Theorem sibfof 27950

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 19234	. . . . . . . . . . . . 13 |- ( W e. TopSp -> B = U. J )
6	2, 5	syl 16	. . . . . . . . . . . 12 |- ( ph -> B = U. J )
7	6, 6	xpeq12d 5024	. . . . . . . . . . 11 |- ( ph -> ( B X. B ) = ( U. J X. U. J ) )
8	7	feq2d 5718	. . . . . . . . . 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 6430	. . . . . . . 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 27946	. . . . . . . 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 27946	. . . . . . . 8 |- ( ph -> G : U. dom M --> U. J )
21		dmexg 6715	. . . . . . . . 9 |- ( M e. U. ran measures -> dom M e. _V )
22		uniexg 6581	. . . . . . . . 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 3707	. . . . . . . 8 |- ( U. dom M i^i U. dom M ) = U. dom M
25	10, 18, 20, 23, 23, 24	off 6538	. . . . . . 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 2467	. . . . . . . . . . 11 |- ( TopOpen ` K ) = ( TopOpen ` K )
29	27, 28	tpsuni 19234	. . . . . . . . . 10 |- ( K e. TopSp -> C = U. ( TopOpen ` K ) )
30	26, 29	syl 16	. . . . . . . . 9 |- ( ph -> C = U. ( TopOpen ` K ) )
31		eqid 2467	. . . . . . . . . . 11 |- (sigaGen ` ( TopOpen ` K ) ) = (sigaGen ` ( TopOpen ` K ) )
32	31	unieqi 4254	. . . . . . . . . 10 |- U. (sigaGen ` ( TopOpen ` K ) ) = U. (sigaGen ` ( TopOpen ` K ) )
33		fvex 5876	. . . . . . . . . . 11 |- ( TopOpen ` K ) e. _V
34		unisg 27811	. . . . . . . . . . 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 2496	. . . . . . . . 9 |- U. (sigaGen ` ( TopOpen ` K ) ) = U. ( TopOpen ` K )
37	30, 36	syl6eqr 2526	. . . . . . . 8 |- ( ph -> C = U. (sigaGen ` ( TopOpen ` K ) ) )
38		feq3 5715	. . . . . . . 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 27810	. . . . . . . . 9 |- ( ph -> (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra )
43	31, 42	syl5eqel 2559	. . . . . . . 8 |- ( ph -> (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra )
44		uniexg 6581	. . . . . . . 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 7433	. . . . . . 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 661	. . . . . 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 3905	. . . . . . . . 9 |- ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) = b
50	49	imaeq2i 5335	. . . . . . . 8 |- ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( `' ( F oF .+ G ) " b )
51		ffun 5733	. . . . . . . . . 10 |- ( ( F oF .+ G ) : U. dom M --> C -> Fun ( F oF .+ G ) )
52		unpreima 6007	. . . . . . . . . 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 465	. . . . . . . 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 2522	. . . . . . 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 27840	. . . . . . . . . 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 465	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> dom M e. U. ran sigAlgebra )
59		imaiun 6145	. . . . . . . . . 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 4380	. . . . . . . . . . 11 |- U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } = ( b i^i ran ( F oF .+ G ) )
61	60	imaeq2i 5335	. . . . . . . . . 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 2498	. . . . . . . . 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 3719	. . . . . . . . . . . 12 |- ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G )
64		feq3 5715	. . . . . . . . . . . . . . . . . 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 5715	. . . . . . . . . . . . . . . . . 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 5731	. . . . . . . . . . . . . . . . 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 27208	. . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> dom M e. U. ran sigAlgebra )
75	57	ad2antrr 725	. . . . . . . . . . . . . . . . 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 753	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
77		inss1 3718	. . . . . . . . . . . . . . . . . . . 20 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( `' .+ " { z } )
78		cnvimass 5357	. . . . . . . . . . . . . . . . . . . . . 22 |- ( `' .+ " { z } ) C_ dom .+
79		fdm 5735	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( .+ : ( B X. B ) --> C -> dom .+ = ( B X. B ) )
80	1, 79	syl 16	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> dom .+ = ( B X. B ) )
81	78, 80	syl5sseq 3552	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> ( `' .+ " { z } ) C_ ( B X. B ) )
82	81	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' .+ " { z } ) C_ ( B X. B ) )
83	77, 82	syl5ss 3515	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( B X. B ) )
84	83	sselda 3504	. . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> dom M e. U. ran sigAlgebra )
86		fvex 5876	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( TopOpen ` W ) e. _V
87	4, 86	eqeltri 2551	. . . . . . . . . . . . . . . . . . . . . . 23 |- J e. _V
88	87	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> J e. _V )
89	88	sgsiga 27810	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
90	11, 89	syl5eqel 2559	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> S e. U. ran sigAlgebra )
91	90	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> S e. U. ran sigAlgebra )
92	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfmbl 27945	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> F e. ( dom MMblFnM S ) )
93	92	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> F e. ( dom MMblFnM S ) )
94	4	tpstop 19235	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( W e. TopSp -> J e. Top )
95		cldssbrsiga 27826	. . . . . . . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( Clsd ` J ) C_ (sigaGen ` J ) )
98		sibfof.4	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> J e. Fre )
99	98	adantr 465	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ p e. ( B X. B ) ) -> J e. Fre )
100		xp1st 6814	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( p e. ( B X. B ) -> ( 1st ` p ) e. B )
101	100	adantl 466	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. B )
102	6	adantr 465	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ph /\ p e. ( B X. B ) ) -> B = U. J )
103	101, 102	eleqtrd 2557	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. U. J )
104		eqid 2467	. . . . . . . . . . . . . . . . . . . . . . 23 |- U. J = U. J
105	104	t1sncld 19621	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( J e. Fre /\ ( 1st ` p ) e. U. J ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
106	99, 103, 105	syl2anc 661	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
107	97, 106	sseldd 3505	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. (sigaGen ` J ) )
108	107, 11	syl6eleqr 2566	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. S )
109	85, 91, 93, 108	mbfmcnvima 27896	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' F " { ( 1st ` p ) } ) e. dom M )
110	76, 84, 109	syl2anc 661	. . . . . . . . . . . . . . . . 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 27945	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> G e. ( dom MMblFnM S ) )
112	111	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> G e. ( dom MMblFnM S ) )
113		xp2nd 6815	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( p e. ( B X. B ) -> ( 2nd ` p ) e. B )
114	113	adantl 466	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. B )
115	114, 102	eleqtrd 2557	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. U. J )
116	104	t1sncld 19621	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( J e. Fre /\ ( 2nd ` p ) e. U. J ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
117	99, 115, 116	syl2anc 661	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
118	97, 117	sseldd 3505	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. (sigaGen ` J ) )
119	118, 11	syl6eleqr 2566	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. S )
120	85, 91, 112, 119	mbfmcnvima 27896	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' G " { ( 2nd ` p ) } ) e. dom M )
121	76, 84, 120	syl2anc 661	. . . . . . . . . . . . . . . . 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 27803	. . . . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . . . 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 2878	. . . . . . . . . . . . . . 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 3719	. . . . . . . . . . . . . . . . . 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 27947	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ran F e. Fin )
127	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 27947	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ran G e. Fin )
128		xpfi 7791	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
129	126, 127, 128	syl2anc 661	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( ran F X. ran G ) e. Fin )
130		ssdomg 7561	. . . . . . . . . . . . . . . . . . 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 8069	. . . . . . . . . . . . . . . . . . 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 7543	. . . . . . . . . . . . . . . . . 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 7568	. . . . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
139	138	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
140		nfcv 2629	. . . . . . . . . . . . . . . 16 |- F/_ p ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) )
141	140	sigaclcuni 27786	. . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 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 2555	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' ( F oF .+ G ) " { z } ) e. dom M )
144	143	ralrimiva 2878	. . . . . . . . . . . 12 |- ( ph -> A. z e. ran ( F oF .+ G ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
145		ssralv 3564	. . . . . . . . . . . 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 465	. . . . . . . . . 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 5733	. . . . . . . . . . . . . . . . 17 |- ( .+ : ( B X. B ) --> C -> Fun .+ )
149	1, 148	syl 16	. . . . . . . . . . . . . . . 16 |- ( ph -> Fun .+ )
150		imafi 7813	. . . . . . . . . . . . . . . 16 |- ( ( Fun .+ /\ ( ran F X. ran G ) e. Fin ) -> ( .+ " ( ran F X. ran G ) ) e. Fin )
151	149, 129, 150	syl2anc 661	. . . . . . . . . . . . . . 15 |- ( ph -> ( .+ " ( ran F X. ran G ) ) e. Fin )
152	18, 20, 9, 23	ofrn2 27181	. . . . . . . . . . . . . . 15 |- ( ph -> ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) )
153		ssfi 7740	. . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . 14 |- ( ph -> ran ( F oF .+ G ) e. Fin )
155		ssdomg 7561	. . . . . . . . . . . . . 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 8069	. . . . . . . . . . . . . 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 7543	. . . . . . . . . . . . 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 7568	. . . . . . . . . . . 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 661	. . . . . . . . . . 11 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
164	163	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
165		nfcv 2629	. . . . . . . . . . 11 |- F/_ z ( b i^i ran ( F oF .+ G ) )
166	165	sigaclcuni 27786	. . . . . . . . . 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 1228	. . . . . . . . 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 2560	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) e. dom M )
169		difpreima 6009	. . . . . . . . . . . 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 5358	. . . . . . . . . . . . 13 |- ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) = dom ( F oF .+ G )
172	171	difeq2i 3619	. . . . . . . . . . . 12 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) )
173		cnvimass 5357	. . . . . . . . . . . . 13 |- ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G )
174		ssdif0 3885	. . . . . . . . . . . . 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 2496	. . . . . . . . . . 11 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = (/)
177	170, 176	syl6eq 2524	. . . . . . . . . 10 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = (/) )
178		0elsiga 27782	. . . . . . . . . . 11 |- ( dom M e. U. ran sigAlgebra -> (/) e. dom M )
179	57, 178	syl 16	. . . . . . . . . 10 |- ( ph -> (/) e. dom M )
180	177, 179	eqeltrd 2555	. . . . . . . . 9 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
181	180	adantr 465	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
182		unelsiga 27802	. . . . . . . 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 1228	. . . . . . 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 2555	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " b ) e. dom M )
185	184	ralrimiva 2878	. . . . 5 |- ( ph -> A. b e. (sigaGen ` ( TopOpen ` K ) ) ( `' ( F oF .+ G ) " b ) e. dom M )
186	48, 185	jca 532	. . . 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 27891	. . . 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 465	. . . . . . . 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 5870	. . . . . . 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 27841	. . . . . . . . . 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 465	. . . . . . . 8 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> M e. (measures ` dom M ) )
194		difss 3631	. . . . . . . . . 10 |- ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) C_ ran ( F oF .+ G )
195	194	sseli 3500	. . . . . . . . 9 |- ( z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) -> z e. ran ( F oF .+ G ) )
196	195, 124	sylan2 474	. . . . . . . 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 465	. . . . . . . . 9 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
198		ffun 5733	. . . . . . . . . . . . . 14 |- ( F : U. dom M --> U. J -> Fun F )
199	18, 198	syl 16	. . . . . . . . . . . . 13 |- ( ph -> Fun F )
200		sndisj 4439	. . . . . . . . . . . . 13 |- Disj x e. ran F { x }
201		disjpreima 27146	. . . . . . . . . . . . 13 |- ( ( Fun F /\ Disj x e. ran F { x } ) -> Disj x e. ran F ( `' F " { x } ) )
202	199, 200, 201	sylancl 662	. . . . . . . . . . . 12 |- ( ph -> Disj x e. ran F ( `' F " { x } ) )
203		ffun 5733	. . . . . . . . . . . . . 14 |- ( G : U. dom M --> U. J -> Fun G )
204	20, 203	syl 16	. . . . . . . . . . . . 13 |- ( ph -> Fun G )
205		sndisj 4439	. . . . . . . . . . . . 13 |- Disj y e. ran G { y }
206		disjpreima 27146	. . . . . . . . . . . . 13 |- ( ( Fun G /\ Disj y e. ran G { y } ) -> Disj y e. ran G ( `' G " { y } ) )
207	204, 205, 206	sylancl 662	. . . . . . . . . . . 12 |- ( ph -> Disj y e. ran G ( `' G " { y } ) )
208		sneq 4037	. . . . . . . . . . . . . 14 |- ( x = ( 1st ` p ) -> { x } = { ( 1st ` p ) } )
209	208	imaeq2d 5337	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( `' F " { x } ) = ( `' F " { ( 1st ` p ) } ) )
210		sneq 4037	. . . . . . . . . . . . . 14 |- ( y = ( 2nd ` p ) -> { y } = { ( 2nd ` p ) } )
211	210	imaeq2d 5337	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` p ) -> ( `' G " { y } ) = ( `' G " { ( 2nd ` p ) } ) )
212		simpl 457	. . . . . . . . . . . . 13 |- ( (Disj x e. ran F ( `' F " { x } ) /\ Disj y e. ran G ( `' G " { y } ) ) -> Disj x e. ran F ( `' F " { x } ) )
213		simpr 461	. . . . . . . . . . . . 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 27148	. . . . . . . . . . . 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 661	. . . . . . . . . . 11 |- ( ph -> Disj p e. ( ran F X. ran G ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
216		disjss1 4423	. . . . . . . . . . 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 465	. . . . . . . . 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 532	. . . . . . . 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 27853	. . . . . . . 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 1228	. . . . . . 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 7740	. . . . . . . . . 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 662	. . . . . . . . 9 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
224	223	adantr 465	. . . . . . . 8 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
225		rge0ssre 11628	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
226	195, 76	sylanl2 651	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
227		simpr 461	. . . . . . . . . . . . 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 3502	. . . . . . . . . . . 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 6814	. . . . . . . . . . . 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 6815	. . . . . . . . . . . 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 6293	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( .0. .+ .0. ) )
234	233	adantl 466	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( x = .0. /\ y = .0. ) ) -> ( x .+ y ) = ( .0. .+ .0. ) )
235		sibfof.5	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( .0. .+ .0. ) = ( 0g ` K ) )
236	235	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( x = .0. /\ y = .0. ) ) -> ( .0. .+ .0. ) = ( 0g ` K ) )
237	234, 236	eqtrd 2508	. . . . . . . . . . . . . . . . . 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 2677	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> -. ( x = .0. /\ y = .0. ) ) )
240		oran 496	. . . . . . . . . . . . . . . . 17 |- ( ( x =/= .0. \/ y =/= .0. ) <-> -. ( -. x =/= .0. /\ -. y =/= .0. ) )
241		nne 2668	. . . . . . . . . . . . . . . . . . 19 |- ( -. x =/= .0. <-> x = .0. )
242		nne 2668	. . . . . . . . . . . . . . . . . . 19 |- ( -. y =/= .0. <-> y = .0. )
243	241, 242	anbi12i 697	. . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
248	247	ralrimivva 2885	. . . . . . . . . . . . 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 3504	. . . . . . . . . . . . . . 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 6002	. . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . 15 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = z )
257		1st2nd2 6821	. . . . . . . . . . . . . . . . . 18 |- ( p e. ( B X. B ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
258	257	fveq2d 5870	. . . . . . . . . . . . . . . . 17 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. ) )
259		df-ov 6287	. . . . . . . . . . . . . . . . 17 |- ( ( 1st ` p ) .+ ( 2nd ` p ) ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. )
260	258, 259	syl6eqr 2526	. . . . . . . . . . . . . . . 16 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
261	260	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
262	256, 261	eqtr3d 2510	. . . . . . . . . . . . . 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 754	. . . . . . . . . . . . . . 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 3489	. . . . . . . . . . . . . 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 4041	. . . . . . . . . . . . . . 15 |- ( z e. { ( 0g ` K ) } <-> z = ( 0g ` K ) )
267	266	necon3bbii 2728	. . . . . . . . . . . . . 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 2761	. . . . . . . . . . . 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 651	. . . . . . . . . . . . . 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 6291	. . . . . . . . . . . . . . . 16 |- ( x = ( 1st ` p ) -> ( x .+ y ) = ( ( 1st ` p ) .+ y ) )
274	273	neeq1d 2744	. . . . . . . . . . . . . . 15 |- ( x = ( 1st ` p ) -> ( ( x .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) ) )
275		neeq1 2748	. . . . . . . . . . . . . . . 16 |- ( x = ( 1st ` p ) -> ( x =/= .0. <-> ( 1st ` p ) =/= .0. ) )
276	275	orbi1d 702	. . . . . . . . . . . . . . 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 6292	. . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` p ) -> ( ( 1st ` p ) .+ y ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
279	278	neeq1d 2744	. . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) ) )
280		neeq1 2748	. . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` p ) -> ( y =/= .0. <-> ( 2nd ` p ) =/= .0. ) )
281	280	orbi2d 701	. . . . . . . . . . . . . . 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 3223	. . . . . . . . . . . . 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 661	. . . . . . . . . . . 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 27949	. . . . . . . . . . 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 1231	. . . . . . . . . 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 3502	. . . . . . . . 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 465	. . . . . . . . . 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 651	. . . . . . . . . 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 27846	. . . . . . . . . 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 661	. . . . . . . . 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 11627	. . . . . . . . 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 664	. . . . . . . 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 27749	. . . . . . 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 2512	. . . . . 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 13519	. . . . . 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 2555	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. RR )
299	224, 288, 292	fsumge0 13572	. . . . . 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 4473	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> 0 <_ ( M ` ( `' ( F oF .+ G ) " { z } ) ) )
301		elrege0 11627	. . . . 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 664	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
303	302	ralrimiva 2878	. . 3 |- ( ph -> A. z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
304	188, 154, 303	3jca 1176	. 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 2467	. . 3 |- ( 0g ` K ) = ( 0g ` K )
306		eqid 2467	. . 3 |- ( .s ` K ) = ( .s ` K )
307		eqid 2467	. . 3 |- (RRHom ` (Scalar ` K ) ) = (RRHom ` (Scalar ` K ) )
308	27, 28, 31, 305, 306, 307, 26, 16	issibf 27943	. 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 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   _Vcvv 3113   \ cdif 3473   u. cun 3474   i^i cin 3475   C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   U_ciun 4325  Disj wdisj 4417   class class class wbr 4447   X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   Fun wfun 5582   Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   oFcof 6522   omcom 6684   1stc1st 6782   2ndc2nd 6783   ^m cmap 7420   ~<_ cdom 7514   ~< csdm 7515   Fincfn 7516   RRcr 9491   0cc0 9492   +oocpnf 9625   <_ cle 9629   [,)cico 11531   sum_csu 13471   Basecbs 14490  Scalarcsca 14558   .scvsca 14559   TopOpenctopn 14677   0gc0g 14695   Topctop 19189   TopSpctps 19192   Clsdccld 19311   Frect1 19602  RRHomcrrh 27638  Σ^*cesum 27708  sigAlgebracsiga 27775  sigaGencsigagen 27806  measurescmeas 27834  MblFnMcmbfm 27889  sitgcsitg 27939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-ac2 8843  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-ac 8497  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-ordt 14756  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-ps 15687  df-tsr 15688  df-mnd 15732  df-plusf 15733  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-cntz 16160  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-subrg 17227  df-abv 17266  df-lmod 17314  df-scaf 17315  df-sra 17618  df-rgmod 17619  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-t1 19609  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-tmd 20334  df-tgp 20335  df-tsms 20388  df-trg 20425  df-xms 20586  df-ms 20587  df-tms 20588  df-nm 20866  df-ngp 20867  df-nrg 20869  df-nlm 20870  df-ii 21144  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700  df-esum 27709  df-siga 27776  df-sigagen 27807  df-meas 27835  df-mbfm 27890  df-sitg 27940

This theorem is referenced by: (None)