Theorem sibfof 34331

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	⊢ · = ( ·𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
sibfmbl.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
sibfof.c	⊢ 𝐶 = (Base‘𝐾)
sibfof.0	⊢ (𝜑 → 𝑊 ∈ TopSp)
sibfof.1	⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶)
sibfof.2	⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))
sibfof.3	⊢ (𝜑 → 𝐾 ∈ TopSp)
sibfof.4	⊢ (𝜑 → 𝐽 ∈ Fre)
sibfof.5	⊢ (𝜑 → ( 0 + 0 ) = (0g‘𝐾))

Assertion

Ref	Expression
sibfof	⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom (𝐾sitg𝑀))

Proof of Theorem sibfof

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

Step	Hyp	Ref	Expression
1		sibfof.1	. . . . . . . 8 ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶)
2		sibfof.0	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ TopSp)
3		sitgval.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑊)
4		sitgval.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (TopOpen‘𝑊)
5	3, 4	tpsuni 22823	. . . . . . . . . . 11 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
6	2, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
7	6	sqxpeqd 5670	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐵) = (∪ 𝐽 × ∪ 𝐽))
8	7	feq2d 6672	. . . . . . . 8 ⊢ (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶 ↔ + :(∪ 𝐽 × ∪ 𝐽)⟶𝐶))
9	1, 8	mpbid 232	. . . . . . 7 ⊢ (𝜑 → + :(∪ 𝐽 × ∪ 𝐽)⟶𝐶)
10	9	fovcdmda 7560	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽)) → (𝑧 + 𝑥) ∈ 𝐶)
11		sitgval.s	. . . . . . 7 ⊢ 𝑆 = (sigaGen‘𝐽)
12		sitgval.0	. . . . . . 7 ⊢ 0 = (0g‘𝑊)
13		sitgval.x	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
14		sitgval.h	. . . . . . 7 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
15		sitgval.1	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
16		sitgval.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
17		sibfmbl.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
18	3, 4, 11, 12, 13, 14, 15, 16, 17	sibff 34327	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)
19		sibfof.2	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))
20	3, 4, 11, 12, 13, 14, 15, 16, 19	sibff 34327	. . . . . 6 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
21		dmexg 7877	. . . . . . 7 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ V)
22		uniexg 7716	. . . . . . 7 ⊢ (dom 𝑀 ∈ V → ∪ dom 𝑀 ∈ V)
23	16, 21, 22	3syl 18	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑀 ∈ V)
24		inidm 4190	. . . . . 6 ⊢ (∪ dom 𝑀 ∩ ∪ dom 𝑀) = ∪ dom 𝑀
25	10, 18, 20, 23, 23, 24	off 7671	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):∪ dom 𝑀⟶𝐶)
26		sibfof.3	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ TopSp)
27		sibfof.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝐾)
28		eqid 2729	. . . . . . . . 9 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
29	27, 28	tpsuni 22823	. . . . . . . 8 ⊢ (𝐾 ∈ TopSp → 𝐶 = ∪ (TopOpen‘𝐾))
30	26, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 = ∪ (TopOpen‘𝐾))
31		fvex 6871	. . . . . . . 8 ⊢ (TopOpen‘𝐾) ∈ V
32		unisg 34133	. . . . . . . 8 ⊢ ((TopOpen‘𝐾) ∈ V → ∪ (sigaGen‘(TopOpen‘𝐾)) = ∪ (TopOpen‘𝐾))
33	31, 32	ax-mp 5	. . . . . . 7 ⊢ ∪ (sigaGen‘(TopOpen‘𝐾)) = ∪ (TopOpen‘𝐾)
34	30, 33	eqtr4di 2782	. . . . . 6 ⊢ (𝜑 → 𝐶 = ∪ (sigaGen‘(TopOpen‘𝐾)))
35	34	feq3d 6673	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘f + 𝐺):∪ dom 𝑀⟶𝐶 ↔ (𝐹 ∘f + 𝐺):∪ dom 𝑀⟶∪ (sigaGen‘(TopOpen‘𝐾))))
36	25, 35	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):∪ dom 𝑀⟶∪ (sigaGen‘(TopOpen‘𝐾)))
37	31	a1i 11	. . . . . . 7 ⊢ (𝜑 → (TopOpen‘𝐾) ∈ V)
38	37	sgsiga 34132	. . . . . 6 ⊢ (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ∪ ran sigAlgebra)
39	38	uniexd 7718	. . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘(TopOpen‘𝐾)) ∈ V)
40	39, 23	elmapd 8813	. . . 4 ⊢ (𝜑 → ((𝐹 ∘f + 𝐺) ∈ (∪ (sigaGen‘(TopOpen‘𝐾)) ↑m ∪ dom 𝑀) ↔ (𝐹 ∘f + 𝐺):∪ dom 𝑀⟶∪ (sigaGen‘(TopOpen‘𝐾))))
41	36, 40	mpbird 257	. . 3 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (∪ (sigaGen‘(TopOpen‘𝐾)) ↑m ∪ dom 𝑀))
42		inundif 4442	. . . . . . 7 ⊢ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) = 𝑏
43	42	imaeq2i 6029	. . . . . 6 ⊢ (◡(𝐹 ∘f + 𝐺) “ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))) = (◡(𝐹 ∘f + 𝐺) “ 𝑏)
44		ffun 6691	. . . . . . . 8 ⊢ ((𝐹 ∘f + 𝐺):∪ dom 𝑀⟶𝐶 → Fun (𝐹 ∘f + 𝐺))
45		unpreima 7035	. . . . . . . 8 ⊢ (Fun (𝐹 ∘f + 𝐺) → (◡(𝐹 ∘f + 𝐺) “ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))) = ((◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∪ (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))))
46	25, 44, 45	3syl 18	. . . . . . 7 ⊢ (𝜑 → (◡(𝐹 ∘f + 𝐺) “ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))) = ((◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∪ (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))))
47	46	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))) = ((◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∪ (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))))
48	43, 47	eqtr3id 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ 𝑏) = ((◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∪ (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))))
49		dmmeas 34191	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
50	16, 49	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
51	50	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → dom 𝑀 ∈ ∪ ran sigAlgebra)
52		imaiun 7219	. . . . . . . 8 ⊢ (◡(𝐹 ∘f + 𝐺) “ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)){𝑧}) = ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧})
53		iunid 5024	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹 ∘f + 𝐺))
54	53	imaeq2i 6029	. . . . . . . 8 ⊢ (◡(𝐹 ∘f + 𝐺) “ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)){𝑧}) = (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)))
55	52, 54	eqtr3i 2754	. . . . . . 7 ⊢ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) = (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)))
56		inss2 4201	. . . . . . . . . 10 ⊢ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ⊆ ran (𝐹 ∘f + 𝐺)
57	6	feq3d 6673	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶𝐵 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽))
58	18, 57	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶𝐵)
59	6	feq3d 6673	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺:∪ dom 𝑀⟶𝐵 ↔ 𝐺:∪ dom 𝑀⟶∪ 𝐽))
60	20, 59	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶𝐵)
61	1	ffnd 6689	. . . . . . . . . . . . . 14 ⊢ (𝜑 → + Fn (𝐵 × 𝐵))
62	58, 60, 23, 61	ofpreima2 32590	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡(𝐹 ∘f + 𝐺) “ {𝑧}) = ∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
63	62	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → (◡(𝐹 ∘f + 𝐺) “ {𝑧}) = ∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
64	50	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → dom 𝑀 ∈ ∪ ran sigAlgebra)
65	50	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → dom 𝑀 ∈ ∪ ran sigAlgebra)
66		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
67		inss1 4200	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (◡ + “ {𝑧})
68		cnvimass 6053	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡ + “ {𝑧}) ⊆ dom +
69	68, 1	fssdm 6707	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (◡ + “ {𝑧}) ⊆ (𝐵 × 𝐵))
70	69	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → (◡ + “ {𝑧}) ⊆ (𝐵 × 𝐵))
71	67, 70	sstrid 3958	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
72	71	sselda 3946	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
73	50	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → dom 𝑀 ∈ ∪ ran sigAlgebra)
74		sibfof.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ Fre)
75	74	sgsiga 34132	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
76	11, 75	eqeltrid 2832	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
77	76	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝑆 ∈ ∪ ran sigAlgebra)
78	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfmbl 34326	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
79	78	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
80	4	tpstop 22824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
81		cldssbrsiga 34177	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
82	2, 80, 81	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
83	82	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
84	74	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐽 ∈ Fre)
85		xp1st 8000	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → (1st ‘𝑝) ∈ 𝐵)
86	85	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (1st ‘𝑝) ∈ 𝐵)
87	6	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐵 = ∪ 𝐽)
88	86, 87	eleqtrd 2830	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (1st ‘𝑝) ∈ ∪ 𝐽)
89		eqid 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ 𝐽 = ∪ 𝐽
90	89	t1sncld 23213	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Fre ∧ (1st ‘𝑝) ∈ ∪ 𝐽) → {(1st ‘𝑝)} ∈ (Clsd‘𝐽))
91	84, 88, 90	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ (Clsd‘𝐽))
92	83, 91	sseldd 3947	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ (sigaGen‘𝐽))
93	92, 11	eleqtrrdi 2839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ 𝑆)
94	73, 77, 79, 93	mbfmcnvima 34246	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (◡𝐹 “ {(1st ‘𝑝)}) ∈ dom 𝑀)
95	66, 72, 94	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (◡𝐹 “ {(1st ‘𝑝)}) ∈ dom 𝑀)
96	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfmbl 34326	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
97	96	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
98		xp2nd 8001	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵)
99	98	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (2nd ‘𝑝) ∈ 𝐵)
100	99, 87	eleqtrd 2830	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (2nd ‘𝑝) ∈ ∪ 𝐽)
101	89	t1sncld 23213	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Fre ∧ (2nd ‘𝑝) ∈ ∪ 𝐽) → {(2nd ‘𝑝)} ∈ (Clsd‘𝐽))
102	84, 100, 101	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ (Clsd‘𝐽))
103	83, 102	sseldd 3947	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ (sigaGen‘𝐽))
104	103, 11	eleqtrrdi 2839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ 𝑆)
105	73, 77, 97, 104	mbfmcnvima 34246	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀)
106	66, 72, 105	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀)
107		inelsiga 34125	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (◡𝐹 “ {(1st ‘𝑝)}) ∈ dom 𝑀 ∧ (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
108	65, 95, 106, 107	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
109	108	ralrimiva 3125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → ∀𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
110	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfrn 34328	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
111	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 34328	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
112		xpfi 9269	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
113	110, 111, 112	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
114		inss2 4201	. . . . . . . . . . . . . . . 16 ⊢ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
115		ssdomg 8971	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝐹 × ran 𝐺) ∈ Fin → (((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺)))
116	113, 114, 115	mpisyl 21	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺))
117		isfinite 9605	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 × ran 𝐺) ∈ Fin ↔ (ran 𝐹 × ran 𝐺) ≺ ω)
118	117	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝐹 × ran 𝐺) ∈ Fin → (ran 𝐹 × ran 𝐺) ≺ ω)
119		sdomdom 8951	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝐹 × ran 𝐺) ≺ ω → (ran 𝐹 × ran 𝐺) ≼ ω)
120	113, 118, 119	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ≼ ω)
121		domtr 8978	. . . . . . . . . . . . . . 15 ⊢ ((((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺) ∧ (ran 𝐹 × ran 𝐺) ≼ ω) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
122	116, 120, 121	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
123	122	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
124		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑝((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
125	124	sigaclcuni 34108	. . . . . . . . . . . . 13 ⊢ ((dom 𝑀 ∈ ∪ ran sigAlgebra ∧ ∀𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀 ∧ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω) → ∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
126	64, 109, 123, 125	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → ∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
127	63, 126	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → (◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
128	127	ralrimiva 3125	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝐹 ∘f + 𝐺)(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
129		ssralv 4015	. . . . . . . . . 10 ⊢ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ⊆ ran (𝐹 ∘f + 𝐺) → (∀𝑧 ∈ ran (𝐹 ∘f + 𝐺)(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀 → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀))
130	56, 128, 129	mpsyl 68	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
131	130	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
132	1	ffund 6692	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun + )
133		imafi 9264	. . . . . . . . . . . . 13 ⊢ ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
134	132, 113, 133	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
135	18, 20, 9, 23	ofrn2 32564	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
136		ssfi 9137	. . . . . . . . . . . 12 ⊢ ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹 ∘f + 𝐺) ∈ Fin)
137	134, 135, 136	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ∈ Fin)
138		ssdomg 8971	. . . . . . . . . . 11 ⊢ (ran (𝐹 ∘f + 𝐺) ∈ Fin → ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ⊆ ran (𝐹 ∘f + 𝐺) → (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ≼ ran (𝐹 ∘f + 𝐺)))
139	137, 56, 138	mpisyl 21	. . . . . . . . . 10 ⊢ (𝜑 → (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ≼ ran (𝐹 ∘f + 𝐺))
140		isfinite 9605	. . . . . . . . . . . 12 ⊢ (ran (𝐹 ∘f + 𝐺) ∈ Fin ↔ ran (𝐹 ∘f + 𝐺) ≺ ω)
141	137, 140	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ≺ ω)
142		sdomdom 8951	. . . . . . . . . . 11 ⊢ (ran (𝐹 ∘f + 𝐺) ≺ ω → ran (𝐹 ∘f + 𝐺) ≼ ω)
143	141, 142	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ≼ ω)
144		domtr 8978	. . . . . . . . . 10 ⊢ (((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ≼ ran (𝐹 ∘f + 𝐺) ∧ ran (𝐹 ∘f + 𝐺) ≼ ω) → (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ≼ ω)
145	139, 143, 144	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ≼ ω)
146	145	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ≼ ω)
147		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑏 ∩ ran (𝐹 ∘f + 𝐺))
148	147	sigaclcuni 34108	. . . . . . . 8 ⊢ ((dom 𝑀 ∈ ∪ ran sigAlgebra ∧ ∀𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀 ∧ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ≼ ω) → ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
149	51, 131, 146, 148	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
150	55, 149	eqeltrrid 2833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∈ dom 𝑀)
151		difpreima 7037	. . . . . . . . . 10 ⊢ (Fun (𝐹 ∘f + 𝐺) → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) = ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺))))
152	25, 44, 151	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) = ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺))))
153		cnvimarndm 6054	. . . . . . . . . . 11 ⊢ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺)) = dom (𝐹 ∘f + 𝐺)
154	153	difeq2i 4086	. . . . . . . . . 10 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺))) = ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘f + 𝐺))
155		cnvimass 6053	. . . . . . . . . . 11 ⊢ (◡(𝐹 ∘f + 𝐺) “ 𝑏) ⊆ dom (𝐹 ∘f + 𝐺)
156		ssdif0 4329	. . . . . . . . . . 11 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ⊆ dom (𝐹 ∘f + 𝐺) ↔ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘f + 𝐺)) = ∅)
157	155, 156	mpbi 230	. . . . . . . . . 10 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘f + 𝐺)) = ∅
158	154, 157	eqtri 2752	. . . . . . . . 9 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺))) = ∅
159	152, 158	eqtrdi 2780	. . . . . . . 8 ⊢ (𝜑 → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) = ∅)
160		0elsiga 34104	. . . . . . . . 9 ⊢ (dom 𝑀 ∈ ∪ ran sigAlgebra → ∅ ∈ dom 𝑀)
161	16, 49, 160	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ dom 𝑀)
162	159, 161	eqeltrd 2828	. . . . . . 7 ⊢ (𝜑 → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) ∈ dom 𝑀)
163	162	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) ∈ dom 𝑀)
164		unelsiga 34124	. . . . . 6 ⊢ ((dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∈ dom 𝑀 ∧ (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) ∈ dom 𝑀) → ((◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∪ (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))) ∈ dom 𝑀)
165	51, 150, 163, 164	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∪ (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))) ∈ dom 𝑀)
166	48, 165	eqeltrd 2828	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ 𝑏) ∈ dom 𝑀)
167	166	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))(◡(𝐹 ∘f + 𝐺) “ 𝑏) ∈ dom 𝑀)
168	50, 38	ismbfm 34241	. . 3 ⊢ (𝜑 → ((𝐹 ∘f + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))) ↔ ((𝐹 ∘f + 𝐺) ∈ (∪ (sigaGen‘(TopOpen‘𝐾)) ↑m ∪ dom 𝑀) ∧ ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))(◡(𝐹 ∘f + 𝐺) “ 𝑏) ∈ dom 𝑀)))
169	41, 167, 168	mpbir2and 713	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))))
170	62	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → (◡(𝐹 ∘f + 𝐺) “ {𝑧}) = ∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
171	170	fveq2d 6862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) = (𝑀‘∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
172		measbasedom 34192	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
173	16, 172	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (measures‘dom 𝑀))
174	173	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → 𝑀 ∈ (measures‘dom 𝑀))
175		eldifi 4094	. . . . . . . 8 ⊢ (𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)}) → 𝑧 ∈ ran (𝐹 ∘f + 𝐺))
176	175, 109	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → ∀𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
177	122	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
178		sneq 4599	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → {𝑥} = {(1st ‘𝑝)})
179	178	imaeq2d 6031	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑝) → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {(1st ‘𝑝)}))
180		sneq 4599	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑝) → {𝑦} = {(2nd ‘𝑝)})
181	180	imaeq2d 6031	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑝) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(2nd ‘𝑝)}))
182	18	ffund 6692	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
183		sndisj 5099	. . . . . . . . . . 11 ⊢ Disj 𝑥 ∈ ran 𝐹{𝑥}
184		disjpreima 32513	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
185	182, 183, 184	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
186	20	ffund 6692	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐺)
187		sndisj 5099	. . . . . . . . . . 11 ⊢ Disj 𝑦 ∈ ran 𝐺{𝑦}
188		disjpreima 32513	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(◡𝐺 “ {𝑦}))
189	186, 187, 188	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑦 ∈ ran 𝐺(◡𝐺 “ {𝑦}))
190	179, 181, 185, 189	disjxpin 32517	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
191		disjss1 5080	. . . . . . . . 9 ⊢ (((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺) → (Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) → Disj 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
192	114, 190, 191	mpsyl 68	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
193	192	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → Disj 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
194		measvuni 34204	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ ∀𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀 ∧ (((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω ∧ Disj 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))) → (𝑀‘∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) = Σ*𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
195	174, 176, 177, 193, 194	syl112anc 1376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) = Σ*𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
196		ssfi 9137	. . . . . . . . 9 ⊢ (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
197	113, 114, 196	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
198	197	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
199		simpll 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
200		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)))
201	114, 200	sselid 3944	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
202		xp1st 8000	. . . . . . . . 9 ⊢ (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (1st ‘𝑝) ∈ ran 𝐹)
203	201, 202	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st ‘𝑝) ∈ ran 𝐹)
204		xp2nd 8001	. . . . . . . . 9 ⊢ (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (2nd ‘𝑝) ∈ ran 𝐺)
205	201, 204	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd ‘𝑝) ∈ ran 𝐺)
206		oveq12 7396	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 0 ∧ 𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
207		sibfof.5	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( 0 + 0 ) = (0g‘𝐾))
208	206, 207	sylan9eqr 2786	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 = 0 ∧ 𝑦 = 0 )) → (𝑥 + 𝑦) = (0g‘𝐾))
209	208	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 = 0 ∧ 𝑦 = 0 ) → (𝑥 + 𝑦) = (0g‘𝐾)))
210	209	necon3ad 2938	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → ¬ (𝑥 = 0 ∧ 𝑦 = 0 )))
211		neorian 3020	. . . . . . . . . . . . 13 ⊢ ((𝑥 ≠ 0 ∨ 𝑦 ≠ 0 ) ↔ ¬ (𝑥 = 0 ∧ 𝑦 = 0 ))
212	210, 211	imbitrrdi 252	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )))
213	212	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )))
214	213	ralrimivva 3180	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )))
215	199, 214	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )))
216	67	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (◡ + “ {𝑧}))
217	216	sselda 3946	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (◡ + “ {𝑧}))
218		fniniseg 7032	. . . . . . . . . . . . 13 ⊢ ( + Fn (𝐵 × 𝐵) → (𝑝 ∈ (◡ + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧)))
219	199, 61, 218	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (◡ + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧)))
220	217, 219	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧))
221		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧) → ( + ‘𝑝) = 𝑧)
222		1st2nd2 8007	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
223	222	fveq2d 6862	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → ( + ‘𝑝) = ( + ‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
224		df-ov 7390	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑝) + (2nd ‘𝑝)) = ( + ‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
225	223, 224	eqtr4di 2782	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → ( + ‘𝑝) = ((1st ‘𝑝) + (2nd ‘𝑝)))
226	225	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧) → ( + ‘𝑝) = ((1st ‘𝑝) + (2nd ‘𝑝)))
227	221, 226	eqtr3d 2766	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧) → 𝑧 = ((1st ‘𝑝) + (2nd ‘𝑝)))
228	220, 227	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 = ((1st ‘𝑝) + (2nd ‘𝑝)))
229		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)}))
230	229	eldifbd 3927	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g‘𝐾)})
231		velsn 4605	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {(0g‘𝐾)} ↔ 𝑧 = (0g‘𝐾))
232	231	necon3bbii 2972	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ {(0g‘𝐾)} ↔ 𝑧 ≠ (0g‘𝐾))
233	230, 232	sylib 218	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ≠ (0g‘𝐾))
234	228, 233	eqnetrrd 2993	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st ‘𝑝) + (2nd ‘𝑝)) ≠ (0g‘𝐾))
235	175, 72	sylanl2 681	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
236	235, 85	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st ‘𝑝) ∈ 𝐵)
237	235, 98	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd ‘𝑝) ∈ 𝐵)
238		oveq1 7394	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥 + 𝑦) = ((1st ‘𝑝) + 𝑦))
239	238	neeq1d 2984	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → ((𝑥 + 𝑦) ≠ (0g‘𝐾) ↔ ((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾)))
240		neeq1 2987	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥 ≠ 0 ↔ (1st ‘𝑝) ≠ 0 ))
241	240	orbi1d 916	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → ((𝑥 ≠ 0 ∨ 𝑦 ≠ 0 ) ↔ ((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 )))
242	239, 241	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → (((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )) ↔ (((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾) → ((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 ))))
243		oveq2 7395	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑝) → ((1st ‘𝑝) + 𝑦) = ((1st ‘𝑝) + (2nd ‘𝑝)))
244	243	neeq1d 2984	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑝) → (((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾) ↔ ((1st ‘𝑝) + (2nd ‘𝑝)) ≠ (0g‘𝐾)))
245		neeq1 2987	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑝) → (𝑦 ≠ 0 ↔ (2nd ‘𝑝) ≠ 0 ))
246	245	orbi2d 915	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑝) → (((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 ) ↔ ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 )))
247	244, 246	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑝) → ((((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾) → ((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 )) ↔ (((1st ‘𝑝) + (2nd ‘𝑝)) ≠ (0g‘𝐾) → ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 ))))
248	242, 247	rspc2v 3599	. . . . . . . . . 10 ⊢ (((1st ‘𝑝) ∈ 𝐵 ∧ (2nd ‘𝑝) ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )) → (((1st ‘𝑝) + (2nd ‘𝑝)) ≠ (0g‘𝐾) → ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 ))))
249	236, 237, 248	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )) → (((1st ‘𝑝) + (2nd ‘𝑝)) ≠ (0g‘𝐾) → ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 ))))
250	215, 234, 249	mp2d 49	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 ))
251	3, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 74	sibfinima 34330	. . . . . . . 8 ⊢ (((𝜑 ∧ (1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺) ∧ ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 )) → (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) ∈ (0[,)+∞))
252	199, 203, 205, 250, 251	syl31anc 1375	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) ∈ (0[,)+∞))
253	198, 252	esumpfinval 34065	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → Σ*𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) = Σ𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
254	171, 195, 253	3eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) = Σ𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
255		rge0ssre 13417	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
256	255, 252	sselid 3944	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) ∈ ℝ)
257	198, 256	fsumrecl 15700	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → Σ𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) ∈ ℝ)
258	254, 257	eqeltrd 2828	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ ℝ)
259	174	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑀 ∈ (measures‘dom 𝑀))
260	175, 108	sylanl2 681	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
261		measge0 34197	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀) → 0 ≤ (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
262	259, 260, 261	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 0 ≤ (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
263	198, 256, 262	fsumge0 15761	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → 0 ≤ Σ𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
264	263, 254	breqtrrd 5135	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → 0 ≤ (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})))
265		elrege0 13415	. . . 4 ⊢ ((𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧}))))
266	258, 264, 265	sylanbrc 583	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
267	266	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})(𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
268		eqid 2729	. . 3 ⊢ (sigaGen‘(TopOpen‘𝐾)) = (sigaGen‘(TopOpen‘𝐾))
269		eqid 2729	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
270		eqid 2729	. . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
271		eqid 2729	. . 3 ⊢ (ℝHom‘(Scalar‘𝐾)) = (ℝHom‘(Scalar‘𝐾))
272	27, 28, 268, 269, 270, 271, 26, 16	issibf 34324	. 2 ⊢ (𝜑 → ((𝐹 ∘f + 𝐺) ∈ dom (𝐾sitg𝑀) ↔ ((𝐹 ∘f + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))) ∧ ran (𝐹 ∘f + 𝐺) ∈ Fin ∧ ∀𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})(𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))))
273	169, 137, 267, 272	mpbir3and 1343	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom (𝐾sitg𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  {csn 4589  ⟨cop 4595  ∪ cuni 4871  ∪ ciun 4955  Disj wdisj 5074   class class class wbr 5107   × cxp 5636  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651  ωcom 7842  1^st c1st 7966  2^nd c2nd 7967   ↑_m cmap 8799   ≼ cdom 8916   ≺ csdm 8917  Fincfn 8918  ℝcr 11067  0cc0 11068  +∞cpnf 11205   ≤ cle 11209  [,)cico 13308  Σcsu 15652  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224  TopOpenctopn 17384  0_gc0g 17402  Topctop 22780  TopSpctps 22819  Clsdccld 22903  Frect1 23194  ℝHomcrrh 33983  Σ^*cesum 34017  sigAlgebracsiga 34098  sigaGencsigagen 34128  measurescmeas 34185  MblFnMcmbfm 34239  sitgcsitg 34320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-ac2 10416  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146  ax-addf 11147  ax-mulf 11148

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-disj 5075  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-fi 9362  df-sup 9393  df-inf 9394  df-oi 9463  df-dju 9854  df-card 9892  df-acn 9895  df-ac 10069  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-q 12908  df-rp 12952  df-xneg 13072  df-xadd 13073  df-xmul 13074  df-ioo 13310  df-ioc 13311  df-ico 13312  df-icc 13313  df-fz 13469  df-fzo 13616  df-fl 13754  df-mod 13832  df-seq 13967  df-exp 14027  df-fac 14239  df-bc 14268  df-hash 14296  df-shft 15033  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-limsup 15437  df-clim 15454  df-rlim 15455  df-sum 15653  df-ef 16033  df-sin 16035  df-cos 16036  df-pi 16038  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-rest 17385  df-topn 17386  df-0g 17404  df-gsum 17405  df-topgen 17406  df-pt 17407  df-prds 17410  df-ordt 17464  df-xrs 17465  df-qtop 17470  df-imas 17471  df-xps 17473  df-mre 17547  df-mrc 17548  df-acs 17550  df-ps 18525  df-tsr 18526  df-plusf 18566  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-cring 20145  df-subrng 20455  df-subrg 20479  df-abv 20718  df-lmod 20768  df-scaf 20769  df-sra 21080  df-rgmod 21081  df-psmet 21256  df-xmet 21257  df-met 21258  df-bl 21259  df-mopn 21260  df-fbas 21261  df-fg 21262  df-cnfld 21265  df-top 22781  df-topon 22798  df-topsp 22820  df-bases 22833  df-cld 22906  df-ntr 22907  df-cls 22908  df-nei 22985  df-lp 23023  df-perf 23024  df-cn 23114  df-cnp 23115  df-t1 23201  df-haus 23202  df-tx 23449  df-hmeo 23642  df-fil 23733  df-fm 23825  df-flim 23826  df-flf 23827  df-tmd 23959  df-tgp 23960  df-tsms 24014  df-trg 24047  df-xms 24208  df-ms 24209  df-tms 24210  df-nm 24470  df-ngp 24471  df-nrg 24473  df-nlm 24474  df-ii 24770  df-cncf 24771  df-limc 25767  df-dv 25768  df-log 26465  df-esum 34018  df-siga 34099  df-sigagen 34129  df-meas 34186  df-mbfm 34240  df-sitg 34321

This theorem is referenced by:  sitmcl  34342