Theorem sibfof 34305

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 22963	. . . . . . . . . . 11 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
6	2, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
7	6	sqxpeqd 5732	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐵) = (∪ 𝐽 × ∪ 𝐽))
8	7	feq2d 6733	. . . . . . . 8 ⊢ (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶 ↔ + :(∪ 𝐽 × ∪ 𝐽)⟶𝐶))
9	1, 8	mpbid 232	. . . . . . 7 ⊢ (𝜑 → + :(∪ 𝐽 × ∪ 𝐽)⟶𝐶)
10	9	fovcdmda 7621	. . . . . 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 34301	. . . . . 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 34301	. . . . . 6 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
21		dmexg 7941	. . . . . . 7 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ V)
22		uniexg 7775	. . . . . . 7 ⊢ (dom 𝑀 ∈ V → ∪ dom 𝑀 ∈ V)
23	16, 21, 22	3syl 18	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑀 ∈ V)
24		inidm 4248	. . . . . 6 ⊢ (∪ dom 𝑀 ∩ ∪ dom 𝑀) = ∪ dom 𝑀
25	10, 18, 20, 23, 23, 24	off 7732	. . . . 5 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):∪ dom 𝑀⟶𝐶)
26		sibfof.3	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ TopSp)
27		sibfof.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝐾)
28		eqid 2740	. . . . . . . . 9 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
29	27, 28	tpsuni 22963	. . . . . . . 8 ⊢ (𝐾 ∈ TopSp → 𝐶 = ∪ (TopOpen‘𝐾))
30	26, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 = ∪ (TopOpen‘𝐾))
31		fvex 6933	. . . . . . . 8 ⊢ (TopOpen‘𝐾) ∈ V
32		unisg 34107	. . . . . . . 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 2798	. . . . . 6 ⊢ (𝜑 → 𝐶 = ∪ (sigaGen‘(TopOpen‘𝐾)))
35	34	feq3d 6734	. . . . 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 34106	. . . . . 6 ⊢ (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ∪ ran sigAlgebra)
39	38	uniexd 7777	. . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘(TopOpen‘𝐾)) ∈ V)
40	39, 23	elmapd 8898	. . . 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 4502	. . . . . . 7 ⊢ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) = 𝑏
43	42	imaeq2i 6087	. . . . . 6 ⊢ (◡(𝐹 ∘f + 𝐺) “ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))) = (◡(𝐹 ∘f + 𝐺) “ 𝑏)
44		ffun 6750	. . . . . . . 8 ⊢ ((𝐹 ∘f + 𝐺):∪ dom 𝑀⟶𝐶 → Fun (𝐹 ∘f + 𝐺))
45		unpreima 7096	. . . . . . . 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 2794	. . . . 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 34165	. . . . . . . 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 7282	. . . . . . . 8 ⊢ (◡(𝐹 ∘f + 𝐺) “ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)){𝑧}) = ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧})
53		iunid 5083	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹 ∘f + 𝐺))
54	53	imaeq2i 6087	. . . . . . . 8 ⊢ (◡(𝐹 ∘f + 𝐺) “ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)){𝑧}) = (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)))
55	52, 54	eqtr3i 2770	. . . . . . 7 ⊢ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) = (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)))
56		inss2 4259	. . . . . . . . . 10 ⊢ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ⊆ ran (𝐹 ∘f + 𝐺)
57	6	feq3d 6734	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶𝐵 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽))
58	18, 57	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶𝐵)
59	6	feq3d 6734	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺:∪ dom 𝑀⟶𝐵 ↔ 𝐺:∪ dom 𝑀⟶∪ 𝐽))
60	20, 59	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶𝐵)
61	1	ffnd 6748	. . . . . . . . . . . . . 14 ⊢ (𝜑 → + Fn (𝐵 × 𝐵))
62	58, 60, 23, 61	ofpreima2 32684	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 4258	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (◡ + “ {𝑧})
68		cnvimass 6111	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡ + “ {𝑧}) ⊆ dom +
69	68, 1	fssdm 6766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (◡ + “ {𝑧}) ⊆ (𝐵 × 𝐵))
70	69	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → (◡ + “ {𝑧}) ⊆ (𝐵 × 𝐵))
71	67, 70	sstrid 4020	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
72	71	sselda 4008	. . . . . . . . . . . . . . . 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 34106	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
76	11, 75	eqeltrid 2848	. . . . . . . . . . . . . . . . . 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 34300	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
79	78	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
80	4	tpstop 22964	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
81		cldssbrsiga 34151	. . . . . . . . . . . . . . . . . . . . 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 8062	. . . . . . . . . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (1st ‘𝑝) ∈ ∪ 𝐽)
89		eqid 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ 𝐽 = ∪ 𝐽
90	89	t1sncld 23355	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Fre ∧ (1st ‘𝑝) ∈ ∪ 𝐽) → {(1st ‘𝑝)} ∈ (Clsd‘𝐽))
91	84, 88, 90	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ (Clsd‘𝐽))
92	83, 91	sseldd 4009	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ (sigaGen‘𝐽))
93	92, 11	eleqtrrdi 2855	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ 𝑆)
94	73, 77, 79, 93	mbfmcnvima 34220	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (◡𝐹 “ {(1st ‘𝑝)}) ∈ dom 𝑀)
95	66, 72, 94	syl2anc 583	. . . . . . . . . . . . . . 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 34300	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
97	96	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
98		xp2nd 8063	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵)
99	98	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (2nd ‘𝑝) ∈ 𝐵)
100	99, 87	eleqtrd 2846	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (2nd ‘𝑝) ∈ ∪ 𝐽)
101	89	t1sncld 23355	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Fre ∧ (2nd ‘𝑝) ∈ ∪ 𝐽) → {(2nd ‘𝑝)} ∈ (Clsd‘𝐽))
102	84, 100, 101	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ (Clsd‘𝐽))
103	83, 102	sseldd 4009	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ (sigaGen‘𝐽))
104	103, 11	eleqtrrdi 2855	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ 𝑆)
105	73, 77, 97, 104	mbfmcnvima 34220	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀)
106	66, 72, 105	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀)
107		inelsiga 34099	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
109	108	ralrimiva 3152	. . . . . . . . . . . . 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 34302	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
111	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 34302	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
112		xpfi 9386	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
113	110, 111, 112	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
114		inss2 4259	. . . . . . . . . . . . . . . 16 ⊢ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
115		ssdomg 9060	. . . . . . . . . . . . . . . 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 9721	. . . . . . . . . . . . . . . . 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 9040	. . . . . . . . . . . . . . . 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 9067	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 2908	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑝((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
125	124	sigaclcuni 34082	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 2844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → (◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
128	127	ralrimiva 3152	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝐹 ∘f + 𝐺)(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
129		ssralv 4077	. . . . . . . . . 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 6751	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun + )
133		imafi 9381	. . . . . . . . . . . . 13 ⊢ ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
134	132, 113, 133	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
135	18, 20, 9, 23	ofrn2 32659	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
136		ssfi 9240	. . . . . . . . . . . 12 ⊢ ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹 ∘f + 𝐺) ∈ Fin)
137	134, 135, 136	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ∈ Fin)
138		ssdomg 9060	. . . . . . . . . . 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 9721	. . . . . . . . . . . 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 9040	. . . . . . . . . . 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 9067	. . . . . . . . . 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 583	. . . . . . . . 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 2908	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑏 ∩ ran (𝐹 ∘f + 𝐺))
148	147	sigaclcuni 34082	. . . . . . . 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 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
150	55, 149	eqeltrrid 2849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∈ dom 𝑀)
151		difpreima 7098	. . . . . . . . . 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 6112	. . . . . . . . . . 11 ⊢ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺)) = dom (𝐹 ∘f + 𝐺)
154	153	difeq2i 4146	. . . . . . . . . 10 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺))) = ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘f + 𝐺))
155		cnvimass 6111	. . . . . . . . . . 11 ⊢ (◡(𝐹 ∘f + 𝐺) “ 𝑏) ⊆ dom (𝐹 ∘f + 𝐺)
156		ssdif0 4389	. . . . . . . . . . 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 2768	. . . . . . . . 9 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺))) = ∅
159	152, 158	eqtrdi 2796	. . . . . . . 8 ⊢ (𝜑 → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) = ∅)
160		0elsiga 34078	. . . . . . . . 9 ⊢ (dom 𝑀 ∈ ∪ ran sigAlgebra → ∅ ∈ dom 𝑀)
161	16, 49, 160	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ dom 𝑀)
162	159, 161	eqeltrd 2844	. . . . . . 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 34098	. . . . . 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 1371	. . . . 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 2844	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ 𝑏) ∈ dom 𝑀)
167	166	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))(◡(𝐹 ∘f + 𝐺) “ 𝑏) ∈ dom 𝑀)
168	50, 38	ismbfm 34215	. . 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 712	. 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 6924	. . . . . 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 34166	. . . . . . . . 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 4154	. . . . . . . 8 ⊢ (𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)}) → 𝑧 ∈ ran (𝐹 ∘f + 𝐺))
176	175, 109	sylan2 592	. . . . . . 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 4658	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → {𝑥} = {(1st ‘𝑝)})
179	178	imaeq2d 6089	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑝) → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {(1st ‘𝑝)}))
180		sneq 4658	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑝) → {𝑦} = {(2nd ‘𝑝)})
181	180	imaeq2d 6089	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑝) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(2nd ‘𝑝)}))
182	18	ffund 6751	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
183		sndisj 5158	. . . . . . . . . . 11 ⊢ Disj 𝑥 ∈ ran 𝐹{𝑥}
184		disjpreima 32606	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
185	182, 183, 184	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
186	20	ffund 6751	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐺)
187		sndisj 5158	. . . . . . . . . . 11 ⊢ Disj 𝑦 ∈ ran 𝐺{𝑦}
188		disjpreima 32606	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(◡𝐺 “ {𝑦}))
189	186, 187, 188	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑦 ∈ ran 𝐺(◡𝐺 “ {𝑦}))
190	179, 181, 185, 189	disjxpin 32610	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
191		disjss1 5139	. . . . . . . . 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 34178	. . . . . . 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 1374	. . . . . 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 9240	. . . . . . . . 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 585	. . . . . . . 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 4006	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
202		xp1st 8062	. . . . . . . . 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 8063	. . . . . . . . 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 7457	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 0 ∧ 𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
207		sibfof.5	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( 0 + 0 ) = (0g‘𝐾))
208	206, 207	sylan9eqr 2802	. . . . . . . . . . . . . . 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 2959	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → ¬ (𝑥 = 0 ∧ 𝑦 = 0 )))
211		neorian 3043	. . . . . . . . . . . . 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 3208	. . . . . . . . . 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 4008	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (◡ + “ {𝑧}))
218		fniniseg 7093	. . . . . . . . . . . . 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 8069	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
223	222	fveq2d 6924	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → ( + ‘𝑝) = ( + ‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
224		df-ov 7451	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑝) + (2nd ‘𝑝)) = ( + ‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
225	223, 224	eqtr4di 2798	. . . . . . . . . . . . 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 2782	. . . . . . . . . . 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 3989	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g‘𝐾)})
231		velsn 4664	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {(0g‘𝐾)} ↔ 𝑧 = (0g‘𝐾))
232	231	necon3bbii 2994	. . . . . . . . . . 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 3015	. . . . . . . . 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 680	. . . . . . . . . . 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 7455	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥 + 𝑦) = ((1st ‘𝑝) + 𝑦))
239	238	neeq1d 3006	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → ((𝑥 + 𝑦) ≠ (0g‘𝐾) ↔ ((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾)))
240		neeq1 3009	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥 ≠ 0 ↔ (1st ‘𝑝) ≠ 0 ))
241	240	orbi1d 915	. . . . . . . . . . . 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 7456	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑝) → ((1st ‘𝑝) + 𝑦) = ((1st ‘𝑝) + (2nd ‘𝑝)))
244	243	neeq1d 3006	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑝) → (((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾) ↔ ((1st ‘𝑝) + (2nd ‘𝑝)) ≠ (0g‘𝐾)))
245		neeq1 3009	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑝) → (𝑦 ≠ 0 ↔ (2nd ‘𝑝) ≠ 0 ))
246	245	orbi2d 914	. . . . . . . . . . . 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 3646	. . . . . . . . . 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 583	. . . . . . . . 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 34304	. . . . . . . 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 1373	. . . . . . 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 34039	. . . . . 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 2784	. . . . 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 13516	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
256	255, 252	sselid 4006	. . . . . 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 15782	. . . . 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 2844	. . . 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 680	. . . . . . 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 34171	. . . . . . 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 583	. . . . . 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 15843	. . . . 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 5194	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → 0 ≤ (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})))
265		elrege0 13514	. . . 4 ⊢ ((𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧}))))
266	258, 264, 265	sylanbrc 582	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
267	266	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})(𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
268		eqid 2740	. . 3 ⊢ (sigaGen‘(TopOpen‘𝐾)) = (sigaGen‘(TopOpen‘𝐾))
269		eqid 2740	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
270		eqid 2740	. . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
271		eqid 2740	. . 3 ⊢ (ℝHom‘(Scalar‘𝐾)) = (ℝHom‘(Scalar‘𝐾))
272	27, 28, 268, 269, 270, 271, 26, 16	issibf 34298	. 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 1342	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom (𝐾sitg𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648  ⟨cop 4654  ∪ cuni 4931  ∪ ciun 5015  Disj wdisj 5133   class class class wbr 5166   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∘_f cof 7712  ωcom 7903  1^st c1st 8028  2^nd c2nd 8029   ↑_m cmap 8884   ≼ cdom 9001   ≺ csdm 9002  Fincfn 9003  ℝcr 11183  0cc0 11184  +∞cpnf 11321   ≤ cle 11325  [,)cico 13409  Σcsu 15734  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  TopOpenctopn 17481  0_gc0g 17499  Topctop 22920  TopSpctps 22959  Clsdccld 23045  Frect1 23336  ℝHomcrrh 33939  Σ^*cesum 33991  sigAlgebracsiga 34072  sigaGencsigagen 34102  measurescmeas 34159  MblFnMcmbfm 34213  sitgcsitg 34294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-ac2 10532  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-addf 11263  ax-mulf 11264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-acn 10011  df-ac 10185  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ioc 13412  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-mod 13921  df-seq 14053  df-exp 14113  df-fac 14323  df-bc 14352  df-hash 14380  df-shft 15116  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-limsup 15517  df-clim 15534  df-rlim 15535  df-sum 15735  df-ef 16115  df-sin 16117  df-cos 16118  df-pi 16120  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-rest 17482  df-topn 17483  df-0g 17501  df-gsum 17502  df-topgen 17503  df-pt 17504  df-prds 17507  df-ordt 17561  df-xrs 17562  df-qtop 17567  df-imas 17568  df-xps 17570  df-mre 17644  df-mrc 17645  df-acs 17647  df-ps 18636  df-tsr 18637  df-plusf 18677  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-cntz 19357  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-cring 20263  df-subrng 20572  df-subrg 20597  df-abv 20832  df-lmod 20882  df-scaf 20883  df-sra 21195  df-rgmod 21196  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-fbas 21384  df-fg 21385  df-cnfld 21388  df-top 22921  df-topon 22938  df-topsp 22960  df-bases 22974  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-lp 23165  df-perf 23166  df-cn 23256  df-cnp 23257  df-t1 23343  df-haus 23344  df-tx 23591  df-hmeo 23784  df-fil 23875  df-fm 23967  df-flim 23968  df-flf 23969  df-tmd 24101  df-tgp 24102  df-tsms 24156  df-trg 24189  df-xms 24351  df-ms 24352  df-tms 24353  df-nm 24616  df-ngp 24617  df-nrg 24619  df-nlm 24620  df-ii 24922  df-cncf 24923  df-limc 25921  df-dv 25922  df-log 26616  df-esum 33992  df-siga 34073  df-sigagen 34103  df-meas 34160  df-mbfm 34214  df-sitg 34295

This theorem is referenced by:  sitmcl  34316