) )
20	3, 4, 11, 12, 13, 14, 15, 16, 19	sibff 28784	. . . . . 6 |- ( ph -> G : U. dom M --> U. J )
21		dmexg 6715	. . . . . . 7 |- ( M e. U. ran measures -> dom M e. _V )
22		uniexg 6579	. . . . . . 7 |- ( dom M e. _V -> U. dom M e. _V )
23	16, 21, 22	3syl 18	. . . . . 6 |- ( ph -> U. dom M e. _V )
24		inidm 3648	. . . . . 6 |- ( U. dom M i^i U. dom M ) = U. dom M
25	10, 18, 20, 23, 23, 24	off 6536	. . . . 5 |- ( ph -> ( F oF .+ G ) : U. dom M --> C )
26		sibfof.3	. . . . . . . 8 |- ( ph -> K e. TopSp )
27		sibfof.c	. . . . . . . . 9 |- C = ( Base ` K )
28		eqid 2402	. . . . . . . . 9 |- ( TopOpen ` K ) = ( TopOpen ` K )
29	27, 28	tpsuni 19731	. . . . . . . 8 |- ( K e. TopSp -> C = U. ( TopOpen ` K ) )
30	26, 29	syl 17	. . . . . . 7 |- ( ph -> C = U. ( TopOpen ` K ) )
31		fvex 5859	. . . . . . . 8 |- ( TopOpen ` K ) e. _V
32		unisg 28591	. . . . . . . 8 |- ( ( TopOpen ` K ) e. _V -> U. (sigaGen ` ( TopOpen ` K ) ) = U. ( TopOpen ` K ) )
33	31, 32	ax-mp 5	. . . . . . 7 |- U. (sigaGen ` ( TopOpen ` K ) ) = U. ( TopOpen ` K )
34	30, 33	syl6eqr 2461	. . . . . 6 |- ( ph -> C = U. (sigaGen ` ( TopOpen ` K ) ) )
35	34	feq3d 5702	. . . . 5 |- ( ph -> ( ( F oF .+ G ) : U. dom M --> C <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
36	25, 35	mpbid 210	. . . 4 |- ( ph -> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) )
37	31	a1i 11	. . . . . . 7 |- ( ph -> ( TopOpen ` K ) e. _V )
38	37	sgsiga 28590	. . . . . 6 |- ( ph -> (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra )
39		uniexg 6579	. . . . . 6 |- ( (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra -> U. (sigaGen ` ( TopOpen ` K ) ) e. _V )
40	38, 39	syl 17	. . . . 5 |- ( ph -> U. (sigaGen ` ( TopOpen ` K ) ) e. _V )
41	40, 23	elmapd 7471	. . . 4 |- ( ph -> ( ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
42	36, 41	mpbird 232	. . 3 |- ( ph -> ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) )
43		inundif 3850	. . . . . . 7 |- ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) = b
44	43	imaeq2i 5155	. . . . . 6 |- ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( `' ( F oF .+ G ) " b )
45		ffun 5716	. . . . . . . 8 |- ( ( F oF .+ G ) : U. dom M --> C -> Fun ( F oF .+ G ) )
46		unpreima 5991	. . . . . . . 8 |- ( Fun ( F oF .+ G ) -> ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) )
47	25, 45, 46	3syl 18	. . . . . . 7 |- ( ph -> ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) )
48	47	adantr 463	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) )
49	44, 48	syl5eqr 2457	. . . . 5 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " b ) = ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) )
50		dmmeas 28649	. . . . . . . 8 |- ( M e. U. ran measures -> dom M e. U. ran sigAlgebra )
51	16, 50	syl 17	. . . . . . 7 |- ( ph -> dom M e. U. ran sigAlgebra )
52	51	adantr 463	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> dom M e. U. ran sigAlgebra )
53		imaiun 6138	. . . . . . . 8 |- ( `' ( F oF .+ G ) " U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } ) = U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } )
54		iunid 4326	. . . . . . . . 9 |- U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } = ( b i^i ran ( F oF .+ G ) )
55	54	imaeq2i 5155	. . . . . . . 8 |- ( `' ( F oF .+ G ) " U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } ) = ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) )
56	53, 55	eqtr3i 2433	. . . . . . 7 |- U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) = ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) )
57		inss2 3660	. . . . . . . . . 10 |- ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G )
58	6	feq3d 5702	. . . . . . . . . . . . . . 15 |- ( ph -> ( F : U. dom M --> B <-> F : U. dom M --> U. J ) )
59	18, 58	mpbird 232	. . . . . . . . . . . . . 14 |- ( ph -> F : U. dom M --> B )
60	6	feq3d 5702	. . . . . . . . . . . . . . 15 |- ( ph -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
61	20, 60	mpbird 232	. . . . . . . . . . . . . 14 |- ( ph -> G : U. dom M --> B )
62		ffn 5714	. . . . . . . . . . . . . . 15 |- ( .+ : ( B X. B ) --> C -> .+ Fn ( B X. B ) )
63	1, 62	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> .+ Fn ( B X. B ) )
64	59, 61, 23, 63	ofpreima2 27951	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( F oF .+ G ) " { z } ) = U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
65	64	adantr 463	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' ( F oF .+ G ) " { z } ) = U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
66	51	adantr 463	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> dom M e. U. ran sigAlgebra )
67	51	ad2antrr 724	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> dom M e. U. ran sigAlgebra )
68		simpll 752	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
69		inss1 3659	. . . . . . . . . . . . . . . . . 18 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( `' .+ " { z } )
70		cnvimass 5177	. . . . . . . . . . . . . . . . . . . 20 |- ( `' .+ " { z } ) C_ dom .+
71		fdm 5718	. . . . . . . . . . . . . . . . . . . . 21 |- ( .+ : ( B X. B ) --> C -> dom .+ = ( B X. B ) )
72	1, 71	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom .+ = ( B X. B ) )
73	70, 72	syl5sseq 3490	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( `' .+ " { z } ) C_ ( B X. B ) )
74	73	adantr 463	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' .+ " { z } ) C_ ( B X. B ) )
75	69, 74	syl5ss 3453	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( B X. B ) )
76	75	sselda 3442	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( B X. B ) )
77	51	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> dom M e. U. ran sigAlgebra )
78		sibfof.4	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> J e. Fre )
79	78	sgsiga 28590	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
80	11, 79	syl5eqel 2494	. . . . . . . . . . . . . . . . . 18 |- ( ph -> S e. U. ran sigAlgebra )
81	80	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> S e. U. ran sigAlgebra )
82	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfmbl 28783	. . . . . . . . . . . . . . . . . 18 |- ( ph -> F e. ( dom MMblFnM S ) )
83	82	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> F e. ( dom MMblFnM S ) )
84	4	tpstop 19732	. . . . . . . . . . . . . . . . . . . . 21 |- ( W e. TopSp -> J e. Top )
85		cldssbrsiga 28635	. . . . . . . . . . . . . . . . . . . . 21 |- ( J e. Top -> ( Clsd ` J ) C_ (sigaGen ` J ) )
86	2, 84, 85	3syl 18	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( Clsd ` J ) C_ (sigaGen ` J ) )
87	86	adantr 463	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> ( Clsd ` J ) C_ (sigaGen ` J ) )
88	78	adantr 463	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> J e. Fre )
89		xp1st 6814	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( B X. B ) -> ( 1st ` p ) e. B )
90	89	adantl 464	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. B )
91	6	adantr 463	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> B = U. J )
92	90, 91	eleqtrd 2492	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. U. J )
93		eqid 2402	. . . . . . . . . . . . . . . . . . . . 21 |- U. J = U. J
94	93	t1sncld 20120	. . . . . . . . . . . . . . . . . . . 20 |- ( ( J e. Fre /\ ( 1st ` p ) e. U. J ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
95	88, 92, 94	syl2anc 659	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
96	87, 95	sseldd 3443	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. (sigaGen ` J ) )
97	96, 11	syl6eleqr 2501	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. S )
98	77, 81, 83, 97	mbfmcnvima 28705	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' F " { ( 1st ` p ) } ) e. dom M )
99	68, 76, 98	syl2anc 659	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( `' F " { ( 1st ` p ) } ) e. dom M )
100	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfmbl 28783	. . . . . . . . . . . . . . . . . 18 |- ( ph -> G e. ( dom MMblFnM S ) )
101	100	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> G e. ( dom MMblFnM S ) )
102		xp2nd 6815	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( B X. B ) -> ( 2nd ` p ) e. B )
103	102	adantl 464	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. B )
104	103, 91	eleqtrd 2492	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. U. J )
105	93	t1sncld 20120	. . . . . . . . . . . . . . . . . . . 20 |- ( ( J e. Fre /\ ( 2nd ` p ) e. U. J ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
106	88, 104, 105	syl2anc 659	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
107	87, 106	sseldd 3443	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. (sigaGen ` J ) )
108	107, 11	syl6eleqr 2501	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. S )
109	77, 81, 101, 108	mbfmcnvima 28705	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' G " { ( 2nd ` p ) } ) e. dom M )
110	68, 76, 109	syl2anc 659	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( `' G " { ( 2nd ` p ) } ) e. dom M )
111		inelsiga 28583	. . . . . . . . . . . . . . 15 |- ( ( dom M e. U. ran sigAlgebra /\ ( `' F " { ( 1st ` p ) } ) e. dom M /\ ( `' G " { ( 2nd ` p ) } ) e. dom M ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
112	67, 99, 110, 111	syl3anc 1230	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
113	112	ralrimiva 2818	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> A. p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
114	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfrn 28785	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran F e. Fin )
115	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 28785	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran G e. Fin )
116		xpfi 7825	. . . . . . . . . . . . . . . . 17 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
117	114, 115, 116	syl2anc 659	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ran F X. ran G ) e. Fin )
118		inss2 3660	. . . . . . . . . . . . . . . 16 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G )
119		ssdomg 7599	. . . . . . . . . . . . . . . 16 |- ( ( ran F X. ran G ) e. Fin -> ( ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ ( ran F X. ran G ) ) )
120	117, 118, 119	mpisyl 21	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ ( ran F X. ran G ) )
121		isfinite 8102	. . . . . . . . . . . . . . . . 17 |- ( ( ran F X. ran G ) e. Fin <-> ( ran F X. ran G ) ~< om )
122	121	biimpi 194	. . . . . . . . . . . . . . . 16 |- ( ( ran F X. ran G ) e. Fin -> ( ran F X. ran G ) ~< om )
123		sdomdom 7581	. . . . . . . . . . . . . . . 16 |- ( ( ran F X. ran G ) ~< om -> ( ran F X. ran G ) ~<_ om )
124	117, 122, 123	3syl 18	. . . . . . . . . . . . . . 15 |- ( ph -> ( ran F X. ran G ) ~<_ om )
125		domtr 7606	. . . . . . . . . . . . . . 15 |- ( ( ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ ( ran F X. ran G ) /\ ( ran F X. ran G ) ~<_ om ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
126	120, 124, 125	syl2anc 659	. . . . . . . . . . . . . 14 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
127	126	adantr 463	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
128		nfcv 2564	. . . . . . . . . . . . . 14 |- F/_ p ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) )
129	128	sigaclcuni 28566	. . . . . . . . . . . . 13 |- ( ( dom M e. U. ran sigAlgebra /\ A. p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M /\ ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om ) -> U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
130	66, 113, 127, 129	syl3anc 1230	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
131	65, 130	eqeltrd 2490	. . . . . . . . . . 11 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' ( F oF .+ G ) " { z } ) e. dom M )
132	131	ralrimiva 2818	. . . . . . . . . 10 |- ( ph -> A. z e. ran ( F oF .+ G ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
133		ssralv 3503	. . . . . . . . . 10 |- ( ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G ) -> ( A. z e. ran ( F oF .+ G ) ( `' ( F oF .+ G ) " { z } ) e. dom M -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M ) )
134	57, 132, 133	mpsyl 62	. . . . . . . . 9 |- ( ph -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
135	134	adantr 463	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
136		ffun 5716	. . . . . . . . . . . . . 14 |- ( .+ : ( B X. B ) --> C -> Fun .+ )
137	1, 136	syl 17	. . . . . . . . . . . . 13 |- ( ph -> Fun .+ )
138		imafi 7847	. . . . . . . . . . . . 13 |- ( ( Fun .+ /\ ( ran F X. ran G ) e. Fin ) -> ( .+ " ( ran F X. ran G ) ) e. Fin )
139	137, 117, 138	syl2anc 659	. . . . . . . . . . . 12 |- ( ph -> ( .+ " ( ran F X. ran G ) ) e. Fin )
140	18, 20, 9, 23	ofrn2 27923	. . . . . . . . . . . 12 |- ( ph -> ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) )
141		ssfi 7775	. . . . . . . . . . . 12 |- ( ( ( .+ " ( ran F X. ran G ) ) e. Fin /\ ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) ) -> ran ( F oF .+ G ) e. Fin )
142	139, 140, 141	syl2anc 659	. . . . . . . . . . 11 |- ( ph -> ran ( F oF .+ G ) e. Fin )
143		ssdomg 7599	. . . . . . . . . . 11 |- ( ran ( F oF .+ G ) e. Fin -> ( ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) ) )
144	142, 57, 143	mpisyl 21	. . . . . . . . . 10 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) )
145		isfinite 8102	. . . . . . . . . . . 12 |- ( ran ( F oF .+ G ) e. Fin <-> ran ( F oF .+ G ) ~< om )
146	142, 145	sylib 196	. . . . . . . . . . 11 |- ( ph -> ran ( F oF .+ G ) ~< om )
147		sdomdom 7581	. . . . . . . . . . 11 |- ( ran ( F oF .+ G ) ~< om -> ran ( F oF .+ G ) ~<_ om )
148	146, 147	syl 17	. . . . . . . . . 10 |- ( ph -> ran ( F oF .+ G ) ~<_ om )
149		domtr 7606	. . . . . . . . . 10 |- ( ( ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) /\ ran ( F oF .+ G ) ~<_ om ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
150	144, 148, 149	syl2anc 659	. . . . . . . . 9 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
151	150	adantr 463	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
152		nfcv 2564	. . . . . . . . 9 |- F/_ z ( b i^i ran ( F oF .+ G ) )
153	152	sigaclcuni 28566	. . . . . . . 8 |- ( ( dom M e. U. ran sigAlgebra /\ A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M /\ ( b i^i ran ( F oF .+ G ) ) ~<_ om ) -> U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
154	52, 135, 151, 153	syl3anc 1230	. . . . . . 7 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
155	56, 154	syl5eqelr 2495	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) e. dom M )
156		difpreima 5993	. . . . . . . . . 10 |- ( Fun ( F oF .+ G ) -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) )
157	25, 45, 156	3syl 18	. . . . . . . . 9 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) )
158		cnvimarndm 5178	. . . . . . . . . . 11 |- ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) = dom ( F oF .+ G )
159	158	difeq2i 3558	. . . . . . . . . 10 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) )
160		cnvimass 5177	. . . . . . . . . . 11 |- ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G )
161		ssdif0 3828	. . . . . . . . . . 11 |- ( ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G ) <-> ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/) )
162	160, 161	mpbi 208	. . . . . . . . . 10 |- ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/)
163	159, 162	eqtri 2431	. . . . . . . . 9 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = (/)
164	157, 163	syl6eq 2459	. . . . . . . 8 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = (/) )
165		0elsiga 28562	. . . . . . . . 9 |- ( dom M e. U. ran sigAlgebra -> (/) e. dom M )
166	16, 50, 165	3syl 18	. . . . . . . 8 |- ( ph -> (/) e. dom M )
167	164, 166	eqeltrd 2490	. . . . . . 7 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
168	167	adantr 463	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
169		unelsiga 28582	. . . . . 6 |- ( ( dom M e. U. ran sigAlgebra /\ ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) e. dom M /\ ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M ) -> ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) e. dom M )
170	52, 155, 168, 169	syl3anc 1230	. . . . 5 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) e. dom M )
171	49, 170	eqeltrd 2490	. . . 4 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " b ) e. dom M )
172	171	ralrimiva 2818	. . 3 |- ( ph -> A. b e. (sigaGen ` ( TopOpen ` K ) ) ( `' ( F oF .+ G ) " b ) e. dom M )
173	51, 38	ismbfm 28700	. . 3 |- ( ph -> ( ( F oF .+ G ) e. ( dom MMblFnM (sigaGen ` ( TopOpen ` K ) ) ) <-> ( ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) /\ A. b e. (sigaGen ` ( TopOpen ` K ) ) ( `' ( F oF .+ G ) " b ) e. dom M ) ) )
174	42, 172, 173	mpbir2and 923	. 2 |- ( ph -> ( F oF .+ G ) e. ( dom MMblFnM (sigaGen ` ( TopOpen ` K ) ) ) )
175	64	adantr 463	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( `' ( F oF .+ G ) " { z } ) = U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
176	175	fveq2d 5853	. . . . . 6 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) = ( M ` U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
177		measbasedom 28650	. . . . . . . . 9 |- ( M e. U. ran measures <-> M e. (measures ` dom M ) )
178	16, 177	sylib 196	. . . . . . . 8 |- ( ph -> M e. (measures ` dom M ) )
179	178	adantr 463	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> M e. (measures ` dom M ) )
180		eldifi 3565	. . . . . . . 8 |- ( z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) -> z e. ran ( F oF .+ G ) )
181	180, 113	sylan2 472	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> A. p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
182	126	adantr 463	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
183		sneq 3982	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> { x } = { ( 1st ` p ) } )
184	183	imaeq2d 5157	. . . . . . . . . 10 |- ( x = ( 1st ` p ) -> ( `' F " { x } ) = ( `' F " { ( 1st ` p ) } ) )
185		sneq 3982	. . . . . . . . . . 11 |- ( y = ( 2nd ` p ) -> { y } = { ( 2nd ` p ) } )
186	185	imaeq2d 5157	. . . . . . . . . 10 |- ( y = ( 2nd ` p ) -> ( `' G " { y } ) = ( `' G " { ( 2nd ` p ) } ) )
187		ffun 5716	. . . . . . . . . . . 12 |- ( F : U. dom M --> U. J -> Fun F )
188	18, 187	syl 17	. . . . . . . . . . 11 |- ( ph -> Fun F )
189		sndisj 4387	. . . . . . . . . . 11 |- Disj x e. ran F { x }
190		disjpreima 27876	. . . . . . . . . . 11 |- ( ( Fun F /\ Disj x e. ran F { x } ) -> Disj x e. ran F ( `' F " { x } ) )
191	188, 189, 190	sylancl 660	. . . . . . . . . 10 |- ( ph -> Disj x e. ran F ( `' F " { x } ) )
192		ffun 5716	. . . . . . . . . . . 12 |- ( G : U. dom M --> U. J -> Fun G )
193	20, 192	syl 17	. . . . . . . . . . 11 |- ( ph -> Fun G )
194		sndisj 4387	. . . . . . . . . . 11 |- Disj y e. ran G { y }
195		disjpreima 27876	. . . . . . . . . . 11 |- ( ( Fun G /\ Disj y e. ran G { y } ) -> Disj y e. ran G ( `' G " { y } ) )
196	193, 194, 195	sylancl 660	. . . . . . . . . 10 |- ( ph -> Disj y e. ran G ( `' G " { y } ) )
197	184, 186, 191, 196	disjxpin 27880	. . . . . . . . 9 |- ( ph -> Disj p e. ( ran F X. ran G ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
198		disjss1 4372	. . . . . . . . 9 |- ( ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G ) -> (Disj p e. ( ran F X. ran G ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) -> Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
199	118, 197, 198	mpsyl 62	. . . . . . . 8 |- ( ph -> Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
200	199	adantr 463	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
201		measvuni 28662	. . . . . . 7 |- ( ( M e. (measures ` dom M ) /\ A. p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M /\ ( ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om /\ Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) -> ( M ` U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) = Σ* p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
202	179, 181, 182, 200, 201	syl112anc 1234	. . . . . 6 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) = Σ* p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
203		ssfi 7775	. . . . . . . . 9 |- ( ( ( ran F X. ran G ) e. Fin /\ ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
204	117, 118, 203	sylancl 660	. . . . . . . 8 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
205	204	adantr 463	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
206		simpll 752	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
207		simpr 459	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) )
208	118, 207	sseldi 3440	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( ran F X. ran G ) )
209		xp1st 6814	. . . . . . . . 9 |- ( p e. ( ran F X. ran G ) -> ( 1st ` p ) e. ran F )
210	208, 209	syl 17	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 1st ` p ) e. ran F )
211		xp2nd 6815	. . . . . . . . 9 |- ( p e. ( ran F X. ran G ) -> ( 2nd ` p ) e. ran G )
212	208, 211	syl 17	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 2nd ` p ) e. ran G )
213		oveq12 6287	. . . . . . . . . . . . . . . 16 |- ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( .0. .+ .0. ) )
214		sibfof.5	. . . . . . . . . . . . . . . 16 |- ( ph -> ( .0. .+ .0. ) = ( 0g ` K ) )
215	213, 214	sylan9eqr 2465	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x = .0. /\ y = .0. ) ) -> ( x .+ y ) = ( 0g ` K ) )
216	215	ex 432	. . . . . . . . . . . . . 14 |- ( ph -> ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( 0g ` K ) ) )
217	216	necon3ad 2613	. . . . . . . . . . . . 13 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> -. ( x = .0. /\ y = .0. ) ) )
218		neorian 2730	. . . . . . . . . . . . 13 |- ( ( x =/= .0. \/ y =/= .0. ) <-> -. ( x = .0. /\ y = .0. ) )
219	217, 218	syl6ibr 227	. . . . . . . . . . . 12 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
220	219	adantr 463	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
221	220	ralrimivva 2825	. . . . . . . . . 10 |- ( ph -> A. x e. B A. y e. B ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
222	206, 221	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> A. x e. B A. y e. B ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
223	69	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( `' .+ " { z } ) )
224	223	sselda 3442	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( `' .+ " { z } ) )
225		fniniseg 5986	. . . . . . . . . . . . 13 |- ( .+ Fn ( B X. B ) -> ( p e. ( `' .+ " { z } ) <-> ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) ) )
226	206, 63, 225	3syl 18	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( p e. ( `' .+ " { z } ) <-> ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) ) )
227	224, 226	mpbid 210	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) )
228		simpr 459	. . . . . . . . . . . 12 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = z )
229		1st2nd2 6821	. . . . . . . . . . . . . . 15 |- ( p e. ( B X. B ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
230	229	fveq2d 5853	. . . . . . . . . . . . . 14 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. ) )
231		df-ov 6281	. . . . . . . . . . . . . 14 |- ( ( 1st ` p ) .+ ( 2nd ` p ) ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. )
232	230, 231	syl6eqr 2461	. . . . . . . . . . . . 13 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
233	232	adantr 463	. . . . . . . . . . . 12 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
234	228, 233	eqtr3d 2445	. . . . . . . . . . 11 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> z = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
235	227, 234	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> z = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
236		simplr 754	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) )
237	236	eldifbd 3427	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> -. z e. { ( 0g ` K ) } )
238		elsn 3986	. . . . . . . . . . . 12 |- ( z e. { ( 0g ` K ) } <-> z = ( 0g ` K ) )
239	238	necon3bbii 2664	. . . . . . . . . . 11 |- ( -. z e. { ( 0g ` K ) } <-> z =/= ( 0g ` K ) )
240	237, 239	sylib 196	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> z =/= ( 0g ` K ) )
241	235, 240	eqnetrrd 2697	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) )
242	180, 76	sylanl2 649	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( B X. B ) )
243	242, 89	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 1st ` p ) e. B )
244	242, 102	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 2nd ` p ) e. B )
245		oveq1 6285	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( x .+ y ) = ( ( 1st ` p ) .+ y ) )
246	245	neeq1d 2680	. . . . . . . . . . . 12 |- ( x = ( 1st ` p ) -> ( ( x .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) ) )
247		neeq1 2684	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( x =/= .0. <-> ( 1st ` p ) =/= .0. ) )
248	247	orbi1d 701	. . . . . . . . . . . 12 |- ( x = ( 1st ` p ) -> ( ( x =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) )
249	246, 248	imbi12d 318	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> ( ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) <-> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) ) )
250		oveq2 6286	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` p ) -> ( ( 1st ` p ) .+ y ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
251	250	neeq1d 2680	. . . . . . . . . . . 12 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) ) )
252		neeq1 2684	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` p ) -> ( y =/= .0. <-> ( 2nd ` p ) =/= .0. ) )
253	252	orbi2d 700	. . . . . . . . . . . 12 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) )
254	251, 253	imbi12d 318	. . . . . . . . . . 11 |- ( y = ( 2nd ` p ) -> ( ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) <-> ( ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) ) )
255	249, 254	rspc2v 3169	. . . . . . . . . 10 |- ( ( ( 1st ` p ) e. B /\ ( 2nd ` p ) e. B ) -> ( A. x e. B A. y e. B ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) -> ( ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) ) )
256	243, 244, 255	syl2anc 659	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( A. x e. B A. y e. B ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) -> ( ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) ) )
257	222, 241, 256	mp2d 43	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) )
258	3, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 78	sibfinima 28787	. . . . . . . 8 |- ( ( ( ph /\ ( 1st ` p ) e. ran F /\ ( 2nd ` p ) e. ran G ) /\ ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) -> ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. ( 0 [,) +oo ) )
259	206, 210, 212, 257, 258	syl31anc 1233	. . . . . . 7 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. ( 0 [,) +oo ) )
260	205, 259	esumpfinval 28522	. . . . . 6 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> Σ* p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) = sum_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
261	176, 202, 260	3eqtrd 2447	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) = sum_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
262		rge0ssre 11682	. . . . . . 7 |- ( 0 [,) +oo ) C_ RR
263	262, 259	sseldi 3440	. . . . . 6 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. RR )
264	205, 263	fsumrecl 13705	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> sum_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. RR )
265	261, 264	eqeltrd 2490	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. RR )
266	179	adantr 463	. . . . . . 7 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> M e. (measures ` dom M ) )
267	180, 112	sylanl2 649	. . . . . . 7 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
268		measge0 28655	. . . . . . 7 |- ( ( M e. (measures ` dom M ) /\ ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M ) -> 0 <_ ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
269	266, 267, 268	syl2anc 659	. . . . . 6 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> 0 <_ ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
270	205, 263, 269	fsumge0 13760	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> 0 <_ sum_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
271	270, 261	breqtrrd 4421	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> 0 <_ ( M ` ( `' ( F oF .+ G ) " { z } ) ) )
272		elrege0 11681	. . . 4 |- ( ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) <-> ( ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. RR /\ 0 <_ ( M ` ( `' ( F oF .+ G ) " { z } ) ) ) )
273	265, 271, 272	sylanbrc 662	. . 3 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
274	273	ralrimiva 2818	. 2 |- ( ph -> A. z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
275		eqid 2402	. . 3 |- (sigaGen ` ( TopOpen ` K ) ) = (sigaGen ` ( TopOpen ` K ) )
276		eqid 2402	. . 3 |- ( 0g ` K ) = ( 0g ` K )
277		eqid 2402	. . 3 |- ( .s ` K ) = ( .s ` K )
278		eqid 2402	. . 3 |- (RRHom ` (Scalar ` K ) ) = (RRHom ` (Scalar ` K ) )
279	27, 28, 275, 276, 277, 278, 26, 16	issibf 28781	. 2 |- ( ph -> ( ( F oF .+ G ) e. dom ( Ksitg M ) <-> ( ( F oF .+ G ) e. ( dom MMblFnM (sigaGen ` ( TopOpen ` K ) ) ) /\ ran ( F oF .+ G ) e. Fin /\ A. z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) ) ) )
280	174, 142, 274, 279	mpbir3and 1180	1 |- ( ph -> ( F oF .+ G ) e. dom ( Ksitg M ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   = wceq 1405   e. wcel 1842   =/= wne 2598   A.wral 2754   _Vcvv 3059   \ cdif 3411   u. cun 3412   i^i cin 3413   C_ wss 3414   (/)c0 3738   {csn 3972   <.cop 3978   U.cuni 4191   U_ciun 4271  Disj wdisj 4366   class class class wbr 4395   X. cxp 4821   `'ccnv 4822   dom cdm 4823   ran crn 4824   "cima 4826   Fun wfun 5563   Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   oFcof 6519   omcom 6683   1stc1st 6782   2ndc2nd 6783   ^m cmap 7457   ~<_ cdom 7552   ~< csdm 7553   Fincfn 7554   RRcr 9521   0cc0 9522   +oocpnf 9655   <_ cle 9659   [,)cico 11584   sum_csu 13657   Basecbs 14841  Scalarcsca 14912   .scvsca 14913   TopOpenctopn 15036   0gc0g 15054   Topctop 19686   TopSpctps 19689   Clsdccld 19809   Frect1 20101  RRHomcrrh 28426  Σ^*cesum 28474  sigAlgebracsiga 28555  sigaGencsigagen 28586  measurescmeas 28643  MblFnMcmbfm 28698  sitgcsitg 28777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-ac2 8875  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-disj 4367  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-acn 8355  df-ac 8529  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012  df-sin 14014  df-cos 14015  df-pi 14017  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-ordt 15115  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-ps 16154  df-tsr 16155  df-plusf 16195  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-mhm 16290  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-mulg 16384  df-subg 16522  df-cntz 16679  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-cring 17521  df-subrg 17747  df-abv 17786  df-lmod 17834  df-scaf 17835  df-sra 18138  df-rgmod 18139  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-t1 20108  df-haus 20109  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-tmd 20863  df-tgp 20864  df-tsms 20917  df-trg 20954  df-xms 21115  df-ms 21116  df-tms 21117  df-nm 21395  df-ngp 21396  df-nrg 21398  df-nlm 21399  df-ii 21673  df-cncf 21674  df-limc 22562  df-dv 22563  df-log 23236  df-esum 28475  df-siga 28556  df-sigagen 28587  df-meas 28644  df-mbfm 28699  df-sitg 28778

This theorem is referenced by: (None)