Theorem sibfof 29246

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 20030	. . . . . . . . . . 11 |- ( W e. TopSp -> B = U. J )
6	2, 5	syl 17	. . . . . . . . . 10 |- ( ph -> B = U. J )
7	6	sqxpeqd 4865	. . . . . . . . 9 |- ( ph -> ( B X. B ) = ( U. J X. U. J ) )
8	7	feq2d 5725	. . . . . . . 8 |- ( ph -> ( .+ : ( B X. B ) --> C <-> .+ : ( U. J X. U. J ) --> C ) )
9	1, 8	mpbid 215	. . . . . . 7 |- ( ph -> .+ : ( U. J X. U. J ) --> C )
10	9	fovrnda 6459	. . . . . 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 29242	. . . . . 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 29242	. . . . . 6 |- ( ph -> G : U. dom M --> U. J )
21		dmexg 6743	. . . . . . 7 |- ( M e. U. ran measures -> dom M e. _V )
22		uniexg 6607	. . . . . . 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 3632	. . . . . 6 |- ( U. dom M i^i U. dom M ) = U. dom M
25	10, 18, 20, 23, 23, 24	off 6565	. . . . 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 2471	. . . . . . . . 9 |- ( TopOpen ` K ) = ( TopOpen ` K )
29	27, 28	tpsuni 20030	. . . . . . . 8 |- ( K e. TopSp -> C = U. ( TopOpen ` K ) )
30	26, 29	syl 17	. . . . . . 7 |- ( ph -> C = U. ( TopOpen ` K ) )
31		fvex 5889	. . . . . . . 8 |- ( TopOpen ` K ) e. _V
32		unisg 29039	. . . . . . . 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 2523	. . . . . 6 |- ( ph -> C = U. (sigaGen ` ( TopOpen ` K ) ) )
35	34	feq3d 5726	. . . . 5 |- ( ph -> ( ( F oF .+ G ) : U. dom M --> C <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
36	25, 35	mpbid 215	. . . 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 29038	. . . . . 6 |- ( ph -> (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra )
39		uniexg 6607	. . . . . 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 7504	. . . 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 240	. . 3 |- ( ph -> ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) )
43		inundif 3836	. . . . . . 7 |- ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) = b
44	43	imaeq2i 5172	. . . . . 6 |- ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( `' ( F oF .+ G ) " b )
45		ffun 5742	. . . . . . . 8 |- ( ( F oF .+ G ) : U. dom M --> C -> Fun ( F oF .+ G ) )
46		unpreima 6021	. . . . . . . 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 472	. . . . . 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 2519	. . . . 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 29097	. . . . . . . 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 472	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> dom M e. U. ran sigAlgebra )
53		imaiun 6168	. . . . . . . 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 4324	. . . . . . . . 9 |- U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } = ( b i^i ran ( F oF .+ G ) )
55	54	imaeq2i 5172	. . . . . . . 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 2495	. . . . . . 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 3644	. . . . . . . . . 10 |- ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G )
58	6	feq3d 5726	. . . . . . . . . . . . . . 15 |- ( ph -> ( F : U. dom M --> B <-> F : U. dom M --> U. J ) )
59	18, 58	mpbird 240	. . . . . . . . . . . . . 14 |- ( ph -> F : U. dom M --> B )
60	6	feq3d 5726	. . . . . . . . . . . . . . 15 |- ( ph -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
61	20, 60	mpbird 240	. . . . . . . . . . . . . 14 |- ( ph -> G : U. dom M --> B )
62		ffn 5739	. . . . . . . . . . . . . . 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 28344	. . . . . . . . . . . . 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 472	. . . . . . . . . . . 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 472	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> dom M e. U. ran sigAlgebra )
67	51	ad2antrr 740	. . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
69		inss1 3643	. . . . . . . . . . . . . . . . . 18 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( `' .+ " { z } )
70		cnvimass 5194	. . . . . . . . . . . . . . . . . . . 20 |- ( `' .+ " { z } ) C_ dom .+
71		fdm 5745	. . . . . . . . . . . . . . . . . . . . 21 |- ( .+ : ( B X. B ) --> C -> dom .+ = ( B X. B ) )
72	1, 71	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom .+ = ( B X. B ) )
73	70, 72	syl5sseq 3466	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( `' .+ " { z } ) C_ ( B X. B ) )
74	73	adantr 472	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' .+ " { z } ) C_ ( B X. B ) )
75	69, 74	syl5ss 3429	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( B X. B ) )
76	75	sselda 3418	. . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> dom M e. U. ran sigAlgebra )
78		sibfof.4	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> J e. Fre )
79	78	sgsiga 29038	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
80	11, 79	syl5eqel 2553	. . . . . . . . . . . . . . . . . 18 |- ( ph -> S e. U. ran sigAlgebra )
81	80	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> S e. U. ran sigAlgebra )
82	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfmbl 29241	. . . . . . . . . . . . . . . . . 18 |- ( ph -> F e. ( dom MMblFnM S ) )
83	82	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> F e. ( dom MMblFnM S ) )
84	4	tpstop 20031	. . . . . . . . . . . . . . . . . . . . 21 |- ( W e. TopSp -> J e. Top )
85		cldssbrsiga 29083	. . . . . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> ( Clsd ` J ) C_ (sigaGen ` J ) )
88	78	adantr 472	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> J e. Fre )
89		xp1st 6842	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( B X. B ) -> ( 1st ` p ) e. B )
90	89	adantl 473	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. B )
91	6	adantr 472	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> B = U. J )
92	90, 91	eleqtrd 2551	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. U. J )
93		eqid 2471	. . . . . . . . . . . . . . . . . . . . 21 |- U. J = U. J
94	93	t1sncld 20419	. . . . . . . . . . . . . . . . . . . 20 |- ( ( J e. Fre /\ ( 1st ` p ) e. U. J ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
95	88, 92, 94	syl2anc 673	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
96	87, 95	sseldd 3419	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. (sigaGen ` J ) )
97	96, 11	syl6eleqr 2560	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. S )
98	77, 81, 83, 97	mbfmcnvima 29152	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' F " { ( 1st ` p ) } ) e. dom M )
99	68, 76, 98	syl2anc 673	. . . . . . . . . . . . . . 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 29241	. . . . . . . . . . . . . . . . . 18 |- ( ph -> G e. ( dom MMblFnM S ) )
101	100	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> G e. ( dom MMblFnM S ) )
102		xp2nd 6843	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( B X. B ) -> ( 2nd ` p ) e. B )
103	102	adantl 473	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. B )
104	103, 91	eleqtrd 2551	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. U. J )
105	93	t1sncld 20419	. . . . . . . . . . . . . . . . . . . 20 |- ( ( J e. Fre /\ ( 2nd ` p ) e. U. J ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
106	88, 104, 105	syl2anc 673	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
107	87, 106	sseldd 3419	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. (sigaGen ` J ) )
108	107, 11	syl6eleqr 2560	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. S )
109	77, 81, 101, 108	mbfmcnvima 29152	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' G " { ( 2nd ` p ) } ) e. dom M )
110	68, 76, 109	syl2anc 673	. . . . . . . . . . . . . . 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 29031	. . . . . . . . . . . . . . 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 1292	. . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . 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 29243	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran F e. Fin )
115	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 29243	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran G e. Fin )
116		xpfi 7860	. . . . . . . . . . . . . . . . 17 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
117	114, 115, 116	syl2anc 673	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ran F X. ran G ) e. Fin )
118		inss2 3644	. . . . . . . . . . . . . . . 16 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G )
119		ssdomg 7633	. . . . . . . . . . . . . . . 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 8175	. . . . . . . . . . . . . . . . 17 |- ( ( ran F X. ran G ) e. Fin <-> ( ran F X. ran G ) ~< om )
122	121	biimpi 199	. . . . . . . . . . . . . . . 16 |- ( ( ran F X. ran G ) e. Fin -> ( ran F X. ran G ) ~< om )
123		sdomdom 7615	. . . . . . . . . . . . . . . 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 7640	. . . . . . . . . . . . . . 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 673	. . . . . . . . . . . . . 14 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
127	126	adantr 472	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
128		nfcv 2612	. . . . . . . . . . . . . 14 |- F/_ p ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) )
129	128	sigaclcuni 29014	. . . . . . . . . . . . 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 1292	. . . . . . . . . . . 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 2549	. . . . . . . . . . 11 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' ( F oF .+ G ) " { z } ) e. dom M )
132	131	ralrimiva 2809	. . . . . . . . . 10 |- ( ph -> A. z e. ran ( F oF .+ G ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
133		ssralv 3479	. . . . . . . . . 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 64	. . . . . . . . 9 |- ( ph -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
135	134	adantr 472	. . . . . . . 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 5742	. . . . . . . . . . . . . 14 |- ( .+ : ( B X. B ) --> C -> Fun .+ )
137	1, 136	syl 17	. . . . . . . . . . . . 13 |- ( ph -> Fun .+ )
138		imafi 7885	. . . . . . . . . . . . 13 |- ( ( Fun .+ /\ ( ran F X. ran G ) e. Fin ) -> ( .+ " ( ran F X. ran G ) ) e. Fin )
139	137, 117, 138	syl2anc 673	. . . . . . . . . . . 12 |- ( ph -> ( .+ " ( ran F X. ran G ) ) e. Fin )
140	18, 20, 9, 23	ofrn2 28317	. . . . . . . . . . . 12 |- ( ph -> ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) )
141		ssfi 7810	. . . . . . . . . . . 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 673	. . . . . . . . . . 11 |- ( ph -> ran ( F oF .+ G ) e. Fin )
143		ssdomg 7633	. . . . . . . . . . 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 8175	. . . . . . . . . . . 12 |- ( ran ( F oF .+ G ) e. Fin <-> ran ( F oF .+ G ) ~< om )
146	142, 145	sylib 201	. . . . . . . . . . 11 |- ( ph -> ran ( F oF .+ G ) ~< om )
147		sdomdom 7615	. . . . . . . . . . 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 7640	. . . . . . . . . 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 673	. . . . . . . . 9 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
151	150	adantr 472	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
152		nfcv 2612	. . . . . . . . 9 |- F/_ z ( b i^i ran ( F oF .+ G ) )
153	152	sigaclcuni 29014	. . . . . . . 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 1292	. . . . . . 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 2554	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) e. dom M )
156		difpreima 6023	. . . . . . . . . 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 5195	. . . . . . . . . . 11 |- ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) = dom ( F oF .+ G )
159	158	difeq2i 3537	. . . . . . . . . 10 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) )
160		cnvimass 5194	. . . . . . . . . . 11 |- ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G )
161		ssdif0 3741	. . . . . . . . . . 11 |- ( ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G ) <-> ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/) )
162	160, 161	mpbi 213	. . . . . . . . . 10 |- ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/)
163	159, 162	eqtri 2493	. . . . . . . . 9 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = (/)
164	157, 163	syl6eq 2521	. . . . . . . 8 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = (/) )
165		0elsiga 29010	. . . . . . . . 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 2549	. . . . . . 7 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
168	167	adantr 472	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
169		unelsiga 29030	. . . . . 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 1292	. . . . 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 2549	. . . 4 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " b ) e. dom M )
172	171	ralrimiva 2809	. . 3 |- ( ph -> A. b e. (sigaGen ` ( TopOpen ` K ) ) ( `' ( F oF .+ G ) " b ) e. dom M )
173	51, 38	ismbfm 29147	. . 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 936	. 2 |- ( ph -> ( F oF .+ G ) e. ( dom MMblFnM (sigaGen ` ( TopOpen ` K ) ) ) )
175	64	adantr 472	. . . . . . 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 5883	. . . . . 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 29098	. . . . . . . . 9 |- ( M e. U. ran measures <-> M e. (measures ` dom M ) )
178	16, 177	sylib 201	. . . . . . . 8 |- ( ph -> M e. (measures ` dom M ) )
179	178	adantr 472	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> M e. (measures ` dom M ) )
180		eldifi 3544	. . . . . . . 8 |- ( z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) -> z e. ran ( F oF .+ G ) )
181	180, 113	sylan2 482	. . . . . . 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 472	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
183		sneq 3969	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> { x } = { ( 1st ` p ) } )
184	183	imaeq2d 5174	. . . . . . . . . 10 |- ( x = ( 1st ` p ) -> ( `' F " { x } ) = ( `' F " { ( 1st ` p ) } ) )
185		sneq 3969	. . . . . . . . . . 11 |- ( y = ( 2nd ` p ) -> { y } = { ( 2nd ` p ) } )
186	185	imaeq2d 5174	. . . . . . . . . 10 |- ( y = ( 2nd ` p ) -> ( `' G " { y } ) = ( `' G " { ( 2nd ` p ) } ) )
187		ffun 5742	. . . . . . . . . . . 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 28271	. . . . . . . . . . 11 |- ( ( Fun F /\ Disj x e. ran F { x } ) -> Disj x e. ran F ( `' F " { x } ) )
191	188, 189, 190	sylancl 675	. . . . . . . . . 10 |- ( ph -> Disj x e. ran F ( `' F " { x } ) )
192		ffun 5742	. . . . . . . . . . . 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 28271	. . . . . . . . . . 11 |- ( ( Fun G /\ Disj y e. ran G { y } ) -> Disj y e. ran G ( `' G " { y } ) )
196	193, 194, 195	sylancl 675	. . . . . . . . . 10 |- ( ph -> Disj y e. ran G ( `' G " { y } ) )
197	184, 186, 191, 196	disjxpin 28275	. . . . . . . . 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 64	. . . . . . . 8 |- ( ph -> Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
200	199	adantr 472	. . . . . . 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 29110	. . . . . . 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 1296	. . . . . 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 7810	. . . . . . . . 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 675	. . . . . . . 8 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
205	204	adantr 472	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
206		simpll 768	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
207		simpr 468	. . . . . . . . . 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 3416	. . . . . . . . 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 6842	. . . . . . . . 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 6843	. . . . . . . . 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 6317	. . . . . . . . . . . . . . . 16 |- ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( .0. .+ .0. ) )
214		sibfof.5	. . . . . . . . . . . . . . . 16 |- ( ph -> ( .0. .+ .0. ) = ( 0g ` K ) )
215	213, 214	sylan9eqr 2527	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x = .0. /\ y = .0. ) ) -> ( x .+ y ) = ( 0g ` K ) )
216	215	ex 441	. . . . . . . . . . . . . 14 |- ( ph -> ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( 0g ` K ) ) )
217	216	necon3ad 2656	. . . . . . . . . . . . 13 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> -. ( x = .0. /\ y = .0. ) ) )
218		neorian 2737	. . . . . . . . . . . . 13 |- ( ( x =/= .0. \/ y =/= .0. ) <-> -. ( x = .0. /\ y = .0. ) )
219	217, 218	syl6ibr 235	. . . . . . . . . . . 12 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
220	219	adantr 472	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
221	220	ralrimivva 2814	. . . . . . . . . 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 3418	. . . . . . . . . . . 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 6018	. . . . . . . . . . . . 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 215	. . . . . . . . . . 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 468	. . . . . . . . . . . 12 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = z )
229		1st2nd2 6849	. . . . . . . . . . . . . . 15 |- ( p e. ( B X. B ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
230	229	fveq2d 5883	. . . . . . . . . . . . . 14 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. ) )
231		df-ov 6311	. . . . . . . . . . . . . 14 |- ( ( 1st ` p ) .+ ( 2nd ` p ) ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. )
232	230, 231	syl6eqr 2523	. . . . . . . . . . . . 13 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
233	232	adantr 472	. . . . . . . . . . . 12 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
234	228, 233	eqtr3d 2507	. . . . . . . . . . 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 770	. . . . . . . . . . . 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 3403	. . . . . . . . . . 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 3973	. . . . . . . . . . . 12 |- ( z e. { ( 0g ` K ) } <-> z = ( 0g ` K ) )
239	238	necon3bbii 2690	. . . . . . . . . . 11 |- ( -. z e. { ( 0g ` K ) } <-> z =/= ( 0g ` K ) )
240	237, 239	sylib 201	. . . . . . . . . 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 2711	. . . . . . . . 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 663	. . . . . . . . . . 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 6315	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( x .+ y ) = ( ( 1st ` p ) .+ y ) )
246	245	neeq1d 2702	. . . . . . . . . . . 12 |- ( x = ( 1st ` p ) -> ( ( x .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) ) )
247		neeq1 2705	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( x =/= .0. <-> ( 1st ` p ) =/= .0. ) )
248	247	orbi1d 717	. . . . . . . . . . . 12 |- ( x = ( 1st ` p ) -> ( ( x =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) )
249	246, 248	imbi12d 327	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> ( ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) <-> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) ) )
250		oveq2 6316	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` p ) -> ( ( 1st ` p ) .+ y ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
251	250	neeq1d 2702	. . . . . . . . . . . 12 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) ) )
252		neeq1 2705	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` p ) -> ( y =/= .0. <-> ( 2nd ` p ) =/= .0. ) )
253	252	orbi2d 716	. . . . . . . . . . . 12 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) )
254	251, 253	imbi12d 327	. . . . . . . . . . 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 3147	. . . . . . . . . 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 673	. . . . . . . . 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 45	. . . . . . . 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 29245	. . . . . . . 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 1295	. . . . . . 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 28970	. . . . . 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 2509	. . . . 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 11766	. . . . . . 7 |- ( 0 [,) +oo ) C_ RR
263	262, 259	sseldi 3416	. . . . . 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 13877	. . . . 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 2549	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. RR )
266	179	adantr 472	. . . . . . 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 663	. . . . . . 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 29103	. . . . . . 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 673	. . . . . 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 13932	. . . . 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 4422	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> 0 <_ ( M ` ( `' ( F oF .+ G ) " { z } ) ) )
272		elrege0 11764	. . . 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 677	. . 3 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
274	273	ralrimiva 2809	. 2 |- ( ph -> A. z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
275		eqid 2471	. . 3 |- (sigaGen ` ( TopOpen ` K ) ) = (sigaGen ` ( TopOpen ` K ) )
276		eqid 2471	. . 3 |- ( 0g ` K ) = ( 0g ` K )
277		eqid 2471	. . 3 |- ( .s ` K ) = ( .s ` K )
278		eqid 2471	. . 3 |- (RRHom ` (Scalar ` K ) ) = (RRHom ` (Scalar ` K ) )
279	27, 28, 275, 276, 277, 278, 26, 16	issibf 29239	. 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 1213	1 |- ( ph -> ( F oF .+ G ) e. dom ( Ksitg M ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   _Vcvv 3031   \ cdif 3387   u. cun 3388   i^i cin 3389   C_ wss 3390   (/)c0 3722   {csn 3959   <.cop 3965   U.cuni 4190   U_ciun 4269  Disj wdisj 4366   class class class wbr 4395   X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842   Fun wfun 5583   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   oFcof 6548   omcom 6711   1stc1st 6810   2ndc2nd 6811   ^m cmap 7490   ~<_ cdom 7585   ~< csdm 7586   Fincfn 7587   RRcr 9556   0cc0 9557   +oocpnf 9690   <_ cle 9694   [,)cico 11662   sum_csu 13829   Basecbs 15199  Scalarcsca 15271   .scvsca 15272   TopOpenctopn 15398   0gc0g 15416   Topctop 19994   TopSpctps 19996   Clsdccld 20108   Frect1 20400  RRHomcrrh 28871  Σ^*cesum 28922  sigAlgebracsiga 29003  sigaGencsigagen 29034  measurescmeas 29091  MblFnMcmbfm 29145  sitgcsitg 29235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-ac2 8911  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-ac 8565  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-ordt 15477  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-ps 16524  df-tsr 16525  df-plusf 16565  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-subrg 18084  df-abv 18123  df-lmod 18171  df-scaf 18172  df-sra 18473  df-rgmod 18474  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-t1 20407  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tmd 21165  df-tgp 21166  df-tsms 21219  df-trg 21252  df-xms 21413  df-ms 21414  df-tms 21415  df-nm 21675  df-ngp 21676  df-nrg 21678  df-nlm 21679  df-ii 21987  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-esum 28923  df-siga 29004  df-sigagen 29035  df-meas 29092  df-mbfm 29146  df-sitg 29236

This theorem is referenced by:  sitmcl  29257