voliunsge0lem - Mathbox for Glauco Siliprandi

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Theorem voliunsge0lem 46463

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 1914	. . . . 5 ⊢ Ⅎ𝑛𝜑
2		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑛vol
3		nfiu1 4977	. . . . . . 7 ⊢ Ⅎ𝑛∪ 𝑛 ∈ ℕ (𝐸‘𝑛)
4	2, 3	nffv 6832	. . . . . 6 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
5	4	nfeq1 2907	. . . . 5 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞
6		iccssxr 13333	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ^*
7		volf 25428	. . . . . . . . . . . 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 7018	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom vol)
11	10	ralrimiva 3121	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
12		iunmbl 25452	. . . . . . . . . . . 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 7019	. . . . . . . . . 10 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ (0[,]+∞))
15	6, 14	sselid 3933	. . . . . . . . 9 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ^*)
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ^*)
17	16	3adant3 1132	. . . . . . 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 2735	. . . . . . . . 9 ⊢ ((vol‘(𝐸‘𝑛)) = +∞ → +∞ = (vol‘(𝐸‘𝑛)))
20	19	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → +∞ = (vol‘(𝐸‘𝑛)))
21	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
22		ssiun2 4996	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
23	22	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
24		volss 25432	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
26	25	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
27	20, 26	eqbrtrd 5114	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
28	17, 27	xrgepnfd 45321	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞)
29	28	3exp 1119	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ → ((vol‘(𝐸‘𝑛)) = +∞ → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞)))
30	1, 5, 29	rexlimd 3236	. . . 4 ⊢ (𝜑 → (∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞))
31	30	imp 406	. . 3 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞)
32		nfre1 3254	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞
33	1, 32	nfan 1899	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
34		nnex 12134	. . . . 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 7019	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,]+∞))
38	37	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,]+∞))
39		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
40	33, 35, 38, 39	sge0pnfmpt 46436	. . 3 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = +∞)
41	31, 40	eqtr4d 2767	. 2 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
42		ralnex 3055	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
43	42	biimpri 228	. . . . 5 ⊢ (¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞ → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞)
44	43	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞)
45	37	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ∈ (0[,]+∞))
46	18	necon3bi 2951	. . . . . . . . . 10 ⊢ (¬ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) ≠ +∞)
47	46	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ≠ +∞)
48		ge0xrre 45522	. . . . . . . . 9 ⊢ (((vol‘(𝐸‘𝑛)) ∈ (0[,]+∞) ∧ (vol‘(𝐸‘𝑛)) ≠ +∞) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
49	45, 47, 48	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
50	49	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) ∈ ℝ))
51		renepnf 11163	. . . . . . . . 9 ⊢ ((vol‘(𝐸‘𝑛)) ∈ ℝ → (vol‘(𝐸‘𝑛)) ≠ +∞)
52	51	neneqd 2930	. . . . . . . 8 ⊢ ((vol‘(𝐸‘𝑛)) ∈ ℝ → ¬ (vol‘(𝐸‘𝑛)) = +∞)
53	52	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐸‘𝑛)) ∈ ℝ → ¬ (vol‘(𝐸‘𝑛)) = +∞))
54	50, 53	impbid 212	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ (vol‘(𝐸‘𝑛)) ∈ ℝ))
55	54	ralbidva 3150	. . . . 5 ⊢ (𝜑 → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ))
56	55	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ))
57	44, 56	mpbid 232	. . 3 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ)
58		nfra1 3253	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ
59	1, 58	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ)
60	10	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom vol)
61		rspa 3218	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
62	61	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
63	60, 62	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) ∈ dom vol ∧ (vol‘(𝐸‘𝑛)) ∈ ℝ))
64	63	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝐸‘𝑛) ∈ dom vol ∧ (vol‘(𝐸‘𝑛)) ∈ ℝ)))
65	59, 64	ralrimi 3227	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝐸‘𝑛) ∈ dom vol ∧ (vol‘(𝐸‘𝑛)) ∈ ℝ))
66		voliunsge0lem.d	. . . . . 6 ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
67	66	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
68		voliunsge0lem.s	. . . . . 6 ⊢ 𝑆 = seq1( + , 𝐺)
69		voliunsge0lem.g	. . . . . 6 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))
70	68, 69	voliun 25453	. . . . 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^*, < ))
71	65, 67, 70	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = sup(ran 𝑆, ℝ^*, < ))
72		1zzd 12506	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → 1 ∈ ℤ)
73		nnuz 12778	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
74		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
75	59, 74	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑛((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)
76		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑛(vol‘(𝐸‘𝑚)) ∈ (0[,)+∞)
77	75, 76	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑛(((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞))
78		eleq1w 2811	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ ℕ ↔ 𝑚 ∈ ℕ))
79	78	anbi2d 630	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) ↔ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)))
80		2fveq3 6827	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (vol‘(𝐸‘𝑛)) = (vol‘(𝐸‘𝑚)))
81	80	eleq1d 2813	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((vol‘(𝐸‘𝑛)) ∈ (0[,)+∞) ↔ (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞)))
82	79, 81	imbi12d 344	. . . . . . 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[,)+∞))))
83		0xr 11162	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
84	83	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
85		pnfxr 11169	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
86	85	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
87	62	rexrd 11165	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ^*)
88		volge0 45952	. . . . . . . . . 10 ⊢ ((𝐸‘𝑛) ∈ dom vol → 0 ≤ (vol‘(𝐸‘𝑛)))
89	10, 88	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸‘𝑛)))
90	89	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸‘𝑛)))
91	62	ltpnfd 13023	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) < +∞)
92	84, 86, 87, 90, 91	elicod 13298	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,)+∞))
93	77, 82, 92	chvarfv 2241	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞))
94	80	cbvmptv 5196	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘(𝐸‘𝑚)))
95	93, 94	fmptd 7048	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))):ℕ⟶(0[,)+∞))
96		seqeq3 13913	. . . . . . 7 ⊢ (𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))) → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
97	69, 96	ax-mp 5	. . . . . 6 ⊢ seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))))
98	68, 97	eqtri 2752	. . . . 5 ⊢ 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))))
99	72, 73, 95, 98	sge0seq 46437	. . . 4 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = sup(ran 𝑆, ℝ^*, < ))
100	71, 99	eqtr4d 2767	. . 3 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
101	57, 100	syldan 591	. 2 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
102	41, 101	pm2.61dan 812	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3436 ⊆ wss 3903 ∪ ciun 4941 Disj wdisj 5059 class class class wbr 5092 ↦ cmpt 5173 dom cdm 5619 ran crn 5620 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 supcsup 9330 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 +∞cpnf 11146 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 ℕcn 12128 [,)cico 13250 [,]cicc 13251 seqcseq 13908 volcvol 25362 Σ^{^}csumge0 46353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-z 12472 df-uz 12736 df-q 12850 df-rp 12894 df-xadd 13015 df-ioo 13252 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-xmet 21254 df-met 21255 df-ovol 25363 df-vol 25364 df-sumge0 46354

This theorem is referenced by: voliunsge0 46464

W3C validator