Theorem sibfof 34212

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 22952	. . . . . . . . . . 11 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
6	2, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
7	6	sqxpeqd 5718	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐵) = (∪ 𝐽 × ∪ 𝐽))
8	7	feq2d 6718	. . . . . . . 8 ⊢ (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶 ↔ + :(∪ 𝐽 × ∪ 𝐽)⟶𝐶))
9	1, 8	mpbid 231	. . . . . . 7 ⊢ (𝜑 → + :(∪ 𝐽 × ∪ 𝐽)⟶𝐶)
10	9	fovcdmda 7602	. . . . . 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 34208	. . . . . 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 34208	. . . . . 6 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
21		dmexg 7922	. . . . . . 7 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ V)
22		uniexg 7756	. . . . . . 7 ⊢ (dom 𝑀 ∈ V → ∪ dom 𝑀 ∈ V)
23	16, 21, 22	3syl 18	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑀 ∈ V)
24		inidm 4230	. . . . . 6 ⊢ (∪ dom 𝑀 ∩ ∪ dom 𝑀) = ∪ dom 𝑀
25	10, 18, 20, 23, 23, 24	off 7713	. . . . 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 22952	. . . . . . . 8 ⊢ (𝐾 ∈ TopSp → 𝐶 = ∪ (TopOpen‘𝐾))
30	26, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 = ∪ (TopOpen‘𝐾))
31		fvex 6918	. . . . . . . 8 ⊢ (TopOpen‘𝐾) ∈ V
32		unisg 34014	. . . . . . . 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 2787	. . . . . 6 ⊢ (𝜑 → 𝐶 = ∪ (sigaGen‘(TopOpen‘𝐾)))
35	34	feq3d 6719	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘f + 𝐺):∪ dom 𝑀⟶𝐶 ↔ (𝐹 ∘f + 𝐺):∪ dom 𝑀⟶∪ (sigaGen‘(TopOpen‘𝐾))))
36	25, 35	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):∪ dom 𝑀⟶∪ (sigaGen‘(TopOpen‘𝐾)))
37	31	a1i 11	. . . . . . 7 ⊢ (𝜑 → (TopOpen‘𝐾) ∈ V)
38	37	sgsiga 34013	. . . . . 6 ⊢ (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ∪ ran sigAlgebra)
39	38	uniexd 7758	. . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘(TopOpen‘𝐾)) ∈ V)
40	39, 23	elmapd 8877	. . . 4 ⊢ (𝜑 → ((𝐹 ∘f + 𝐺) ∈ (∪ (sigaGen‘(TopOpen‘𝐾)) ↑m ∪ dom 𝑀) ↔ (𝐹 ∘f + 𝐺):∪ dom 𝑀⟶∪ (sigaGen‘(TopOpen‘𝐾))))
41	36, 40	mpbird 256	. . 3 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (∪ (sigaGen‘(TopOpen‘𝐾)) ↑m ∪ dom 𝑀))
42		inundif 4486	. . . . . . 7 ⊢ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) = 𝑏
43	42	imaeq2i 6072	. . . . . 6 ⊢ (◡(𝐹 ∘f + 𝐺) “ ((𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘f + 𝐺)))) = (◡(𝐹 ∘f + 𝐺) “ 𝑏)
44		ffun 6735	. . . . . . . 8 ⊢ ((𝐹 ∘f + 𝐺):∪ dom 𝑀⟶𝐶 → Fun (𝐹 ∘f + 𝐺))
45		unpreima 7081	. . . . . . . 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 479	. . . . . 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 2783	. . . . 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 34072	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
50	16, 49	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
51	50	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → dom 𝑀 ∈ ∪ ran sigAlgebra)
52		imaiun 7265	. . . . . . . 8 ⊢ (◡(𝐹 ∘f + 𝐺) “ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)){𝑧}) = ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧})
53		iunid 5073	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹 ∘f + 𝐺))
54	53	imaeq2i 6072	. . . . . . . 8 ⊢ (◡(𝐹 ∘f + 𝐺) “ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)){𝑧}) = (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)))
55	52, 54	eqtr3i 2759	. . . . . . 7 ⊢ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) = (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)))
56		inss2 4241	. . . . . . . . . 10 ⊢ (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ⊆ ran (𝐹 ∘f + 𝐺)
57	6	feq3d 6719	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶𝐵 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽))
58	18, 57	mpbird 256	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶𝐵)
59	6	feq3d 6719	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺:∪ dom 𝑀⟶𝐵 ↔ 𝐺:∪ dom 𝑀⟶∪ 𝐽))
60	20, 59	mpbird 256	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶𝐵)
61	1	ffnd 6733	. . . . . . . . . . . . . 14 ⊢ (𝜑 → + Fn (𝐵 × 𝐵))
62	58, 60, 23, 61	ofpreima2 32609	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡(𝐹 ∘f + 𝐺) “ {𝑧}) = ∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
63	62	adantr 479	. . . . . . . . . . . 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 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → dom 𝑀 ∈ ∪ ran sigAlgebra)
65	50	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → dom 𝑀 ∈ ∪ ran sigAlgebra)
66		simpll 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
67		inss1 4240	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (◡ + “ {𝑧})
68		cnvimass 6096	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡ + “ {𝑧}) ⊆ dom +
69	68, 1	fssdm 6751	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (◡ + “ {𝑧}) ⊆ (𝐵 × 𝐵))
70	69	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → (◡ + “ {𝑧}) ⊆ (𝐵 × 𝐵))
71	67, 70	sstrid 4003	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
72	71	sselda 3991	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
73	50	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → dom 𝑀 ∈ ∪ ran sigAlgebra)
74		sibfof.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ Fre)
75	74	sgsiga 34013	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
76	11, 75	eqeltrid 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
77	76	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝑆 ∈ ∪ ran sigAlgebra)
78	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfmbl 34207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
79	78	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
80	4	tpstop 22953	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
81		cldssbrsiga 34058	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
82	2, 80, 81	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
83	82	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
84	74	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐽 ∈ Fre)
85		xp1st 8043	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → (1st ‘𝑝) ∈ 𝐵)
86	85	adantl 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (1st ‘𝑝) ∈ 𝐵)
87	6	adantr 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐵 = ∪ 𝐽)
88	86, 87	eleqtrd 2831	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (1st ‘𝑝) ∈ ∪ 𝐽)
89		eqid 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ 𝐽 = ∪ 𝐽
90	89	t1sncld 23344	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Fre ∧ (1st ‘𝑝) ∈ ∪ 𝐽) → {(1st ‘𝑝)} ∈ (Clsd‘𝐽))
91	84, 88, 90	syl2anc 582	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ (Clsd‘𝐽))
92	83, 91	sseldd 3992	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ (sigaGen‘𝐽))
93	92, 11	eleqtrrdi 2840	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ 𝑆)
94	73, 77, 79, 93	mbfmcnvima 34127	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (◡𝐹 “ {(1st ‘𝑝)}) ∈ dom 𝑀)
95	66, 72, 94	syl2anc 582	. . . . . . . . . . . . . . 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 34207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
97	96	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
98		xp2nd 8044	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵)
99	98	adantl 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (2nd ‘𝑝) ∈ 𝐵)
100	99, 87	eleqtrd 2831	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (2nd ‘𝑝) ∈ ∪ 𝐽)
101	89	t1sncld 23344	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Fre ∧ (2nd ‘𝑝) ∈ ∪ 𝐽) → {(2nd ‘𝑝)} ∈ (Clsd‘𝐽))
102	84, 100, 101	syl2anc 582	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ (Clsd‘𝐽))
103	83, 102	sseldd 3992	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ (sigaGen‘𝐽))
104	103, 11	eleqtrrdi 2840	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ 𝑆)
105	73, 77, 97, 104	mbfmcnvima 34127	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀)
106	66, 72, 105	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀)
107		inelsiga 34006	. . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
109	108	ralrimiva 3139	. . . . . . . . . . . . 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 34209	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
111	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 34209	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
112		xpfi 9367	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
113	110, 111, 112	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
114		inss2 4241	. . . . . . . . . . . . . . . 16 ⊢ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
115		ssdomg 9039	. . . . . . . . . . . . . . . 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 9702	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 × ran 𝐺) ∈ Fin ↔ (ran 𝐹 × ran 𝐺) ≺ ω)
118	117	biimpi 215	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝐹 × ran 𝐺) ∈ Fin → (ran 𝐹 × ran 𝐺) ≺ ω)
119		sdomdom 9019	. . . . . . . . . . . . . . . 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 9046	. . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
123	122	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
124		nfcv 2895	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑝((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
125	124	sigaclcuni 33989	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 2829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘f + 𝐺)) → (◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
128	127	ralrimiva 3139	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝐹 ∘f + 𝐺)(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
129		ssralv 4060	. . . . . . . . . 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 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
132	1	ffund 6736	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun + )
133		imafi 9362	. . . . . . . . . . . . 13 ⊢ ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
134	132, 113, 133	syl2anc 582	. . . . . . . . . . . 12 ⊢ (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
135	18, 20, 9, 23	ofrn2 32583	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
136		ssfi 9219	. . . . . . . . . . . 12 ⊢ ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹 ∘f + 𝐺) ∈ Fin)
137	134, 135, 136	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ∈ Fin)
138		ssdomg 9039	. . . . . . . . . . 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 9702	. . . . . . . . . . . 12 ⊢ (ran (𝐹 ∘f + 𝐺) ∈ Fin ↔ ran (𝐹 ∘f + 𝐺) ≺ ω)
141	137, 140	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ≺ ω)
142		sdomdom 9019	. . . . . . . . . . 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 9046	. . . . . . . . . 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 582	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ≼ ω)
146	145	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (𝑏 ∩ ran (𝐹 ∘f + 𝐺)) ≼ ω)
147		nfcv 2895	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑏 ∩ ran (𝐹 ∘f + 𝐺))
148	147	sigaclcuni 33989	. . . . . . . 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 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))(◡(𝐹 ∘f + 𝐺) “ {𝑧}) ∈ dom 𝑀)
150	55, 149	eqeltrrid 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘f + 𝐺))) ∈ dom 𝑀)
151		difpreima 7083	. . . . . . . . . 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 6097	. . . . . . . . . . 11 ⊢ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺)) = dom (𝐹 ∘f + 𝐺)
154	153	difeq2i 4128	. . . . . . . . . 10 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺))) = ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘f + 𝐺))
155		cnvimass 6096	. . . . . . . . . . 11 ⊢ (◡(𝐹 ∘f + 𝐺) “ 𝑏) ⊆ dom (𝐹 ∘f + 𝐺)
156		ssdif0 4372	. . . . . . . . . . 11 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ⊆ dom (𝐹 ∘f + 𝐺) ↔ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘f + 𝐺)) = ∅)
157	155, 156	mpbi 229	. . . . . . . . . 10 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘f + 𝐺)) = ∅
158	154, 157	eqtri 2757	. . . . . . . . 9 ⊢ ((◡(𝐹 ∘f + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘f + 𝐺) “ ran (𝐹 ∘f + 𝐺))) = ∅
159	152, 158	eqtrdi 2785	. . . . . . . 8 ⊢ (𝜑 → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) = ∅)
160		0elsiga 33985	. . . . . . . . 9 ⊢ (dom 𝑀 ∈ ∪ ran sigAlgebra → ∅ ∈ dom 𝑀)
161	16, 49, 160	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ dom 𝑀)
162	159, 161	eqeltrd 2829	. . . . . . 7 ⊢ (𝜑 → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) ∈ dom 𝑀)
163	162	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘f + 𝐺))) ∈ dom 𝑀)
164		unelsiga 34005	. . . . . 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 1368	. . . . 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 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘f + 𝐺) “ 𝑏) ∈ dom 𝑀)
167	166	ralrimiva 3139	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))(◡(𝐹 ∘f + 𝐺) “ 𝑏) ∈ dom 𝑀)
168	50, 38	ismbfm 34122	. . 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 711	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))))
170	62	adantr 479	. . . . . . 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 6909	. . . . . 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 34073	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
173	16, 172	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (measures‘dom 𝑀))
174	173	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → 𝑀 ∈ (measures‘dom 𝑀))
175		eldifi 4136	. . . . . . . 8 ⊢ (𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)}) → 𝑧 ∈ ran (𝐹 ∘f + 𝐺))
176	175, 109	sylan2 591	. . . . . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
178		sneq 4646	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → {𝑥} = {(1st ‘𝑝)})
179	178	imaeq2d 6074	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑝) → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {(1st ‘𝑝)}))
180		sneq 4646	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑝) → {𝑦} = {(2nd ‘𝑝)})
181	180	imaeq2d 6074	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑝) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(2nd ‘𝑝)}))
182	18	ffund 6736	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
183		sndisj 5148	. . . . . . . . . . 11 ⊢ Disj 𝑥 ∈ ran 𝐹{𝑥}
184		disjpreima 32530	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
185	182, 183, 184	sylancl 584	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
186	20	ffund 6736	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐺)
187		sndisj 5148	. . . . . . . . . . 11 ⊢ Disj 𝑦 ∈ ran 𝐺{𝑦}
188		disjpreima 32530	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(◡𝐺 “ {𝑦}))
189	186, 187, 188	sylancl 584	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑦 ∈ ran 𝐺(◡𝐺 “ {𝑦}))
190	179, 181, 185, 189	disjxpin 32534	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
191		disjss1 5129	. . . . . . . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → Disj 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
194		measvuni 34085	. . . . . . 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 1371	. . . . . 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 9219	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
198	197	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
199		simpll 765	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
200		simpr 483	. . . . . . . . . 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 3989	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
202		xp1st 8043	. . . . . . . . 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 8044	. . . . . . . . 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 7439	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 0 ∧ 𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
207		sibfof.5	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( 0 + 0 ) = (0g‘𝐾))
208	206, 207	sylan9eqr 2791	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 = 0 ∧ 𝑦 = 0 )) → (𝑥 + 𝑦) = (0g‘𝐾))
209	208	ex 411	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 = 0 ∧ 𝑦 = 0 ) → (𝑥 + 𝑦) = (0g‘𝐾)))
210	209	necon3ad 2946	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → ¬ (𝑥 = 0 ∧ 𝑦 = 0 )))
211		neorian 3030	. . . . . . . . . . . . 13 ⊢ ((𝑥 ≠ 0 ∨ 𝑦 ≠ 0 ) ↔ ¬ (𝑥 = 0 ∧ 𝑦 = 0 ))
212	210, 211	imbitrrdi 251	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )))
213	212	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )))
214	213	ralrimivva 3195	. . . . . . . . . 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 3991	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (◡ + “ {𝑧}))
218		fniniseg 7078	. . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧))
221		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧) → ( + ‘𝑝) = 𝑧)
222		1st2nd2 8050	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
223	222	fveq2d 6909	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → ( + ‘𝑝) = ( + ‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
224		df-ov 7433	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑝) + (2nd ‘𝑝)) = ( + ‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
225	223, 224	eqtr4di 2787	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → ( + ‘𝑝) = ((1st ‘𝑝) + (2nd ‘𝑝)))
226	225	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧) → ( + ‘𝑝) = ((1st ‘𝑝) + (2nd ‘𝑝)))
227	221, 226	eqtr3d 2771	. . . . . . . . . . 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 767	. . . . . . . . . . . 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 3972	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g‘𝐾)})
231		velsn 4652	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {(0g‘𝐾)} ↔ 𝑧 = (0g‘𝐾))
232	231	necon3bbii 2981	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ {(0g‘𝐾)} ↔ 𝑧 ≠ (0g‘𝐾))
233	230, 232	sylib 217	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ≠ (0g‘𝐾))
234	228, 233	eqnetrrd 3002	. . . . . . . . 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 679	. . . . . . . . . . 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 7437	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥 + 𝑦) = ((1st ‘𝑝) + 𝑦))
239	238	neeq1d 2993	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → ((𝑥 + 𝑦) ≠ (0g‘𝐾) ↔ ((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾)))
240		neeq1 2996	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥 ≠ 0 ↔ (1st ‘𝑝) ≠ 0 ))
241	240	orbi1d 914	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → ((𝑥 ≠ 0 ∨ 𝑦 ≠ 0 ) ↔ ((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 )))
242	239, 241	imbi12d 343	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → (((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )) ↔ (((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾) → ((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 ))))
243		oveq2 7438	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑝) → ((1st ‘𝑝) + 𝑦) = ((1st ‘𝑝) + (2nd ‘𝑝)))
244	243	neeq1d 2993	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑝) → (((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾) ↔ ((1st ‘𝑝) + (2nd ‘𝑝)) ≠ (0g‘𝐾)))
245		neeq1 2996	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑝) → (𝑦 ≠ 0 ↔ (2nd ‘𝑝) ≠ 0 ))
246	245	orbi2d 913	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑝) → (((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 ) ↔ ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 )))
247	244, 246	imbi12d 343	. . . . . . . . . . 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 3631	. . . . . . . . . 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 582	. . . . . . . . 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 34211	. . . . . . . 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 1370	. . . . . . 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 33946	. . . . . 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 2773	. . . . 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 13497	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
256	255, 252	sselid 3989	. . . . . 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 15759	. . . . 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 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ ℝ)
259	174	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑀 ∈ (measures‘dom 𝑀))
260	175, 108	sylanl2 679	. . . . . . 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 34078	. . . . . . 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 582	. . . . . 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 15820	. . . . 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 5184	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → 0 ≤ (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})))
265		elrege0 13495	. . . 4 ⊢ ((𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧}))))
266	258, 264, 265	sylanbrc 581	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘f + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘(◡(𝐹 ∘f + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
267	266	ralrimiva 3139	. 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 34205	. 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 1339	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom (𝐾sitg𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1534   ∈ wcel 2100   ≠ wne 2933  ∀wral 3054  Vcvv 3472   ∖ cdif 3956   ∪ cun 3957   ∩ cin 3958   ⊆ wss 3959  ∅c0 4335  {csn 4636  ⟨cop 4642  ∪ cuni 4918  ∪ ciun 5006  Disj wdisj 5123   class class class wbr 5156   × cxp 5684  ^◡ccnv 5685  dom cdm 5686  ran crn 5687   “ cima 5689  Fun wfun 6552   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7430   ∘_f cof 7693  ωcom 7884  1^st c1st 8009  2^nd c2nd 8010   ↑_m cmap 8863   ≼ cdom 8980   ≺ csdm 8981  Fincfn 8982  ℝcr 11164  0cc0 11165  +∞cpnf 11302   ≤ cle 11306  [,)cico 13390  Σcsu 15711  Basecbs 17234  Scalarcsca 17290   ·_𝑠 cvsca 17291  TopOpenctopn 17457  0_gc0g 17475  Topctop 22909  TopSpctps 22948  Clsdccld 23034  Frect1 23325  ℝHomcrrh 33846  Σ^*cesum 33898  sigAlgebracsiga 33979  sigaGencsigagen 34009  measurescmeas 34066  MblFnMcmbfm 34120  sitgcsitg 34201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5293  ax-sep 5307  ax-nul 5314  ax-pow 5373  ax-pr 5437  ax-un 7751  ax-inf2 9691  ax-ac2 10513  ax-cnex 11221  ax-resscn 11222  ax-1cn 11223  ax-icn 11224  ax-addcl 11225  ax-addrcl 11226  ax-mulcl 11227  ax-mulrcl 11228  ax-mulcom 11229  ax-addass 11230  ax-mulass 11231  ax-distr 11232  ax-i2m1 11233  ax-1ne0 11234  ax-1rid 11235  ax-rnegex 11236  ax-rrecex 11237  ax-cnre 11238  ax-pre-lttri 11239  ax-pre-lttrn 11240  ax-pre-ltadd 11241  ax-pre-mulgt0 11242  ax-pre-sup 11243  ax-addf 11244  ax-mulf 11245

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-nel 3040  df-ral 3055  df-rex 3064  df-rmo 3373  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-pss 3979  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-tp 4641  df-op 4643  df-uni 4919  df-int 4960  df-iun 5008  df-iin 5009  df-disj 5124  df-br 5157  df-opab 5219  df-mpt 5240  df-tr 5274  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5641  df-se 5642  df-we 5643  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-pred 6317  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7386  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7695  df-om 7885  df-1st 8011  df-2nd 8012  df-supp 8183  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8742  df-map 8865  df-pm 8866  df-ixp 8935  df-en 8983  df-dom 8984  df-sdom 8985  df-fin 8986  df-fsupp 9413  df-fi 9461  df-sup 9492  df-inf 9493  df-oi 9560  df-dju 9951  df-card 9989  df-acn 9992  df-ac 10166  df-pnf 11307  df-mnf 11308  df-xr 11309  df-ltxr 11310  df-le 11311  df-sub 11503  df-neg 11504  df-div 11929  df-nn 12275  df-2 12337  df-3 12338  df-4 12339  df-5 12340  df-6 12341  df-7 12342  df-8 12343  df-9 12344  df-n0 12535  df-z 12621  df-dec 12740  df-uz 12885  df-q 12995  df-rp 13039  df-xneg 13156  df-xadd 13157  df-xmul 13158  df-ioo 13392  df-ioc 13393  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13692  df-fl 13823  df-mod 13901  df-seq 14033  df-exp 14093  df-fac 14303  df-bc 14332  df-hash 14360  df-shft 15093  df-cj 15125  df-re 15126  df-im 15127  df-sqrt 15261  df-abs 15262  df-limsup 15494  df-clim 15511  df-rlim 15512  df-sum 15712  df-ef 16090  df-sin 16092  df-cos 16093  df-pi 16095  df-struct 17170  df-sets 17187  df-slot 17205  df-ndx 17217  df-base 17235  df-ress 17264  df-plusg 17300  df-mulr 17301  df-starv 17302  df-sca 17303  df-vsca 17304  df-ip 17305  df-tset 17306  df-ple 17307  df-ds 17309  df-unif 17310  df-hom 17311  df-cco 17312  df-rest 17458  df-topn 17459  df-0g 17477  df-gsum 17478  df-topgen 17479  df-pt 17480  df-prds 17483  df-ordt 17537  df-xrs 17538  df-qtop 17543  df-imas 17544  df-xps 17546  df-mre 17620  df-mrc 17621  df-acs 17623  df-ps 18612  df-tsr 18613  df-plusf 18653  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18794  df-submnd 18795  df-grp 18952  df-minusg 18953  df-sbg 18954  df-mulg 19084  df-subg 19139  df-cntz 19333  df-cmn 19802  df-abl 19803  df-mgp 20140  df-rng 20158  df-ur 20187  df-ring 20240  df-cring 20241  df-subrng 20550  df-subrg 20575  df-abv 20810  df-lmod 20860  df-scaf 20861  df-sra 21173  df-rgmod 21174  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-fbas 21362  df-fg 21363  df-cnfld 21366  df-top 22910  df-topon 22927  df-topsp 22949  df-bases 22963  df-cld 23037  df-ntr 23038  df-cls 23039  df-nei 23116  df-lp 23154  df-perf 23155  df-cn 23245  df-cnp 23246  df-t1 23332  df-haus 23333  df-tx 23580  df-hmeo 23773  df-fil 23864  df-fm 23956  df-flim 23957  df-flf 23958  df-tmd 24090  df-tgp 24091  df-tsms 24145  df-trg 24178  df-xms 24340  df-ms 24341  df-tms 24342  df-nm 24605  df-ngp 24606  df-nrg 24608  df-nlm 24609  df-ii 24911  df-cncf 24912  df-limc 25909  df-dv 25910  df-log 26606  df-esum 33899  df-siga 33980  df-sigagen 34010  df-meas 34067  df-mbfm 34121  df-sitg 34202

This theorem is referenced by:  sitmcl  34223