Theorem sibfof 26640

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	. . . . . . . . . 10 |- ( ph -> .+ : ( B X. B ) --> C )
2		sibfof.0	. . . . . . . . . . . . 13 |- ( ph -> W e. TopSp )
3		sitgval.b	. . . . . . . . . . . . . 14 |- B = ( Base ` W )
4		sitgval.j	. . . . . . . . . . . . . 14 |- J = ( TopOpen ` W )
5	3, 4	tpsuni 18443	. . . . . . . . . . . . 13 |- ( W e. TopSp -> B = U. J )
6	2, 5	syl 16	. . . . . . . . . . . 12 |- ( ph -> B = U. J )
7	6, 6	xpeq12d 4861	. . . . . . . . . . 11 |- ( ph -> ( B X. B ) = ( U. J X. U. J ) )
8	7	feq2d 5544	. . . . . . . . . 10 |- ( ph -> ( .+ : ( B X. B ) --> C <-> .+ : ( U. J X. U. J ) --> C ) )
9	1, 8	mpbid 210	. . . . . . . . 9 |- ( ph -> .+ : ( U. J X. U. J ) --> C )
10	9	fovrnda 6233	. . . . . . . 8 |- ( ( ph /\ ( z e. U. J /\ x e. U. J ) ) -> ( z .+ x ) e. C )
11		sitgval.s	. . . . . . . . 9 |- S = (sigaGen ` J )
12		sitgval.0	. . . . . . . . 9 |- .0. = ( 0g ` W )
13		sitgval.x	. . . . . . . . 9 |- .x. = ( .s ` W )
14		sitgval.h	. . . . . . . . 9 |- H = (RRHom ` (Scalar ` W ) )
15		sitgval.1	. . . . . . . . 9 |- ( ph -> W e. V )
16		sitgval.2	. . . . . . . . 9 |- ( ph -> M e. U. ran measures )
17		sibfmbl.1	. . . . . . . . 9 |- ( ph -> F e. dom ( Wsitg M ) )
18	3, 4, 11, 12, 13, 14, 15, 16, 17	sibff 26636	. . . . . . . 8 |- ( ph -> F : U. dom M --> U. J )
19		sibfof.2	. . . . . . . . 9 |- ( ph -> G e. dom ( Wsitg M ) )
20	3, 4, 11, 12, 13, 14, 15, 16, 19	sibff 26636	. . . . . . . 8 |- ( ph -> G : U. dom M --> U. J )
21		dmexg 6508	. . . . . . . . 9 |- ( M e. U. ran measures -> dom M e. _V )
22		uniexg 6376	. . . . . . . . 9 |- ( dom M e. _V -> U. dom M e. _V )
23	16, 21, 22	3syl 20	. . . . . . . 8 |- ( ph -> U. dom M e. _V )
24		inidm 3556	. . . . . . . 8 |- ( U. dom M i^i U. dom M ) = U. dom M
25	10, 18, 20, 23, 23, 24	off 6333	. . . . . . 7 |- ( ph -> ( F oF .+ G ) : U. dom M --> C )
26		sibfof.3	. . . . . . . . . 10 |- ( ph -> K e. TopSp )
27		sibfof.c	. . . . . . . . . . 11 |- C = ( Base ` K )
28		eqid 2441	. . . . . . . . . . 11 |- ( TopOpen ` K ) = ( TopOpen ` K )
29	27, 28	tpsuni 18443	. . . . . . . . . 10 |- ( K e. TopSp -> C = U. ( TopOpen ` K ) )
30	26, 29	syl 16	. . . . . . . . 9 |- ( ph -> C = U. ( TopOpen ` K ) )
31		eqid 2441	. . . . . . . . . . 11 |- (sigaGen ` ( TopOpen ` K ) ) = (sigaGen ` ( TopOpen ` K ) )
32	31	unieqi 4097	. . . . . . . . . 10 |- U. (sigaGen ` ( TopOpen ` K ) ) = U. (sigaGen ` ( TopOpen ` K ) )
33		fvex 5698	. . . . . . . . . . 11 |- ( TopOpen ` K ) e. _V
34		unisg 26506	. . . . . . . . . . 11 |- ( ( TopOpen ` K ) e. _V -> U. (sigaGen ` ( TopOpen ` K ) ) = U. ( TopOpen ` K ) )
35	33, 34	ax-mp 5	. . . . . . . . . 10 |- U. (sigaGen ` ( TopOpen ` K ) ) = U. ( TopOpen ` K )
36	32, 35	eqtri 2461	. . . . . . . . 9 |- U. (sigaGen ` ( TopOpen ` K ) ) = U. ( TopOpen ` K )
37	30, 36	syl6eqr 2491	. . . . . . . 8 |- ( ph -> C = U. (sigaGen ` ( TopOpen ` K ) ) )
38		feq3 5541	. . . . . . . 8 |- ( C = U. (sigaGen ` ( TopOpen ` K ) ) -> ( ( F oF .+ G ) : U. dom M --> C <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
39	37, 38	syl 16	. . . . . . 7 |- ( ph -> ( ( F oF .+ G ) : U. dom M --> C <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
40	25, 39	mpbid 210	. . . . . 6 |- ( ph -> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) )
41	33	a1i 11	. . . . . . . . . 10 |- ( ph -> ( TopOpen ` K ) e. _V )
42	41	sgsiga 26505	. . . . . . . . 9 |- ( ph -> (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra )
43	31, 42	syl5eqel 2525	. . . . . . . 8 |- ( ph -> (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra )
44		uniexg 6376	. . . . . . . 8 |- ( (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra -> U. (sigaGen ` ( TopOpen ` K ) ) e. _V )
45	43, 44	syl 16	. . . . . . 7 |- ( ph -> U. (sigaGen ` ( TopOpen ` K ) ) e. _V )
46		elmapg 7223	. . . . . . 7 |- ( ( U. (sigaGen ` ( TopOpen ` K ) ) e. _V /\ U. dom M e. _V ) -> ( ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
47	45, 23, 46	syl2anc 656	. . . . . 6 |- ( ph -> ( ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
48	40, 47	mpbird 232	. . . . 5 |- ( ph -> ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) )
49		inundif 3754	. . . . . . . . 9 |- ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) = b
50	49	imaeq2i 5164	. . . . . . . 8 |- ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( `' ( F oF .+ G ) " b )
51		ffun 5558	. . . . . . . . . 10 |- ( ( F oF .+ G ) : U. dom M --> C -> Fun ( F oF .+ G ) )
52		unpreima 5826	. . . . . . . . . 10 |- ( 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 ) ) ) ) )
53	25, 51, 52	3syl 20	. . . . . . . . 9 |- ( 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 ) ) ) ) )
54	53	adantr 462	. . . . . . . 8 |- ( ( 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 ) ) ) ) )
55	50, 54	syl5eqr 2487	. . . . . . 7 |- ( ( 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 ) ) ) ) )
56		dmmeas 26535	. . . . . . . . . 10 |- ( M e. U. ran measures -> dom M e. U. ran sigAlgebra )
57	16, 56	syl 16	. . . . . . . . 9 |- ( ph -> dom M e. U. ran sigAlgebra )
58	57	adantr 462	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> dom M e. U. ran sigAlgebra )
59		imaiun 5959	. . . . . . . . . 10 |- ( `' ( 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 } )
60		iunid 4222	. . . . . . . . . . 11 |- U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } = ( b i^i ran ( F oF .+ G ) )
61	60	imaeq2i 5164	. . . . . . . . . 10 |- ( `' ( F oF .+ G ) " U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } ) = ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) )
62	59, 61	eqtr3i 2463	. . . . . . . . 9 |- U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) = ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) )
63		inss2 3568	. . . . . . . . . . . 12 |- ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G )
64		feq3 5541	. . . . . . . . . . . . . . . . . 18 |- ( B = U. J -> ( F : U. dom M --> B <-> F : U. dom M --> U. J ) )
65	6, 64	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( F : U. dom M --> B <-> F : U. dom M --> U. J ) )
66	18, 65	mpbird 232	. . . . . . . . . . . . . . . 16 |- ( ph -> F : U. dom M --> B )
67		feq3 5541	. . . . . . . . . . . . . . . . . 18 |- ( B = U. J -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
68	6, 67	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
69	20, 68	mpbird 232	. . . . . . . . . . . . . . . 16 |- ( ph -> G : U. dom M --> B )
70		ffn 5556	. . . . . . . . . . . . . . . . 17 |- ( .+ : ( B X. B ) --> C -> .+ Fn ( B X. B ) )
71	1, 70	syl 16	. . . . . . . . . . . . . . . 16 |- ( ph -> .+ Fn ( B X. B ) )
72	66, 69, 23, 71	ofpreima2 25904	. . . . . . . . . . . . . . 15 |- ( ph -> ( `' ( F oF .+ G ) " { z } ) = U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
73	72	adantr 462	. . . . . . . . . . . . . 14 |- ( ( 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 ) } ) ) )
74	57	adantr 462	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> dom M e. U. ran sigAlgebra )
75	57	ad2antrr 720	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> dom M e. U. ran sigAlgebra )
76		simpll 748	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
77		inss1 3567	. . . . . . . . . . . . . . . . . . . 20 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( `' .+ " { z } )
78		cnvimass 5186	. . . . . . . . . . . . . . . . . . . . . 22 |- ( `' .+ " { z } ) C_ dom .+
79		fdm 5560	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( .+ : ( B X. B ) --> C -> dom .+ = ( B X. B ) )
80	1, 79	syl 16	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> dom .+ = ( B X. B ) )
81	78, 80	syl5sseq 3401	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> ( `' .+ " { z } ) C_ ( B X. B ) )
82	81	adantr 462	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' .+ " { z } ) C_ ( B X. B ) )
83	77, 82	syl5ss 3364	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( B X. B ) )
84	83	sselda 3353	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( B X. B ) )
85	57	adantr 462	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> dom M e. U. ran sigAlgebra )
86		fvex 5698	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( TopOpen ` W ) e. _V
87	4, 86	eqeltri 2511	. . . . . . . . . . . . . . . . . . . . . . 23 |- J e. _V
88	87	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> J e. _V )
89	88	sgsiga 26505	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
90	11, 89	syl5eqel 2525	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> S e. U. ran sigAlgebra )
91	90	adantr 462	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> S e. U. ran sigAlgebra )
92	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfmbl 26635	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> F e. ( dom MMblFnM S ) )
93	92	adantr 462	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> F e. ( dom MMblFnM S ) )
94	4	tpstop 18444	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( W e. TopSp -> J e. Top )
95		cldssbrsiga 26521	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( J e. Top -> ( Clsd ` J ) C_ (sigaGen ` J ) )
96	2, 94, 95	3syl 20	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( Clsd ` J ) C_ (sigaGen ` J ) )
97	96	adantr 462	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( Clsd ` J ) C_ (sigaGen ` J ) )
98		sibfof.4	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> J e. Fre )
99	98	adantr 462	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ p e. ( B X. B ) ) -> J e. Fre )
100		xp1st 6605	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( p e. ( B X. B ) -> ( 1st ` p ) e. B )
101	100	adantl 463	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. B )
102	6	adantr 462	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ph /\ p e. ( B X. B ) ) -> B = U. J )
103	101, 102	eleqtrd 2517	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. U. J )
104		eqid 2441	. . . . . . . . . . . . . . . . . . . . . . 23 |- U. J = U. J
105	104	t1sncld 18830	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( J e. Fre /\ ( 1st ` p ) e. U. J ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
106	99, 103, 105	syl2anc 656	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
107	97, 106	sseldd 3354	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. (sigaGen ` J ) )
108	107, 11	syl6eleqr 2532	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. S )
109	85, 91, 93, 108	mbfmcnvima 26592	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' F " { ( 1st ` p ) } ) e. dom M )
110	76, 84, 109	syl2anc 656	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( `' F " { ( 1st ` p ) } ) e. dom M )
111	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfmbl 26635	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> G e. ( dom MMblFnM S ) )
112	111	adantr 462	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> G e. ( dom MMblFnM S ) )
113		xp2nd 6606	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( p e. ( B X. B ) -> ( 2nd ` p ) e. B )
114	113	adantl 463	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. B )
115	114, 102	eleqtrd 2517	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. U. J )
116	104	t1sncld 18830	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( J e. Fre /\ ( 2nd ` p ) e. U. J ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
117	99, 115, 116	syl2anc 656	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
118	97, 117	sseldd 3354	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. (sigaGen ` J ) )
119	118, 11	syl6eleqr 2532	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. S )
120	85, 91, 112, 119	mbfmcnvima 26592	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' G " { ( 2nd ` p ) } ) e. dom M )
121	76, 84, 120	syl2anc 656	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( `' G " { ( 2nd ` p ) } ) e. dom M )
122		inelsiga 26498	. . . . . . . . . . . . . . . . 17 |- ( ( 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 )
123	75, 110, 121, 122	syl3anc 1213	. . . . . . . . . . . . . . . 16 |- ( ( ( 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 )
124	123	ralrimiva 2797	. . . . . . . . . . . . . . 15 |- ( ( 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 )
125		inss2 3568	. . . . . . . . . . . . . . . . . 18 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G )
126	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfrn 26637	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ran F e. Fin )
127	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 26637	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ran G e. Fin )
128		xpfi 7579	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
129	126, 127, 128	syl2anc 656	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( ran F X. ran G ) e. Fin )
130		ssdomg 7351	. . . . . . . . . . . . . . . . . . 19 |- ( ( 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 ) ) )
131	129, 130	syl 16	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( ( ( `' .+ " { 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 ) ) )
132	125, 131	mpi 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ ( ran F X. ran G ) )
133		isfinite 7854	. . . . . . . . . . . . . . . . . . 19 |- ( ( ran F X. ran G ) e. Fin <-> ( ran F X. ran G ) ~< om )
134	133	biimpi 194	. . . . . . . . . . . . . . . . . 18 |- ( ( ran F X. ran G ) e. Fin -> ( ran F X. ran G ) ~< om )
135		sdomdom 7333	. . . . . . . . . . . . . . . . . 18 |- ( ( ran F X. ran G ) ~< om -> ( ran F X. ran G ) ~<_ om )
136	129, 134, 135	3syl 20	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ran F X. ran G ) ~<_ om )
137		domtr 7358	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( `' .+ " { 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 )
138	132, 136, 137	syl2anc 656	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
139	138	adantr 462	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
140		nfcv 2577	. . . . . . . . . . . . . . . 16 |- F/_ p ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) )
141	140	sigaclcuni 26481	. . . . . . . . . . . . . . 15 |- ( ( 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 )
142	74, 124, 139, 141	syl3anc 1213	. . . . . . . . . . . . . 14 |- ( ( 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 )
143	73, 142	eqeltrd 2515	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' ( F oF .+ G ) " { z } ) e. dom M )
144	143	ralrimiva 2797	. . . . . . . . . . . 12 |- ( ph -> A. z e. ran ( F oF .+ G ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
145		ssralv 3413	. . . . . . . . . . . 12 |- ( ( 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 ) )
146	63, 144, 145	mpsyl 63	. . . . . . . . . . 11 |- ( ph -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
147	146	adantr 462	. . . . . . . . . 10 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
148		ffun 5558	. . . . . . . . . . . . . . . . 17 |- ( .+ : ( B X. B ) --> C -> Fun .+ )
149	1, 148	syl 16	. . . . . . . . . . . . . . . 16 |- ( ph -> Fun .+ )
150		imafi 7600	. . . . . . . . . . . . . . . 16 |- ( ( Fun .+ /\ ( ran F X. ran G ) e. Fin ) -> ( .+ " ( ran F X. ran G ) ) e. Fin )
151	149, 129, 150	syl2anc 656	. . . . . . . . . . . . . . 15 |- ( ph -> ( .+ " ( ran F X. ran G ) ) e. Fin )
152	18, 20, 9, 23	ofrn2 25877	. . . . . . . . . . . . . . 15 |- ( ph -> ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) )
153		ssfi 7529	. . . . . . . . . . . . . . 15 |- ( ( ( .+ " ( ran F X. ran G ) ) e. Fin /\ ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) ) -> ran ( F oF .+ G ) e. Fin )
154	151, 152, 153	syl2anc 656	. . . . . . . . . . . . . 14 |- ( ph -> ran ( F oF .+ G ) e. Fin )
155		ssdomg 7351	. . . . . . . . . . . . . 14 |- ( 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 ) ) )
156	154, 155	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) ) )
157	63, 156	mpi 17	. . . . . . . . . . . 12 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) )
158		isfinite 7854	. . . . . . . . . . . . . 14 |- ( ran ( F oF .+ G ) e. Fin <-> ran ( F oF .+ G ) ~< om )
159	154, 158	sylib 196	. . . . . . . . . . . . 13 |- ( ph -> ran ( F oF .+ G ) ~< om )
160		sdomdom 7333	. . . . . . . . . . . . 13 |- ( ran ( F oF .+ G ) ~< om -> ran ( F oF .+ G ) ~<_ om )
161	159, 160	syl 16	. . . . . . . . . . . 12 |- ( ph -> ran ( F oF .+ G ) ~<_ om )
162		domtr 7358	. . . . . . . . . . . 12 |- ( ( ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) /\ ran ( F oF .+ G ) ~<_ om ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
163	157, 161, 162	syl2anc 656	. . . . . . . . . . 11 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
164	163	adantr 462	. . . . . . . . . 10 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
165		nfcv 2577	. . . . . . . . . . 11 |- F/_ z ( b i^i ran ( F oF .+ G ) )
166	165	sigaclcuni 26481	. . . . . . . . . 10 |- ( ( 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 )
167	58, 147, 164, 166	syl3anc 1213	. . . . . . . . 9 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
168	62, 167	syl5eqelr 2526	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) e. dom M )
169		difpreima 5828	. . . . . . . . . . . 12 |- ( Fun ( F oF .+ G ) -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) )
170	25, 51, 169	3syl 20	. . . . . . . . . . 11 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) )
171		cnvimarndm 5187	. . . . . . . . . . . . 13 |- ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) = dom ( F oF .+ G )
172	171	difeq2i 3468	. . . . . . . . . . . 12 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) )
173		cnvimass 5186	. . . . . . . . . . . . 13 |- ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G )
174		ssdif0 3734	. . . . . . . . . . . . 13 |- ( ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G ) <-> ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/) )
175	173, 174	mpbi 208	. . . . . . . . . . . 12 |- ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/)
176	172, 175	eqtri 2461	. . . . . . . . . . 11 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = (/)
177	170, 176	syl6eq 2489	. . . . . . . . . 10 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = (/) )
178		0elsiga 26477	. . . . . . . . . . 11 |- ( dom M e. U. ran sigAlgebra -> (/) e. dom M )
179	57, 178	syl 16	. . . . . . . . . 10 |- ( ph -> (/) e. dom M )
180	177, 179	eqeltrd 2515	. . . . . . . . 9 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
181	180	adantr 462	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
182		unelsiga 26497	. . . . . . . 8 |- ( ( 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 )
183	58, 168, 181, 182	syl3anc 1213	. . . . . . 7 |- ( ( 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 )
184	55, 183	eqeltrd 2515	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " b ) e. dom M )
185	184	ralrimiva 2797	. . . . 5 |- ( ph -> A. b e. (sigaGen ` ( TopOpen ` K ) ) ( `' ( F oF .+ G ) " b ) e. dom M )
186	48, 185	jca 529	. . . 4 |- ( ph -> ( ( 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 ) )
187	57, 43	ismbfm 26587	. . . 4 |- ( 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 ) ) )
188	186, 187	mpbird 232	. . 3 |- ( ph -> ( F oF .+ G ) e. ( dom MMblFnM (sigaGen ` ( TopOpen ` K ) ) ) )
189	72	adantr 462	. . . . . . . 8 |- ( ( 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 ) } ) ) )
190	189	fveq2d 5692	. . . . . . 7 |- ( ( 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 ) } ) ) ) )
191		measbasedom 26536	. . . . . . . . . 10 |- ( M e. U. ran measures <-> M e. (measures ` dom M ) )
192	16, 191	sylib 196	. . . . . . . . 9 |- ( ph -> M e. (measures ` dom M ) )
193	192	adantr 462	. . . . . . . 8 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> M e. (measures ` dom M ) )
194		difss 3480	. . . . . . . . . 10 |- ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) C_ ran ( F oF .+ G )
195	194	sseli 3349	. . . . . . . . 9 |- ( z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) -> z e. ran ( F oF .+ G ) )
196	195, 124	sylan2 471	. . . . . . . 8 |- ( ( 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 )
197	138	adantr 462	. . . . . . . . 9 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
198		ffun 5558	. . . . . . . . . . . . . 14 |- ( F : U. dom M --> U. J -> Fun F )
199	18, 198	syl 16	. . . . . . . . . . . . 13 |- ( ph -> Fun F )
200		sndisj 4281	. . . . . . . . . . . . 13 |- Disj x e. ran F { x }
201		disjpreima 25847	. . . . . . . . . . . . 13 |- ( ( Fun F /\ Disj x e. ran F { x } ) -> Disj x e. ran F ( `' F " { x } ) )
202	199, 200, 201	sylancl 657	. . . . . . . . . . . 12 |- ( ph -> Disj x e. ran F ( `' F " { x } ) )
203		ffun 5558	. . . . . . . . . . . . . 14 |- ( G : U. dom M --> U. J -> Fun G )
204	20, 203	syl 16	. . . . . . . . . . . . 13 |- ( ph -> Fun G )
205		sndisj 4281	. . . . . . . . . . . . 13 |- Disj y e. ran G { y }
206		disjpreima 25847	. . . . . . . . . . . . 13 |- ( ( Fun G /\ Disj y e. ran G { y } ) -> Disj y e. ran G ( `' G " { y } ) )
207	204, 205, 206	sylancl 657	. . . . . . . . . . . 12 |- ( ph -> Disj y e. ran G ( `' G " { y } ) )
208		sneq 3884	. . . . . . . . . . . . . 14 |- ( x = ( 1st ` p ) -> { x } = { ( 1st ` p ) } )
209	208	imaeq2d 5166	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( `' F " { x } ) = ( `' F " { ( 1st ` p ) } ) )
210		sneq 3884	. . . . . . . . . . . . . 14 |- ( y = ( 2nd ` p ) -> { y } = { ( 2nd ` p ) } )
211	210	imaeq2d 5166	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` p ) -> ( `' G " { y } ) = ( `' G " { ( 2nd ` p ) } ) )
212		simpl 454	. . . . . . . . . . . . 13 |- ( (Disj x e. ran F ( `' F " { x } ) /\ Disj y e. ran G ( `' G " { y } ) ) -> Disj x e. ran F ( `' F " { x } ) )
213		simpr 458	. . . . . . . . . . . . 13 |- ( (Disj x e. ran F ( `' F " { x } ) /\ Disj y e. ran G ( `' G " { y } ) ) -> Disj y e. ran G ( `' G " { y } ) )
214	209, 211, 212, 213	disjxpin 25849	. . . . . . . . . . . 12 |- ( (Disj x e. ran F ( `' F " { x } ) /\ Disj y e. ran G ( `' G " { y } ) ) -> Disj p e. ( ran F X. ran G ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
215	202, 207, 214	syl2anc 656	. . . . . . . . . . 11 |- ( ph -> Disj p e. ( ran F X. ran G ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
216		disjss1 4265	. . . . . . . . . . 11 |- ( ( ( `' .+ " { 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 ) } ) ) ) )
217	125, 215, 216	mpsyl 63	. . . . . . . . . 10 |- ( ph -> Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
218	217	adantr 462	. . . . . . . . 9 |- ( ( 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 ) } ) ) )
219	197, 218	jca 529	. . . . . . . 8 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( ( `' .+ " { 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 ) } ) ) ) )
220		measvuni 26548	. . . . . . . 8 |- ( ( 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 ) } ) ) ) )
221	193, 196, 219, 220	syl3anc 1213	. . . . . . 7 |- ( ( 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 ) } ) ) ) )
222		ssfi 7529	. . . . . . . . . 10 |- ( ( ( 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 )
223	129, 125, 222	sylancl 657	. . . . . . . . 9 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
224	223	adantr 462	. . . . . . . 8 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
225		rge0ssre 11389	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
226	195, 76	sylanl2 646	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
227		simpr 458	. . . . . . . . . . . . 13 |- ( ( ( 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 ) ) )
228	125, 227	sseldi 3351	. . . . . . . . . . . 12 |- ( ( ( 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 ) )
229		xp1st 6605	. . . . . . . . . . . 12 |- ( p e. ( ran F X. ran G ) -> ( 1st ` p ) e. ran F )
230	228, 229	syl 16	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 1st ` p ) e. ran F )
231		xp2nd 6606	. . . . . . . . . . . 12 |- ( p e. ( ran F X. ran G ) -> ( 2nd ` p ) e. ran G )
232	228, 231	syl 16	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 2nd ` p ) e. ran G )
233		oveq12 6099	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( .0. .+ .0. ) )
234	233	adantl 463	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( x = .0. /\ y = .0. ) ) -> ( x .+ y ) = ( .0. .+ .0. ) )
235		sibfof.5	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( .0. .+ .0. ) = ( 0g ` K ) )
236	235	adantr 462	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( x = .0. /\ y = .0. ) ) -> ( .0. .+ .0. ) = ( 0g ` K ) )
237	234, 236	eqtrd 2473	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( x = .0. /\ y = .0. ) ) -> ( x .+ y ) = ( 0g ` K ) )
238	237	ex 434	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( 0g ` K ) ) )
239	238	necon3ad 2642	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> -. ( x = .0. /\ y = .0. ) ) )
240		oran 493	. . . . . . . . . . . . . . . . 17 |- ( ( x =/= .0. \/ y =/= .0. ) <-> -. ( -. x =/= .0. /\ -. y =/= .0. ) )
241		nne 2610	. . . . . . . . . . . . . . . . . . 19 |- ( -. x =/= .0. <-> x = .0. )
242		nne 2610	. . . . . . . . . . . . . . . . . . 19 |- ( -. y =/= .0. <-> y = .0. )
243	241, 242	anbi12i 692	. . . . . . . . . . . . . . . . . 18 |- ( ( -. x =/= .0. /\ -. y =/= .0. ) <-> ( x = .0. /\ y = .0. ) )
244	243	notbii 296	. . . . . . . . . . . . . . . . 17 |- ( -. ( -. x =/= .0. /\ -. y =/= .0. ) <-> -. ( x = .0. /\ y = .0. ) )
245	240, 244	bitri 249	. . . . . . . . . . . . . . . 16 |- ( ( x =/= .0. \/ y =/= .0. ) <-> -. ( x = .0. /\ y = .0. ) )
246	239, 245	syl6ibr 227	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
247	246	adantr 462	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
248	247	ralrimivva 2806	. . . . . . . . . . . . 13 |- ( ph -> A. x e. B A. y e. B ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
249	226, 248	syl 16	. . . . . . . . . . . 12 |- ( ( ( 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. ) ) )
250	77	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( `' .+ " { z } ) )
251	250	sselda 3353	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( `' .+ " { z } ) )
252	226, 71	syl 16	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> .+ Fn ( B X. B ) )
253		fniniseg 5821	. . . . . . . . . . . . . . . 16 |- ( .+ Fn ( B X. B ) -> ( p e. ( `' .+ " { z } ) <-> ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) ) )
254	252, 253	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( 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 ) ) )
255	251, 254	mpbid 210	. . . . . . . . . . . . . 14 |- ( ( ( 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 ) )
256		simpr 458	. . . . . . . . . . . . . . 15 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = z )
257		1st2nd2 6612	. . . . . . . . . . . . . . . . . 18 |- ( p e. ( B X. B ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
258	257	fveq2d 5692	. . . . . . . . . . . . . . . . 17 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. ) )
259		df-ov 6093	. . . . . . . . . . . . . . . . 17 |- ( ( 1st ` p ) .+ ( 2nd ` p ) ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. )
260	258, 259	syl6eqr 2491	. . . . . . . . . . . . . . . 16 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
261	260	adantr 462	. . . . . . . . . . . . . . 15 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
262	256, 261	eqtr3d 2475	. . . . . . . . . . . . . 14 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> z = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
263	255, 262	syl 16	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> z = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
264		simplr 749	. . . . . . . . . . . . . . 15 |- ( ( ( 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 ) } ) )
265	264	eldifbd 3338	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> -. z e. { ( 0g ` K ) } )
266		elsn 3888	. . . . . . . . . . . . . . 15 |- ( z e. { ( 0g ` K ) } <-> z = ( 0g ` K ) )
267	266	necon3bbii 2637	. . . . . . . . . . . . . 14 |- ( -. z e. { ( 0g ` K ) } <-> z =/= ( 0g ` K ) )
268	265, 267	sylib 196	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> z =/= ( 0g ` K ) )
269	263, 268	eqnetrrd 2626	. . . . . . . . . . . 12 |- ( ( ( 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 ) )
270	195, 84	sylanl2 646	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( B X. B ) )
271	270, 100	syl 16	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 1st ` p ) e. B )
272	270, 113	syl 16	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 2nd ` p ) e. B )
273		oveq1 6097	. . . . . . . . . . . . . . . 16 |- ( x = ( 1st ` p ) -> ( x .+ y ) = ( ( 1st ` p ) .+ y ) )
274	273	neeq1d 2619	. . . . . . . . . . . . . . 15 |- ( x = ( 1st ` p ) -> ( ( x .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) ) )
275		neeq1 2614	. . . . . . . . . . . . . . . 16 |- ( x = ( 1st ` p ) -> ( x =/= .0. <-> ( 1st ` p ) =/= .0. ) )
276	275	orbi1d 697	. . . . . . . . . . . . . . 15 |- ( x = ( 1st ` p ) -> ( ( x =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) )
277	274, 276	imbi12d 320	. . . . . . . . . . . . . 14 |- ( x = ( 1st ` p ) -> ( ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) <-> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) ) )
278		oveq2 6098	. . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` p ) -> ( ( 1st ` p ) .+ y ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
279	278	neeq1d 2619	. . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) ) )
280		neeq1 2614	. . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` p ) -> ( y =/= .0. <-> ( 2nd ` p ) =/= .0. ) )
281	280	orbi2d 696	. . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) )
282	279, 281	imbi12d 320	. . . . . . . . . . . . . 14 |- ( 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. ) ) ) )
283	277, 282	rspc2v 3076	. . . . . . . . . . . . 13 |- ( ( ( 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. ) ) ) )
284	271, 272, 283	syl2anc 656	. . . . . . . . . . . 12 |- ( ( ( 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. ) ) ) )
285	249, 269, 284	mp2d 45	. . . . . . . . . . 11 |- ( ( ( 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. ) )
286	3, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 98	sibfinima 26639	. . . . . . . . . . 11 |- ( ( ( 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 ) )
287	226, 230, 232, 285, 286	syl31anc 1216	. . . . . . . . . 10 |- ( ( ( 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 ) )
288	225, 287	sseldi 3351	. . . . . . . . 9 |- ( ( ( 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 )
289	193	adantr 462	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> M e. (measures ` dom M ) )
290	195, 123	sylanl2 646	. . . . . . . . . 10 |- ( ( ( 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 )
291		measge0 26541	. . . . . . . . . 10 |- ( ( 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 ) } ) ) ) )
292	289, 290, 291	syl2anc 656	. . . . . . . . 9 |- ( ( ( 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 ) } ) ) ) )
293		elrege0 11388	. . . . . . . . 9 |- ( ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. ( 0 [,) +oo ) <-> ( ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. RR /\ 0 <_ ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) )
294	288, 292, 293	sylanbrc 659	. . . . . . . 8 |- ( ( ( 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 ) )
295	224, 294	esumpfinval 26444	. . . . . . 7 |- ( ( 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 ) } ) ) ) )
296	190, 221, 295	3eqtrd 2477	. . . . . 6 |- ( ( 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 ) } ) ) ) )
297	224, 288	fsumrecl 13207	. . . . . 6 |- ( ( 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 )
298	296, 297	eqeltrd 2515	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. RR )
299	224, 288, 292	fsumge0 13254	. . . . . 6 |- ( ( 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 ) } ) ) ) )
300	299, 296	breqtrrd 4315	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> 0 <_ ( M ` ( `' ( F oF .+ G ) " { z } ) ) )
301		elrege0 11388	. . . . 5 |- ( ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) <-> ( ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. RR /\ 0 <_ ( M ` ( `' ( F oF .+ G ) " { z } ) ) ) )
302	298, 300, 301	sylanbrc 659	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
303	302	ralrimiva 2797	. . 3 |- ( ph -> A. z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
304	188, 154, 303	3jca 1163	. 2 |- ( ph -> ( ( 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 ) ) )
305		eqid 2441	. . 3 |- ( 0g ` K ) = ( 0g ` K )
306		eqid 2441	. . 3 |- ( .s ` K ) = ( .s ` K )
307		eqid 2441	. . 3 |- (RRHom ` (Scalar ` K ) ) = (RRHom ` (Scalar ` K ) )
308	27, 28, 31, 305, 306, 307, 26, 16	issibf 26633	. 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 ) ) ) )
309	304, 308	mpbird 232	1 |- ( ph -> ( F oF .+ G ) e. dom ( Ksitg M ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   =/= wne 2604   A.wral 2713   _Vcvv 2970   \ cdif 3322   u. cun 3323   i^i cin 3324   C_ wss 3325   (/)c0 3634   {csn 3874   <.cop 3880   U.cuni 4088   U_ciun 4168  Disj wdisj 4259   class class class wbr 4289   X. cxp 4834   `'ccnv 4835   dom cdm 4836   ran crn 4837   "cima 4839   Fun wfun 5409   Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   oFcof 6317   omcom 6475   1stc1st 6574   2ndc2nd 6575   ^m cmap 7210   ~<_ cdom 7304   ~< csdm 7305   Fincfn 7306   RRcr 9277   0cc0 9278   +oocpnf 9411   <_ cle 9415   [,)cico 11298   sum_csu 13159   Basecbs 14170  Scalarcsca 14237   .scvsca 14238   TopOpenctopn 14356   0gc0g 14374   Topctop 18398   TopSpctps 18401   Clsdccld 18520   Frect1 18811  RRHomcrrh 26342  Σ^*cesum 26403  sigAlgebracsiga 26470  sigaGencsigagen 26501  measurescmeas 26529  MblFnMcmbfm 26585  sitgcsitg 26629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-ac2 8628  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-ac 8282  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-ordt 14435  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-ps 15366  df-tsr 15367  df-mnd 15411  df-plusf 15412  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-subrg 16843  df-abv 16882  df-lmod 16930  df-scaf 16931  df-sra 17231  df-rgmod 17232  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-t1 18818  df-haus 18819  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-tmd 19543  df-tgp 19544  df-tsms 19597  df-trg 19634  df-xms 19795  df-ms 19796  df-tms 19797  df-nm 20075  df-ngp 20076  df-nrg 20078  df-nlm 20079  df-ii 20353  df-cncf 20354  df-limc 21241  df-dv 21242  df-log 21951  df-esum 26404  df-siga 26471  df-sigagen 26502  df-meas 26530  df-mbfm 26586  df-sitg 26630

This theorem is referenced by: (None)