Theorem sibfof 29175

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 19945	. . . . . . . . . . 11 |- ( W e. TopSp -> B = U. J )
6	2, 5	syl 17	. . . . . . . . . 10 |- ( ph -> B = U. J )
7	6	sqxpeqd 4877	. . . . . . . . 9 |- ( ph -> ( B X. B ) = ( U. J X. U. J ) )
8	7	feq2d 5731	. . . . . . . 8 |- ( ph -> ( .+ : ( B X. B ) --> C <-> .+ : ( U. J X. U. J ) --> C ) )
9	1, 8	mpbid 214	. . . . . . 7 |- ( ph -> .+ : ( U. J X. U. J ) --> C )
10	9	fovrnda 6452	. . . . . 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 29171	. . . . . 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 29171	. . . . . 6 |- ( ph -> G : U. dom M --> U. J )
21		dmexg 6736	. . . . . . 7 |- ( M e. U. ran measures -> dom M e. _V )
22		uniexg 6600	. . . . . . 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 3672	. . . . . 6 |- ( U. dom M i^i U. dom M ) = U. dom M
25	10, 18, 20, 23, 23, 24	off 6558	. . . . 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 2423	. . . . . . . . 9 |- ( TopOpen ` K ) = ( TopOpen ` K )
29	27, 28	tpsuni 19945	. . . . . . . 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 28967	. . . . . . . 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 2482	. . . . . 6 |- ( ph -> C = U. (sigaGen ` ( TopOpen ` K ) ) )
35	34	feq3d 5732	. . . . 5 |- ( ph -> ( ( F oF .+ G ) : U. dom M --> C <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
36	25, 35	mpbid 214	. . . 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 28966	. . . . . 6 |- ( ph -> (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra )
39		uniexg 6600	. . . . . 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 7492	. . . 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 236	. . 3 |- ( ph -> ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) )
43		inundif 3874	. . . . . . 7 |- ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) = b
44	43	imaeq2i 5183	. . . . . 6 |- ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( `' ( F oF .+ G ) " b )
45		ffun 5746	. . . . . . . 8 |- ( ( F oF .+ G ) : U. dom M --> C -> Fun ( F oF .+ G ) )
46		unpreima 6019	. . . . . . . 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 467	. . . . . 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 2478	. . . . 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 29025	. . . . . . . 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 467	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> dom M e. U. ran sigAlgebra )
53		imaiun 6163	. . . . . . . 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 4352	. . . . . . . . 9 |- U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } = ( b i^i ran ( F oF .+ G ) )
55	54	imaeq2i 5183	. . . . . . . 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 2454	. . . . . . 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 3684	. . . . . . . . . 10 |- ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G )
58	6	feq3d 5732	. . . . . . . . . . . . . . 15 |- ( ph -> ( F : U. dom M --> B <-> F : U. dom M --> U. J ) )
59	18, 58	mpbird 236	. . . . . . . . . . . . . 14 |- ( ph -> F : U. dom M --> B )
60	6	feq3d 5732	. . . . . . . . . . . . . . 15 |- ( ph -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
61	20, 60	mpbird 236	. . . . . . . . . . . . . 14 |- ( ph -> G : U. dom M --> B )
62		ffn 5744	. . . . . . . . . . . . . . 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 28265	. . . . . . . . . . . . 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 467	. . . . . . . . . . . 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 467	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> dom M e. U. ran sigAlgebra )
67	51	ad2antrr 731	. . . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
69		inss1 3683	. . . . . . . . . . . . . . . . . 18 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( `' .+ " { z } )
70		cnvimass 5205	. . . . . . . . . . . . . . . . . . . 20 |- ( `' .+ " { z } ) C_ dom .+
71		fdm 5748	. . . . . . . . . . . . . . . . . . . . 21 |- ( .+ : ( B X. B ) --> C -> dom .+ = ( B X. B ) )
72	1, 71	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom .+ = ( B X. B ) )
73	70, 72	syl5sseq 3513	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( `' .+ " { z } ) C_ ( B X. B ) )
74	73	adantr 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' .+ " { z } ) C_ ( B X. B ) )
75	69, 74	syl5ss 3476	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( B X. B ) )
76	75	sselda 3465	. . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> dom M e. U. ran sigAlgebra )
78		sibfof.4	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> J e. Fre )
79	78	sgsiga 28966	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
80	11, 79	syl5eqel 2515	. . . . . . . . . . . . . . . . . 18 |- ( ph -> S e. U. ran sigAlgebra )
81	80	adantr 467	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> S e. U. ran sigAlgebra )
82	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfmbl 29170	. . . . . . . . . . . . . . . . . 18 |- ( ph -> F e. ( dom MMblFnM S ) )
83	82	adantr 467	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> F e. ( dom MMblFnM S ) )
84	4	tpstop 19946	. . . . . . . . . . . . . . . . . . . . 21 |- ( W e. TopSp -> J e. Top )
85		cldssbrsiga 29011	. . . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> ( Clsd ` J ) C_ (sigaGen ` J ) )
88	78	adantr 467	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> J e. Fre )
89		xp1st 6835	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( B X. B ) -> ( 1st ` p ) e. B )
90	89	adantl 468	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. B )
91	6	adantr 467	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> B = U. J )
92	90, 91	eleqtrd 2513	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. U. J )
93		eqid 2423	. . . . . . . . . . . . . . . . . . . . 21 |- U. J = U. J
94	93	t1sncld 20334	. . . . . . . . . . . . . . . . . . . 20 |- ( ( J e. Fre /\ ( 1st ` p ) e. U. J ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
95	88, 92, 94	syl2anc 666	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
96	87, 95	sseldd 3466	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. (sigaGen ` J ) )
97	96, 11	syl6eleqr 2522	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. S )
98	77, 81, 83, 97	mbfmcnvima 29081	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' F " { ( 1st ` p ) } ) e. dom M )
99	68, 76, 98	syl2anc 666	. . . . . . . . . . . . . . 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 29170	. . . . . . . . . . . . . . . . . 18 |- ( ph -> G e. ( dom MMblFnM S ) )
101	100	adantr 467	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> G e. ( dom MMblFnM S ) )
102		xp2nd 6836	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( B X. B ) -> ( 2nd ` p ) e. B )
103	102	adantl 468	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. B )
104	103, 91	eleqtrd 2513	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. U. J )
105	93	t1sncld 20334	. . . . . . . . . . . . . . . . . . . 20 |- ( ( J e. Fre /\ ( 2nd ` p ) e. U. J ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
106	88, 104, 105	syl2anc 666	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
107	87, 106	sseldd 3466	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. (sigaGen ` J ) )
108	107, 11	syl6eleqr 2522	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. S )
109	77, 81, 101, 108	mbfmcnvima 29081	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' G " { ( 2nd ` p ) } ) e. dom M )
110	68, 76, 109	syl2anc 666	. . . . . . . . . . . . . . 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 28959	. . . . . . . . . . . . . . 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 1265	. . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . 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 29172	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran F e. Fin )
115	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 29172	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran G e. Fin )
116		xpfi 7846	. . . . . . . . . . . . . . . . 17 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
117	114, 115, 116	syl2anc 666	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ran F X. ran G ) e. Fin )
118		inss2 3684	. . . . . . . . . . . . . . . 16 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G )
119		ssdomg 7620	. . . . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ ( ran F X. ran G ) )
121		isfinite 8161	. . . . . . . . . . . . . . . . 17 |- ( ( ran F X. ran G ) e. Fin <-> ( ran F X. ran G ) ~< om )
122	121	biimpi 198	. . . . . . . . . . . . . . . 16 |- ( ( ran F X. ran G ) e. Fin -> ( ran F X. ran G ) ~< om )
123		sdomdom 7602	. . . . . . . . . . . . . . . 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 7627	. . . . . . . . . . . . . . 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 666	. . . . . . . . . . . . . 14 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
127	126	adantr 467	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
128		nfcv 2585	. . . . . . . . . . . . . 14 |- F/_ p ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) )
129	128	sigaclcuni 28942	. . . . . . . . . . . . 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 1265	. . . . . . . . . . . 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 2511	. . . . . . . . . . 11 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' ( F oF .+ G ) " { z } ) e. dom M )
132	131	ralrimiva 2840	. . . . . . . . . 10 |- ( ph -> A. z e. ran ( F oF .+ G ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
133		ssralv 3526	. . . . . . . . . 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 66	. . . . . . . . 9 |- ( ph -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
135	134	adantr 467	. . . . . . . 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 5746	. . . . . . . . . . . . . 14 |- ( .+ : ( B X. B ) --> C -> Fun .+ )
137	1, 136	syl 17	. . . . . . . . . . . . 13 |- ( ph -> Fun .+ )
138		imafi 7871	. . . . . . . . . . . . 13 |- ( ( Fun .+ /\ ( ran F X. ran G ) e. Fin ) -> ( .+ " ( ran F X. ran G ) ) e. Fin )
139	137, 117, 138	syl2anc 666	. . . . . . . . . . . 12 |- ( ph -> ( .+ " ( ran F X. ran G ) ) e. Fin )
140	18, 20, 9, 23	ofrn2 28237	. . . . . . . . . . . 12 |- ( ph -> ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) )
141		ssfi 7796	. . . . . . . . . . . 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 666	. . . . . . . . . . 11 |- ( ph -> ran ( F oF .+ G ) e. Fin )
143		ssdomg 7620	. . . . . . . . . . 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 22	. . . . . . . . . 10 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) )
145		isfinite 8161	. . . . . . . . . . . 12 |- ( ran ( F oF .+ G ) e. Fin <-> ran ( F oF .+ G ) ~< om )
146	142, 145	sylib 200	. . . . . . . . . . 11 |- ( ph -> ran ( F oF .+ G ) ~< om )
147		sdomdom 7602	. . . . . . . . . . 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 7627	. . . . . . . . . 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 666	. . . . . . . . 9 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
151	150	adantr 467	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
152		nfcv 2585	. . . . . . . . 9 |- F/_ z ( b i^i ran ( F oF .+ G ) )
153	152	sigaclcuni 28942	. . . . . . . 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 1265	. . . . . . 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 2516	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) e. dom M )
156		difpreima 6021	. . . . . . . . . 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 5206	. . . . . . . . . . 11 |- ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) = dom ( F oF .+ G )
159	158	difeq2i 3581	. . . . . . . . . 10 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) )
160		cnvimass 5205	. . . . . . . . . . 11 |- ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G )
161		ssdif0 3852	. . . . . . . . . . 11 |- ( ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G ) <-> ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/) )
162	160, 161	mpbi 212	. . . . . . . . . 10 |- ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/)
163	159, 162	eqtri 2452	. . . . . . . . 9 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = (/)
164	157, 163	syl6eq 2480	. . . . . . . 8 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = (/) )
165		0elsiga 28938	. . . . . . . . 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 2511	. . . . . . 7 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
168	167	adantr 467	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
169		unelsiga 28958	. . . . . 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 1265	. . . . 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 2511	. . . 4 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " b ) e. dom M )
172	171	ralrimiva 2840	. . 3 |- ( ph -> A. b e. (sigaGen ` ( TopOpen ` K ) ) ( `' ( F oF .+ G ) " b ) e. dom M )
173	51, 38	ismbfm 29076	. . 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 931	. 2 |- ( ph -> ( F oF .+ G ) e. ( dom MMblFnM (sigaGen ` ( TopOpen ` K ) ) ) )
175	64	adantr 467	. . . . . . 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 29026	. . . . . . . . 9 |- ( M e. U. ran measures <-> M e. (measures ` dom M ) )
178	16, 177	sylib 200	. . . . . . . 8 |- ( ph -> M e. (measures ` dom M ) )
179	178	adantr 467	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> M e. (measures ` dom M ) )
180		eldifi 3588	. . . . . . . 8 |- ( z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) -> z e. ran ( F oF .+ G ) )
181	180, 113	sylan2 477	. . . . . . 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 467	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
183		sneq 4007	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> { x } = { ( 1st ` p ) } )
184	183	imaeq2d 5185	. . . . . . . . . 10 |- ( x = ( 1st ` p ) -> ( `' F " { x } ) = ( `' F " { ( 1st ` p ) } ) )
185		sneq 4007	. . . . . . . . . . 11 |- ( y = ( 2nd ` p ) -> { y } = { ( 2nd ` p ) } )
186	185	imaeq2d 5185	. . . . . . . . . 10 |- ( y = ( 2nd ` p ) -> ( `' G " { y } ) = ( `' G " { ( 2nd ` p ) } ) )
187		ffun 5746	. . . . . . . . . . . 12 |- ( F : U. dom M --> U. J -> Fun F )
188	18, 187	syl 17	. . . . . . . . . . 11 |- ( ph -> Fun F )
189		sndisj 4413	. . . . . . . . . . 11 |- Disj x e. ran F { x }
190		disjpreima 28190	. . . . . . . . . . 11 |- ( ( Fun F /\ Disj x e. ran F { x } ) -> Disj x e. ran F ( `' F " { x } ) )
191	188, 189, 190	sylancl 667	. . . . . . . . . 10 |- ( ph -> Disj x e. ran F ( `' F " { x } ) )
192		ffun 5746	. . . . . . . . . . . 12 |- ( G : U. dom M --> U. J -> Fun G )
193	20, 192	syl 17	. . . . . . . . . . 11 |- ( ph -> Fun G )
194		sndisj 4413	. . . . . . . . . . 11 |- Disj y e. ran G { y }
195		disjpreima 28190	. . . . . . . . . . 11 |- ( ( Fun G /\ Disj y e. ran G { y } ) -> Disj y e. ran G ( `' G " { y } ) )
196	193, 194, 195	sylancl 667	. . . . . . . . . 10 |- ( ph -> Disj y e. ran G ( `' G " { y } ) )
197	184, 186, 191, 196	disjxpin 28194	. . . . . . . . 9 |- ( ph -> Disj p e. ( ran F X. ran G ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
198		disjss1 4398	. . . . . . . . 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 66	. . . . . . . 8 |- ( ph -> Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
200	199	adantr 467	. . . . . . 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 29038	. . . . . . 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 1269	. . . . . 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 7796	. . . . . . . . 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 667	. . . . . . . 8 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
205	204	adantr 467	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
206		simpll 759	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
207		simpr 463	. . . . . . . . . 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 3463	. . . . . . . . 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 6835	. . . . . . . . 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 6836	. . . . . . . . 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 6312	. . . . . . . . . . . . . . . 16 |- ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( .0. .+ .0. ) )
214		sibfof.5	. . . . . . . . . . . . . . . 16 |- ( ph -> ( .0. .+ .0. ) = ( 0g ` K ) )
215	213, 214	sylan9eqr 2486	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x = .0. /\ y = .0. ) ) -> ( x .+ y ) = ( 0g ` K ) )
216	215	ex 436	. . . . . . . . . . . . . 14 |- ( ph -> ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( 0g ` K ) ) )
217	216	necon3ad 2635	. . . . . . . . . . . . 13 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> -. ( x = .0. /\ y = .0. ) ) )
218		neorian 2752	. . . . . . . . . . . . 13 |- ( ( x =/= .0. \/ y =/= .0. ) <-> -. ( x = .0. /\ y = .0. ) )
219	217, 218	syl6ibr 231	. . . . . . . . . . . 12 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
220	219	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
221	220	ralrimivva 2847	. . . . . . . . . 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 3465	. . . . . . . . . . . 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 6016	. . . . . . . . . . . . 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 214	. . . . . . . . . . 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 463	. . . . . . . . . . . 12 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = z )
229		1st2nd2 6842	. . . . . . . . . . . . . . 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 6306	. . . . . . . . . . . . . 14 |- ( ( 1st ` p ) .+ ( 2nd ` p ) ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. )
232	230, 231	syl6eqr 2482	. . . . . . . . . . . . 13 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
233	232	adantr 467	. . . . . . . . . . . 12 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
234	228, 233	eqtr3d 2466	. . . . . . . . . . 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 761	. . . . . . . . . . . 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 3450	. . . . . . . . . . 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 4011	. . . . . . . . . . . 12 |- ( z e. { ( 0g ` K ) } <-> z = ( 0g ` K ) )
239	238	necon3bbii 2686	. . . . . . . . . . 11 |- ( -. z e. { ( 0g ` K ) } <-> z =/= ( 0g ` K ) )
240	237, 239	sylib 200	. . . . . . . . . 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 2719	. . . . . . . . 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 656	. . . . . . . . . . 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 6310	. . . . . . . . . . . . 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 2706	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( x =/= .0. <-> ( 1st ` p ) =/= .0. ) )
248	247	orbi1d 708	. . . . . . . . . . . 12 |- ( x = ( 1st ` p ) -> ( ( x =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) )
249	246, 248	imbi12d 322	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> ( ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) <-> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) ) )
250		oveq2 6311	. . . . . . . . . . . . 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 2706	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` p ) -> ( y =/= .0. <-> ( 2nd ` p ) =/= .0. ) )
253	252	orbi2d 707	. . . . . . . . . . . 12 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) )
254	251, 253	imbi12d 322	. . . . . . . . . . 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 3192	. . . . . . . . . 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 666	. . . . . . . . 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 47	. . . . . . . 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 29174	. . . . . . . 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 1268	. . . . . . 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 28898	. . . . . 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 2468	. . . . 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 11742	. . . . . . 7 |- ( 0 [,) +oo ) C_ RR
263	262, 259	sseldi 3463	. . . . . 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 13793	. . . . 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 2511	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. RR )
266	179	adantr 467	. . . . . . 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 656	. . . . . . 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 29031	. . . . . . 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 666	. . . . . 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 13848	. . . . 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 4448	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> 0 <_ ( M ` ( `' ( F oF .+ G ) " { z } ) ) )
272		elrege0 11740	. . . 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 669	. . 3 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
274	273	ralrimiva 2840	. 2 |- ( ph -> A. z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
275		eqid 2423	. . 3 |- (sigaGen ` ( TopOpen ` K ) ) = (sigaGen ` ( TopOpen ` K ) )
276		eqid 2423	. . 3 |- ( 0g ` K ) = ( 0g ` K )
277		eqid 2423	. . 3 |- ( .s ` K ) = ( .s ` K )
278		eqid 2423	. . 3 |- (RRHom ` (Scalar ` K ) ) = (RRHom ` (Scalar ` K ) )
279	27, 28, 275, 276, 277, 278, 26, 16	issibf 29168	. 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 1189	1 |- ( ph -> ( F oF .+ G ) e. dom ( Ksitg M ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1438   e. wcel 1869   =/= wne 2619   A.wral 2776   _Vcvv 3082   \ cdif 3434   u. cun 3435   i^i cin 3436   C_ wss 3437   (/)c0 3762   {csn 3997   <.cop 4003   U.cuni 4217   U_ciun 4297  Disj wdisj 4392   class class class wbr 4421   X. cxp 4849   `'ccnv 4850   dom cdm 4851   ran crn 4852   "cima 4854   Fun wfun 5593   Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303   oFcof 6541   omcom 6704   1stc1st 6803   2ndc2nd 6804   ^m cmap 7478   ~<_ cdom 7573   ~< csdm 7574   Fincfn 7575   RRcr 9540   0cc0 9541   +oocpnf 9674   <_ cle 9678   [,)cico 11639   sum_csu 13745   Basecbs 15114  Scalarcsca 15186   .scvsca 15187   TopOpenctopn 15313   0gc0g 15331   Topctop 19909   TopSpctps 19911   Clsdccld 20023   Frect1 20315  RRHomcrrh 28799  Σ^*cesum 28850  sigAlgebracsiga 28931  sigaGencsigagen 28962  measurescmeas 29019  MblFnMcmbfm 29074  sitgcsitg 29164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-ac2 8895  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-disj 4393  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-acn 8379  df-ac 8549  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ioc 11642  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-ordt 15392  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-ps 16439  df-tsr 16440  df-plusf 16480  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-mulg 16669  df-subg 16807  df-cntz 16964  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-cring 17776  df-subrg 17999  df-abv 18038  df-lmod 18086  df-scaf 18087  df-sra 18388  df-rgmod 18389  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-t1 20322  df-haus 20323  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-tmd 21079  df-tgp 21080  df-tsms 21133  df-trg 21166  df-xms 21327  df-ms 21328  df-tms 21329  df-nm 21589  df-ngp 21590  df-nrg 21592  df-nlm 21593  df-ii 21901  df-cncf 21902  df-limc 22813  df-dv 22814  df-log 23498  df-esum 28851  df-siga 28932  df-sigagen 28963  df-meas 29020  df-mbfm 29075  df-sitg 29165

This theorem is referenced by:  sitmcl  29186