Theorem sibfof 30402

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 20740	. . . . . . . . . . 11 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
6	2, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
7	6	sqxpeqd 5141	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐵) = (∪ 𝐽 × ∪ 𝐽))
8	7	feq2d 6031	. . . . . . . 8 ⊢ (𝜑 → ( + :(𝐵 × 𝐵)⟶𝐶 ↔ + :(∪ 𝐽 × ∪ 𝐽)⟶𝐶))
9	1, 8	mpbid 222	. . . . . . 7 ⊢ (𝜑 → + :(∪ 𝐽 × ∪ 𝐽)⟶𝐶)
10	9	fovrnda 6805	. . . . . 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 30398	. . . . . 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 30398	. . . . . 6 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
21		dmexg 7097	. . . . . . 7 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ V)
22		uniexg 6955	. . . . . . 7 ⊢ (dom 𝑀 ∈ V → ∪ dom 𝑀 ∈ V)
23	16, 21, 22	3syl 18	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑀 ∈ V)
24		inidm 3822	. . . . . 6 ⊢ (∪ dom 𝑀 ∩ ∪ dom 𝑀) = ∪ dom 𝑀
25	10, 18, 20, 23, 23, 24	off 6912	. . . . 5 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺):∪ dom 𝑀⟶𝐶)
26		sibfof.3	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ TopSp)
27		sibfof.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝐾)
28		eqid 2622	. . . . . . . . 9 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
29	27, 28	tpsuni 20740	. . . . . . . 8 ⊢ (𝐾 ∈ TopSp → 𝐶 = ∪ (TopOpen‘𝐾))
30	26, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 = ∪ (TopOpen‘𝐾))
31		fvex 6201	. . . . . . . 8 ⊢ (TopOpen‘𝐾) ∈ V
32		unisg 30206	. . . . . . . 8 ⊢ ((TopOpen‘𝐾) ∈ V → ∪ (sigaGen‘(TopOpen‘𝐾)) = ∪ (TopOpen‘𝐾))
33	31, 32	ax-mp 5	. . . . . . 7 ⊢ ∪ (sigaGen‘(TopOpen‘𝐾)) = ∪ (TopOpen‘𝐾)
34	30, 33	syl6eqr 2674	. . . . . 6 ⊢ (𝜑 → 𝐶 = ∪ (sigaGen‘(TopOpen‘𝐾)))
35	34	feq3d 6032	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘𝑓 + 𝐺):∪ dom 𝑀⟶𝐶 ↔ (𝐹 ∘𝑓 + 𝐺):∪ dom 𝑀⟶∪ (sigaGen‘(TopOpen‘𝐾))))
36	25, 35	mpbid 222	. . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺):∪ dom 𝑀⟶∪ (sigaGen‘(TopOpen‘𝐾)))
37	31	a1i 11	. . . . . . 7 ⊢ (𝜑 → (TopOpen‘𝐾) ∈ V)
38	37	sgsiga 30205	. . . . . 6 ⊢ (𝜑 → (sigaGen‘(TopOpen‘𝐾)) ∈ ∪ ran sigAlgebra)
39		uniexg 6955	. . . . . 6 ⊢ ((sigaGen‘(TopOpen‘𝐾)) ∈ ∪ ran sigAlgebra → ∪ (sigaGen‘(TopOpen‘𝐾)) ∈ V)
40	38, 39	syl 17	. . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘(TopOpen‘𝐾)) ∈ V)
41	40, 23	elmapd 7871	. . . 4 ⊢ (𝜑 → ((𝐹 ∘𝑓 + 𝐺) ∈ (∪ (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 ∪ dom 𝑀) ↔ (𝐹 ∘𝑓 + 𝐺):∪ dom 𝑀⟶∪ (sigaGen‘(TopOpen‘𝐾))))
42	36, 41	mpbird 247	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (∪ (sigaGen‘(TopOpen‘𝐾)) ↑𝑚 ∪ dom 𝑀))
43		inundif 4046	. . . . . . 7 ⊢ ((𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘𝑓 + 𝐺))) = 𝑏
44	43	imaeq2i 5464	. . . . . 6 ⊢ (◡(𝐹 ∘𝑓 + 𝐺) “ ((𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ∪ (𝑏 ∖ ran (𝐹 ∘𝑓 + 𝐺)))) = (◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏)
45		ffun 6048	. . . . . . . 8 ⊢ ((𝐹 ∘𝑓 + 𝐺):∪ dom 𝑀⟶𝐶 → Fun (𝐹 ∘𝑓 + 𝐺))
46		unpreima 6341	. . . . . . . 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)))))
47	25, 45, 46	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)))))
48	47	adantr 481	. . . . . 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)))))
49	44, 48	syl5eqr 2670	. . . . 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)))))
50		dmmeas 30264	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
51	16, 50	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
52	51	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → dom 𝑀 ∈ ∪ ran sigAlgebra)
53		imaiun 6503	. . . . . . . 8 ⊢ (◡(𝐹 ∘𝑓 + 𝐺) “ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)){𝑧}) = ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺))(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧})
54		iunid 4575	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)){𝑧} = (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺))
55	54	imaeq2i 5464	. . . . . . . 8 ⊢ (◡(𝐹 ∘𝑓 + 𝐺) “ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)){𝑧}) = (◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)))
56	53, 55	eqtr3i 2646	. . . . . . 7 ⊢ ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺))(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧}) = (◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)))
57		inss2 3834	. . . . . . . . . 10 ⊢ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ⊆ ran (𝐹 ∘𝑓 + 𝐺)
58	6	feq3d 6032	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹:∪ dom 𝑀⟶𝐵 ↔ 𝐹:∪ dom 𝑀⟶∪ 𝐽))
59	18, 58	mpbird 247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶𝐵)
60	6	feq3d 6032	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺:∪ dom 𝑀⟶𝐵 ↔ 𝐺:∪ dom 𝑀⟶∪ 𝐽))
61	20, 60	mpbird 247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶𝐵)
62		ffn 6045	. . . . . . . . . . . . . . 15 ⊢ ( + :(𝐵 × 𝐵)⟶𝐶 → + Fn (𝐵 × 𝐵))
63	1, 62	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → + Fn (𝐵 × 𝐵))
64	59, 61, 23, 63	ofpreima2 29466	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧}) = ∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
65	64	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) → (◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧}) = ∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
66	51	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) → dom 𝑀 ∈ ∪ ran sigAlgebra)
67	51	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → dom 𝑀 ∈ ∪ ran sigAlgebra)
68		simpll 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
69		inss1 3833	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (◡ + “ {𝑧})
70		cnvimass 5485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡ + “ {𝑧}) ⊆ dom +
71		fdm 6051	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( + :(𝐵 × 𝐵)⟶𝐶 → dom + = (𝐵 × 𝐵))
72	1, 71	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom + = (𝐵 × 𝐵))
73	70, 72	syl5sseq 3653	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (◡ + “ {𝑧}) ⊆ (𝐵 × 𝐵))
74	73	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) → (◡ + “ {𝑧}) ⊆ (𝐵 × 𝐵))
75	69, 74	syl5ss 3614	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐵))
76	75	sselda 3603	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
77	51	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → dom 𝑀 ∈ ∪ ran sigAlgebra)
78		sibfof.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ Fre)
79	78	sgsiga 30205	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
80	11, 79	syl5eqel 2705	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
81	80	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝑆 ∈ ∪ ran sigAlgebra)
82	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfmbl 30397	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
83	82	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
84	4	tpstop 20741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
85		cldssbrsiga 30250	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
86	2, 84, 85	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
87	86	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
88	78	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐽 ∈ Fre)
89		xp1st 7198	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → (1st ‘𝑝) ∈ 𝐵)
90	89	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (1st ‘𝑝) ∈ 𝐵)
91	6	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐵 = ∪ 𝐽)
92	90, 91	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (1st ‘𝑝) ∈ ∪ 𝐽)
93		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ 𝐽 = ∪ 𝐽
94	93	t1sncld 21130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Fre ∧ (1st ‘𝑝) ∈ ∪ 𝐽) → {(1st ‘𝑝)} ∈ (Clsd‘𝐽))
95	88, 92, 94	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ (Clsd‘𝐽))
96	87, 95	sseldd 3604	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ (sigaGen‘𝐽))
97	96, 11	syl6eleqr 2712	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(1st ‘𝑝)} ∈ 𝑆)
98	77, 81, 83, 97	mbfmcnvima 30319	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (◡𝐹 “ {(1st ‘𝑝)}) ∈ dom 𝑀)
99	68, 76, 98	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (◡𝐹 “ {(1st ‘𝑝)}) ∈ dom 𝑀)
100	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfmbl 30397	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
101	100	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
102		xp2nd 7199	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵)
103	102	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (2nd ‘𝑝) ∈ 𝐵)
104	103, 91	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (2nd ‘𝑝) ∈ ∪ 𝐽)
105	93	t1sncld 21130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Fre ∧ (2nd ‘𝑝) ∈ ∪ 𝐽) → {(2nd ‘𝑝)} ∈ (Clsd‘𝐽))
106	88, 104, 105	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ (Clsd‘𝐽))
107	87, 106	sseldd 3604	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ (sigaGen‘𝐽))
108	107, 11	syl6eleqr 2712	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → {(2nd ‘𝑝)} ∈ 𝑆)
109	77, 81, 101, 108	mbfmcnvima 30319	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝐵)) → (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀)
110	68, 76, 109	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀)
111		inelsiga 30198	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (◡𝐹 “ {(1st ‘𝑝)}) ∈ dom 𝑀 ∧ (◡𝐺 “ {(2nd ‘𝑝)}) ∈ dom 𝑀) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
112	67, 99, 110, 111	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
113	112	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) → ∀𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
114	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfrn 30399	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
115	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 30399	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
116		xpfi 8231	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
117	114, 115, 116	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
118		inss2 3834	. . . . . . . . . . . . . . . 16 ⊢ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)
119		ssdomg 8001	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝐹 × ran 𝐺) ∈ Fin → (((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺)))
120	117, 118, 119	mpisyl 21	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺))
121		isfinite 8549	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 × ran 𝐺) ∈ Fin ↔ (ran 𝐹 × ran 𝐺) ≺ ω)
122	121	biimpi 206	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝐹 × ran 𝐺) ∈ Fin → (ran 𝐹 × ran 𝐺) ≺ ω)
123		sdomdom 7983	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝐹 × ran 𝐺) ≺ ω → (ran 𝐹 × ran 𝐺) ≼ ω)
124	117, 122, 123	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ≼ ω)
125		domtr 8009	. . . . . . . . . . . . . . 15 ⊢ ((((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ (ran 𝐹 × ran 𝐺) ∧ (ran 𝐹 × ran 𝐺) ≼ ω) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
126	120, 124, 125	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
127	126	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
128		nfcv 2764	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑝((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))
129	128	sigaclcuni 30181	. . . . . . . . . . . . 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)
130	66, 113, 127, 129	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) → ∪ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
131	65, 130	eqeltrd 2701	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)) → (◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
132	131	ralrimiva 2966	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺)(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
133		ssralv 3666	. . . . . . . . . 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))
134	57, 132, 133	mpsyl 68	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺))(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
135	134	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∀𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺))(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
136		ffun 6048	. . . . . . . . . . . . . 14 ⊢ ( + :(𝐵 × 𝐵)⟶𝐶 → Fun + )
137	1, 136	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun + )
138		imafi 8259	. . . . . . . . . . . . 13 ⊢ ((Fun + ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
139	137, 117, 138	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin)
140	18, 20, 9, 23	ofrn2 29442	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝐹 ∘𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
141		ssfi 8180	. . . . . . . . . . . 12 ⊢ ((( + “ (ran 𝐹 × ran 𝐺)) ∈ Fin ∧ ran (𝐹 ∘𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺))) → ran (𝐹 ∘𝑓 + 𝐺) ∈ Fin)
142	139, 140, 141	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝐹 ∘𝑓 + 𝐺) ∈ Fin)
143		ssdomg 8001	. . . . . . . . . . 11 ⊢ (ran (𝐹 ∘𝑓 + 𝐺) ∈ Fin → ((𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ⊆ ran (𝐹 ∘𝑓 + 𝐺) → (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ≼ ran (𝐹 ∘𝑓 + 𝐺)))
144	142, 57, 143	mpisyl 21	. . . . . . . . . 10 ⊢ (𝜑 → (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ≼ ran (𝐹 ∘𝑓 + 𝐺))
145		isfinite 8549	. . . . . . . . . . . 12 ⊢ (ran (𝐹 ∘𝑓 + 𝐺) ∈ Fin ↔ ran (𝐹 ∘𝑓 + 𝐺) ≺ ω)
146	142, 145	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝐹 ∘𝑓 + 𝐺) ≺ ω)
147		sdomdom 7983	. . . . . . . . . . 11 ⊢ (ran (𝐹 ∘𝑓 + 𝐺) ≺ ω → ran (𝐹 ∘𝑓 + 𝐺) ≼ ω)
148	146, 147	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝐹 ∘𝑓 + 𝐺) ≼ ω)
149		domtr 8009	. . . . . . . . . 10 ⊢ (((𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ≼ ran (𝐹 ∘𝑓 + 𝐺) ∧ ran (𝐹 ∘𝑓 + 𝐺) ≼ ω) → (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ≼ ω)
150	144, 148, 149	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ≼ ω)
151	150	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺)) ≼ ω)
152		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺))
153	152	sigaclcuni 30181	. . . . . . . 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)
154	52, 135, 151, 153	syl3anc 1326	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ∪ 𝑧 ∈ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺))(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧}) ∈ dom 𝑀)
155	56, 154	syl5eqelr 2706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺))) ∈ dom 𝑀)
156		difpreima 6343	. . . . . . . . . 10 ⊢ (Fun (𝐹 ∘𝑓 + 𝐺) → (◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘𝑓 + 𝐺))) = ((◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘𝑓 + 𝐺) “ ran (𝐹 ∘𝑓 + 𝐺))))
157	25, 45, 156	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘𝑓 + 𝐺))) = ((◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘𝑓 + 𝐺) “ ran (𝐹 ∘𝑓 + 𝐺))))
158		cnvimarndm 5486	. . . . . . . . . . 11 ⊢ (◡(𝐹 ∘𝑓 + 𝐺) “ ran (𝐹 ∘𝑓 + 𝐺)) = dom (𝐹 ∘𝑓 + 𝐺)
159	158	difeq2i 3725	. . . . . . . . . 10 ⊢ ((◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘𝑓 + 𝐺) “ ran (𝐹 ∘𝑓 + 𝐺))) = ((◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘𝑓 + 𝐺))
160		cnvimass 5485	. . . . . . . . . . 11 ⊢ (◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹 ∘𝑓 + 𝐺)
161		ssdif0 3942	. . . . . . . . . . 11 ⊢ ((◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ⊆ dom (𝐹 ∘𝑓 + 𝐺) ↔ ((◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘𝑓 + 𝐺)) = ∅)
162	160, 161	mpbi 220	. . . . . . . . . 10 ⊢ ((◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ∖ dom (𝐹 ∘𝑓 + 𝐺)) = ∅
163	159, 162	eqtri 2644	. . . . . . . . 9 ⊢ ((◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ∖ (◡(𝐹 ∘𝑓 + 𝐺) “ ran (𝐹 ∘𝑓 + 𝐺))) = ∅
164	157, 163	syl6eq 2672	. . . . . . . 8 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘𝑓 + 𝐺))) = ∅)
165		0elsiga 30177	. . . . . . . . 9 ⊢ (dom 𝑀 ∈ ∪ ran sigAlgebra → ∅ ∈ dom 𝑀)
166	16, 50, 165	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ dom 𝑀)
167	164, 166	eqeltrd 2701	. . . . . . 7 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘𝑓 + 𝐺))) ∈ dom 𝑀)
168	167	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘𝑓 + 𝐺))) ∈ dom 𝑀)
169		unelsiga 30197	. . . . . 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)
170	52, 155, 168, 169	syl3anc 1326	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → ((◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∩ ran (𝐹 ∘𝑓 + 𝐺))) ∪ (◡(𝐹 ∘𝑓 + 𝐺) “ (𝑏 ∖ ran (𝐹 ∘𝑓 + 𝐺)))) ∈ dom 𝑀)
171	49, 170	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))) → (◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
172	171	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ (sigaGen‘(TopOpen‘𝐾))(◡(𝐹 ∘𝑓 + 𝐺) “ 𝑏) ∈ dom 𝑀)
173	51, 38	ismbfm 30314	. . 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)))
174	42, 172, 173	mpbir2and 957	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (dom 𝑀MblFnM(sigaGen‘(TopOpen‘𝐾))))
175	64	adantr 481	. . . . . . 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)})))
176	175	fveq2d 6195	. . . . . 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)}))))
177		measbasedom 30265	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
178	16, 177	sylib 208	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (measures‘dom 𝑀))
179	178	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → 𝑀 ∈ (measures‘dom 𝑀))
180		eldifi 3732	. . . . . . . 8 ⊢ (𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)}) → 𝑧 ∈ ran (𝐹 ∘𝑓 + 𝐺))
181	180, 113	sylan2 491	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → ∀𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
182	126	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ≼ ω)
183		sneq 4187	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → {𝑥} = {(1st ‘𝑝)})
184	183	imaeq2d 5466	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑝) → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {(1st ‘𝑝)}))
185		sneq 4187	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑝) → {𝑦} = {(2nd ‘𝑝)})
186	185	imaeq2d 5466	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑝) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {(2nd ‘𝑝)}))
187		ffun 6048	. . . . . . . . . . . 12 ⊢ (𝐹:∪ dom 𝑀⟶∪ 𝐽 → Fun 𝐹)
188	18, 187	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
189		sndisj 4644	. . . . . . . . . . 11 ⊢ Disj 𝑥 ∈ ran 𝐹{𝑥}
190		disjpreima 29397	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ ran 𝐹{𝑥}) → Disj 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
191	188, 189, 190	sylancl 694	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑥 ∈ ran 𝐹(◡𝐹 “ {𝑥}))
192		ffun 6048	. . . . . . . . . . . 12 ⊢ (𝐺:∪ dom 𝑀⟶∪ 𝐽 → Fun 𝐺)
193	20, 192	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐺)
194		sndisj 4644	. . . . . . . . . . 11 ⊢ Disj 𝑦 ∈ ran 𝐺{𝑦}
195		disjpreima 29397	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ Disj 𝑦 ∈ ran 𝐺{𝑦}) → Disj 𝑦 ∈ ran 𝐺(◡𝐺 “ {𝑦}))
196	193, 194, 195	sylancl 694	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑦 ∈ ran 𝐺(◡𝐺 “ {𝑦}))
197	184, 186, 191, 196	disjxpin 29401	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑝 ∈ (ran 𝐹 × ran 𝐺)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
198		disjss1 4626	. . . . . . . . 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)}))))
199	118, 197, 198	mpsyl 68	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
200	199	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → Disj 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
201		measvuni 30277	. . . . . . 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)}))))
202	179, 181, 182, 200, 201	syl112anc 1330	. . . . . 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)}))))
203		ssfi 8180	. . . . . . . . 9 ⊢ (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (ran 𝐹 × ran 𝐺)) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
204	117, 118, 203	sylancl 694	. . . . . . . 8 ⊢ (𝜑 → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
205	204	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ∈ Fin)
206		simpll 790	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝜑)
207		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)))
208	118, 207	sseldi 3601	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (ran 𝐹 × ran 𝐺))
209		xp1st 7198	. . . . . . . . 9 ⊢ (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (1st ‘𝑝) ∈ ran 𝐹)
210	208, 209	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st ‘𝑝) ∈ ran 𝐹)
211		xp2nd 7199	. . . . . . . . 9 ⊢ (𝑝 ∈ (ran 𝐹 × ran 𝐺) → (2nd ‘𝑝) ∈ ran 𝐺)
212	208, 211	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd ‘𝑝) ∈ ran 𝐺)
213		oveq12 6659	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 0 ∧ 𝑦 = 0 ) → (𝑥 + 𝑦) = ( 0 + 0 ))
214		sibfof.5	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( 0 + 0 ) = (0g‘𝐾))
215	213, 214	sylan9eqr 2678	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 = 0 ∧ 𝑦 = 0 )) → (𝑥 + 𝑦) = (0g‘𝐾))
216	215	ex 450	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 = 0 ∧ 𝑦 = 0 ) → (𝑥 + 𝑦) = (0g‘𝐾)))
217	216	necon3ad 2807	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → ¬ (𝑥 = 0 ∧ 𝑦 = 0 )))
218		neorian 2888	. . . . . . . . . . . . 13 ⊢ ((𝑥 ≠ 0 ∨ 𝑦 ≠ 0 ) ↔ ¬ (𝑥 = 0 ∧ 𝑦 = 0 ))
219	217, 218	syl6ibr 242	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )))
220	219	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )))
221	220	ralrimivva 2971	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )))
222	206, 221	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 )))
223	69	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺)) ⊆ (◡ + “ {𝑧}))
224	223	sselda 3603	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (◡ + “ {𝑧}))
225		fniniseg 6338	. . . . . . . . . . . . 13 ⊢ ( + Fn (𝐵 × 𝐵) → (𝑝 ∈ (◡ + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧)))
226	206, 63, 225	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (◡ + “ {𝑧}) ↔ (𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧)))
227	224, 226	mpbid 222	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧))
228		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧) → ( + ‘𝑝) = 𝑧)
229		1st2nd2 7205	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
230	229	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → ( + ‘𝑝) = ( + ‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
231		df-ov 6653	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑝) + (2nd ‘𝑝)) = ( + ‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
232	230, 231	syl6eqr 2674	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (𝐵 × 𝐵) → ( + ‘𝑝) = ((1st ‘𝑝) + (2nd ‘𝑝)))
233	232	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧) → ( + ‘𝑝) = ((1st ‘𝑝) + (2nd ‘𝑝)))
234	228, 233	eqtr3d 2658	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ (𝐵 × 𝐵) ∧ ( + ‘𝑝) = 𝑧) → 𝑧 = ((1st ‘𝑝) + (2nd ‘𝑝)))
235	227, 234	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 = ((1st ‘𝑝) + (2nd ‘𝑝)))
236		simplr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)}))
237	236	eldifbd 3587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ¬ 𝑧 ∈ {(0g‘𝐾)})
238		velsn 4193	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {(0g‘𝐾)} ↔ 𝑧 = (0g‘𝐾))
239	238	necon3bbii 2841	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ {(0g‘𝐾)} ↔ 𝑧 ≠ (0g‘𝐾))
240	237, 239	sylib 208	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑧 ≠ (0g‘𝐾))
241	235, 240	eqnetrrd 2862	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st ‘𝑝) + (2nd ‘𝑝)) ≠ (0g‘𝐾))
242	180, 76	sylanl2 683	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐵))
243	242, 89	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (1st ‘𝑝) ∈ 𝐵)
244	242, 102	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (2nd ‘𝑝) ∈ 𝐵)
245		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥 + 𝑦) = ((1st ‘𝑝) + 𝑦))
246	245	neeq1d 2853	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → ((𝑥 + 𝑦) ≠ (0g‘𝐾) ↔ ((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾)))
247		neeq1 2856	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥 ≠ 0 ↔ (1st ‘𝑝) ≠ 0 ))
248	247	orbi1d 739	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → ((𝑥 ≠ 0 ∨ 𝑦 ≠ 0 ) ↔ ((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 )))
249	246, 248	imbi12d 334	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → (((𝑥 + 𝑦) ≠ (0g‘𝐾) → (𝑥 ≠ 0 ∨ 𝑦 ≠ 0 )) ↔ (((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾) → ((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 ))))
250		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑝) → ((1st ‘𝑝) + 𝑦) = ((1st ‘𝑝) + (2nd ‘𝑝)))
251	250	neeq1d 2853	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑝) → (((1st ‘𝑝) + 𝑦) ≠ (0g‘𝐾) ↔ ((1st ‘𝑝) + (2nd ‘𝑝)) ≠ (0g‘𝐾)))
252		neeq1 2856	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑝) → (𝑦 ≠ 0 ↔ (2nd ‘𝑝) ≠ 0 ))
253	252	orbi2d 738	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑝) → (((1st ‘𝑝) ≠ 0 ∨ 𝑦 ≠ 0 ) ↔ ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 )))
254	251, 253	imbi12d 334	. . . . . . . . . . 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 ))))
255	249, 254	rspc2v 3322	. . . . . . . . . 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 ))))
256	243, 244, 255	syl2anc 693	. . . . . . . . 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 ))))
257	222, 241, 256	mp2d 49	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 ))
258	3, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 78	sibfinima 30401	. . . . . . . 8 ⊢ (((𝜑 ∧ (1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺) ∧ ((1st ‘𝑝) ≠ 0 ∨ (2nd ‘𝑝) ≠ 0 )) → (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) ∈ (0[,)+∞))
259	206, 210, 212, 257, 258	syl31anc 1329	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) ∈ (0[,)+∞))
260	205, 259	esumpfinval 30137	. . . . . 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)}))))
261	176, 202, 260	3eqtrd 2660	. . . . 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)}))))
262		rge0ssre 12280	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
263	262, 259	sseldi 3601	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) ∈ ℝ)
264	205, 263	fsumrecl 14465	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → Σ𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) ∈ ℝ)
265	261, 264	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧})) ∈ ℝ)
266	179	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 𝑀 ∈ (measures‘dom 𝑀))
267	180, 112	sylanl2 683	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀)
268		measge0 30270	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∈ dom 𝑀) → 0 ≤ (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
269	266, 267, 268	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) ∧ 𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))) → 0 ≤ (𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
270	205, 263, 269	fsumge0 14527	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → 0 ≤ Σ𝑝 ∈ ((◡ + “ {𝑧}) ∩ (ran 𝐹 × ran 𝐺))(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
271	270, 261	breqtrrd 4681	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → 0 ≤ (𝑀‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧})))
272		elrege0 12278	. . . 4 ⊢ ((𝑀‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧}))))
273	265, 271, 272	sylanbrc 698	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})) → (𝑀‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
274	273	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (ran (𝐹 ∘𝑓 + 𝐺) ∖ {(0g‘𝐾)})(𝑀‘(◡(𝐹 ∘𝑓 + 𝐺) “ {𝑧})) ∈ (0[,)+∞))
275		eqid 2622	. . 3 ⊢ (sigaGen‘(TopOpen‘𝐾)) = (sigaGen‘(TopOpen‘𝐾))
276		eqid 2622	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
277		eqid 2622	. . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
278		eqid 2622	. . 3 ⊢ (ℝHom‘(Scalar‘𝐾)) = (ℝHom‘(Scalar‘𝐾))
279	27, 28, 275, 276, 277, 278, 26, 16	issibf 30395	. 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[,)+∞))))
280	174, 142, 274, 279	mpbir3and 1245	1 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ dom (𝐾sitg𝑀))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ cuni 4436  ∪ ciun 4520  Disj wdisj 4620   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  ωcom 7065  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  ℝcr 9935  0cc0 9936  +∞cpnf 10071   ≤ cle 10075  [,)cico 12177  Σcsu 14416  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  TopOpenctopn 16082  0_gc0g 16100  Topctop 20698  TopSpctps 20736  Clsdccld 20820  Frect1 21111  ℝHomcrrh 30037  Σ^*cesum 30089  sigAlgebracsiga 30170  sigaGencsigagen 30201  measurescmeas 30258  MblFnMcmbfm 30312  sitgcsitg 30391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-ordt 16161  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tmd 21876  df-tgp 21877  df-tsms 21930  df-trg 21963  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-ii 22680  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-esum 30090  df-siga 30171  df-sigagen 30202  df-meas 30259  df-mbfm 30313  df-sitg 30392

This theorem is referenced by:  sitmcl  30413