voliunsge0lem - Mathbox for Glauco Siliprandi

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Theorem voliunsge0lem 46915

Description: The Lebesgue measure function is countably additive. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
voliunsge0lem.s	⊢ 𝑆 = seq1( + , 𝐺)
voliunsge0lem.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))
voliunsge0lem.e	⊢ (𝜑 → 𝐸:ℕ⟶dom vol)
voliunsge0lem.d	⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))

**Assertion**
Ref	Expression
voliunsge0lem	⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))

Distinct variable groups: 𝑛,𝐸 𝜑,𝑛 Allowed substitution hints: 𝑆(𝑛) 𝐺(𝑛)

Proof of Theorem voliunsge0lem Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1921	. . . . 5 ⊢ Ⅎ𝑛𝜑
2		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑛vol
3		nfiu1 4957	. . . . . . 7 ⊢ Ⅎ𝑛∪ 𝑛 ∈ ℕ (𝐸‘𝑛)
4	2, 3	nffv 6837	. . . . . 6 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
5	4	nfeq1 2916	. . . . 5 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞
6		iccssxr 13374	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ^*
7		volf 25514	. . . . . . . . . . . 12 ⊢ vol:dom vol⟶(0[,]+∞)
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞))
9		voliunsge0lem.e	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸:ℕ⟶dom vol)
10	9	ffvelcdmda 7025	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom vol)
11	10	ralrimiva 3131	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
12		iunmbl 25538	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
13	11, 12	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
14	8, 13	ffvelcdmd 7026	. . . . . . . . . 10 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ (0[,]+∞))
15	6, 14	sselid 3913	. . . . . . . . 9 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ^*)
16	15	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ^*)
17	16	3adant3 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ^*)
18		id 22	. . . . . . . . . 10 ⊢ ((vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) = +∞)
19	18	eqcomd 2745	. . . . . . . . 9 ⊢ ((vol‘(𝐸‘𝑛)) = +∞ → +∞ = (vol‘(𝐸‘𝑛)))
20	19	3ad2ant3 1141	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → +∞ = (vol‘(𝐸‘𝑛)))
21	13	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
22		ssiun2 4977	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
23	22	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
24		volss 25518	. . . . . . . . . 10 ⊢ (((𝐸‘𝑛) ∈ dom vol ∧ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol ∧ (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) → (vol‘(𝐸‘𝑛)) ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
25	10, 21, 23, 24	syl3anc 1379	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
26	25	3adant3 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
27	20, 26	eqbrtrd 5094	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
28	17, 27	xrgepnfd 45776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞)
29	28	3exp 1125	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ → ((vol‘(𝐸‘𝑛)) = +∞ → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞)))
30	1, 5, 29	rexlimd 3246	. . . 4 ⊢ (𝜑 → (∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞))
31	30	imp 407	. . 3 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞)
32		nfre1 3264	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞
33	1, 32	nfan 1906	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
34		nnex 12171	. . . . 5 ⊢ ℕ ∈ V
35	34	a1i 11	. . . 4 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ℕ ∈ V)
36	7	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
37	36, 10	ffvelcdmd 7026	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,]+∞))
38	37	adantlr 721	. . . 4 ⊢ (((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,]+∞))
39		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
40	33, 35, 38, 39	sge0pnfmpt 46888	. . 3 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = +∞)
41	31, 40	eqtr4d 2777	. 2 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
42		ralnex 3065	. . . . 5 ⊢ (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
43	42	bilanri 507	. . . 4 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞)
44	37	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ∈ (0[,]+∞))
45	18	necon3bi 2960	. . . . . . . . . 10 ⊢ (¬ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) ≠ +∞)
46	45	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ≠ +∞)
47		ge0xrre 45976	. . . . . . . . 9 ⊢ (((vol‘(𝐸‘𝑛)) ∈ (0[,]+∞) ∧ (vol‘(𝐸‘𝑛)) ≠ +∞) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
48	44, 46, 47	syl2anc 590	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
49	48	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) ∈ ℝ))
50		renepnf 11184	. . . . . . . . 9 ⊢ ((vol‘(𝐸‘𝑛)) ∈ ℝ → (vol‘(𝐸‘𝑛)) ≠ +∞)
51	50	neneqd 2939	. . . . . . . 8 ⊢ ((vol‘(𝐸‘𝑛)) ∈ ℝ → ¬ (vol‘(𝐸‘𝑛)) = +∞)
52	51	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐸‘𝑛)) ∈ ℝ → ¬ (vol‘(𝐸‘𝑛)) = +∞))
53	49, 52	impbid 213	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ (vol‘(𝐸‘𝑛)) ∈ ℝ))
54	53	ralbidva 3160	. . . . 5 ⊢ (𝜑 → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ))
55	54	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ))
56	43, 55	mpbid 233	. . 3 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ)
57		nfra1 3263	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ
58	1, 57	nfan 1906	. . . . . 6 ⊢ Ⅎ𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ)
59	10	adantlr 721	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom vol)
60		rspa 3228	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
61	60	adantll 720	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
62	59, 61	jca 516	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) ∈ dom vol ∧ (vol‘(𝐸‘𝑛)) ∈ ℝ))
63	62	ex 413	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝐸‘𝑛) ∈ dom vol ∧ (vol‘(𝐸‘𝑛)) ∈ ℝ)))
64	58, 63	ralrimi 3237	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝐸‘𝑛) ∈ dom vol ∧ (vol‘(𝐸‘𝑛)) ∈ ℝ))
65		voliunsge0lem.d	. . . . . 6 ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
66	65	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
67		voliunsge0lem.s	. . . . . 6 ⊢ 𝑆 = seq1( + , 𝐺)
68		voliunsge0lem.g	. . . . . 6 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))
69	67, 68	voliun 25539	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ ((𝐸‘𝑛) ∈ dom vol ∧ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝐸‘𝑛)) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = sup(ran 𝑆, ℝ^*, < ))
70	64, 66, 69	syl2anc 590	. . . 4 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = sup(ran 𝑆, ℝ^*, < ))
71		1zzd 12549	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → 1 ∈ ℤ)
72		nnuz 12818	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
73		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
74	58, 73	nfan 1906	. . . . . . . 8 ⊢ Ⅎ𝑛((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)
75		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑛(vol‘(𝐸‘𝑚)) ∈ (0[,)+∞)
76	74, 75	nfim 1903	. . . . . . 7 ⊢ Ⅎ𝑛(((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞))
77		eleq1w 2822	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ ℕ ↔ 𝑚 ∈ ℕ))
78	77	anbi2d 636	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) ↔ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)))
79		2fveq3 6832	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (vol‘(𝐸‘𝑛)) = (vol‘(𝐸‘𝑚)))
80	79	eleq1d 2824	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((vol‘(𝐸‘𝑛)) ∈ (0[,)+∞) ↔ (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞)))
81	78, 80	imbi12d 345	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,)+∞)) ↔ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞))))
82		0xr 11183	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
83	82	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
84		pnfxr 11190	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
85	84	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
86	61	rexrd 11186	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ^*)
87		volge0 46404	. . . . . . . . . 10 ⊢ ((𝐸‘𝑛) ∈ dom vol → 0 ≤ (vol‘(𝐸‘𝑛)))
88	10, 87	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸‘𝑛)))
89	88	adantlr 721	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸‘𝑛)))
90	61	ltpnfd 13063	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) < +∞)
91	83, 85, 86, 89, 90	elicod 13339	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,)+∞))
92	76, 81, 91	chvarfv 2252	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞))
93	79	cbvmptv 5176	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘(𝐸‘𝑚)))
94	92, 93	fmptd 7055	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))):ℕ⟶(0[,)+∞))
95		seqeq3 13959	. . . . . . 7 ⊢ (𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))) → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
96	68, 95	ax-mp 5	. . . . . 6 ⊢ seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))))
97	67, 96	eqtri 2762	. . . . 5 ⊢ 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))))
98	71, 72, 94, 97	sge0seq 46889	. . . 4 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = sup(ran 𝑆, ℝ^*, < ))
99	70, 98	eqtr4d 2777	. . 3 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
100	56, 99	syldan 597	. 2 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
101	41, 100	pm2.61dan 818	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ⊆ wss 3883 ∪ ciun 4921 Disj wdisj 5039 class class class wbr 5072 ↦ cmpt 5153 dom cdm 5618 ran crn 5619 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 supcsup 9343 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 ℕcn 12165 [,)cico 13291 [,]cicc 13292 seqcseq 13954 volcvol 25448 Σ^{^}csumge0 46805

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-disj 5040 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-q 12890 df-rp 12934 df-xadd 13055 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-xmet 21340 df-met 21341 df-ovol 25449 df-vol 25450 df-sumge0 46806

This theorem is referenced by: voliunsge0 46916

W3C validator