voliunsge0lem - Mathbox for Glauco Siliprandi

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Theorem voliunsge0lem 47481

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 1947	. . . . 5 ⊢ Ⅎ𝑛𝜑
2		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑛vol
3		nfiu1 4986	. . . . . . 7 ⊢ Ⅎ𝑛∪ 𝑛 ∈ ℕ (𝐸‘𝑛)
4	2, 3	nffv 6895	. . . . . 6 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
5	4	nfeq1 2938	. . . . 5 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞
6		iccssxr 13561	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ^*
7		volf 25850	. . . . . . . . . . . 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 7084	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom vol)
11	10	ralrimiva 3155	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
12		iunmbl 25874	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
13	11, 12	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
14	8, 13	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ (0[,]+∞))
15	6, 14	sselid 3929	. . . . . . . . 9 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ^*)
16	15	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ^*)
17	16	3adant3 1150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ^*)
18		id 23	. . . . . . . . . 10 ⊢ ((vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) = +∞)
19	18	eqcomd 2767	. . . . . . . . 9 ⊢ ((vol‘(𝐸‘𝑛)) = +∞ → +∞ = (vol‘(𝐸‘𝑛)))
20	19	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → +∞ = (vol‘(𝐸‘𝑛)))
21	13	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
22		ssiun2 5006	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
23	22	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
24		volss 25854	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
26	25	3adant3 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
27	20, 26	eqbrtrd 5127	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
28	17, 27	xrgepnfd 46342	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞)
29	28	3exp 1137	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ → ((vol‘(𝐸‘𝑛)) = +∞ → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞)))
30	1, 5, 29	rexlimd 3270	. . . 4 ⊢ (𝜑 → (∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞))
31	30	imp 412	. . 3 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞)
32		nfre1 3288	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞
33	1, 32	nfan 1932	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
34		nnex 12341	. . . . 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 7085	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,]+∞))
38	37	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,]+∞))
39		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
40	33, 35, 38, 39	sge0pnfmpt 47454	. . 3 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = +∞)
41	31, 40	eqtr4d 2799	. 2 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
42		ralnex 3089	. . . . 5 ⊢ (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
43	42	bilanri 512	. . . 4 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞)
44	37	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ∈ (0[,]+∞))
45	18	necon3bi 2982	. . . . . . . . . 10 ⊢ (¬ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) ≠ +∞)
46	45	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ≠ +∞)
47		ge0xrre 46542	. . . . . . . . 9 ⊢ (((vol‘(𝐸‘𝑛)) ∈ (0[,]+∞) ∧ (vol‘(𝐸‘𝑛)) ≠ +∞) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
48	44, 46, 47	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
49	48	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) ∈ ℝ))
50		renepnf 11357	. . . . . . . . 9 ⊢ ((vol‘(𝐸‘𝑛)) ∈ ℝ → (vol‘(𝐸‘𝑛)) ≠ +∞)
51	50	neneqd 2961	. . . . . . . 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 215	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ (vol‘(𝐸‘𝑛)) ∈ ℝ))
54	53	ralbidva 3184	. . . . 5 ⊢ (𝜑 → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ))
55	54	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸‘𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ))
56	43, 55	mpbid 235	. . 3 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ)
57		nfra1 3287	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ
58	1, 57	nfan 1932	. . . . . 6 ⊢ Ⅎ𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ)
59	10	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom vol)
60		rspa 3252	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
61	60	adantll 727	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ)
62	59, 61	jca 521	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) ∈ dom vol ∧ (vol‘(𝐸‘𝑛)) ∈ ℝ))
63	62	ex 418	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝐸‘𝑛) ∈ dom vol ∧ (vol‘(𝐸‘𝑛)) ∈ ℝ)))
64	58, 63	ralrimi 3261	. . . . 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 486	. . . . 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 25875	. . . . 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 596	. . . 4 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = sup(ran 𝑆, ℝ^*, < ))
71		1zzd 12727	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → 1 ∈ ℤ)
72		nnuz 13004	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
73		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
74	58, 73	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑛((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)
75		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑛(vol‘(𝐸‘𝑚)) ∈ (0[,)+∞)
76	74, 75	nfim 1929	. . . . . . 7 ⊢ Ⅎ𝑛(((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞))
77		eleq1w 2844	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ ℕ ↔ 𝑚 ∈ ℕ))
78	77	anbi2d 642	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) ↔ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)))
79		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (vol‘(𝐸‘𝑛)) = (vol‘(𝐸‘𝑚)))
80	79	eleq1d 2846	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ((vol‘(𝐸‘𝑛)) ∈ (0[,)+∞) ↔ (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞)))
81	78, 80	imbi12d 347	. . . . . . 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 11356	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
83	82	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
84		pnfxr 11363	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
85	84	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
86	61	rexrd 11359	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ^*)
87		volge0 46970	. . . . . . . . . 10 ⊢ ((𝐸‘𝑛) ∈ dom vol → 0 ≤ (vol‘(𝐸‘𝑛)))
88	10, 87	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸‘𝑛)))
89	88	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸‘𝑛)))
90	61	ltpnfd 13250	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) < +∞)
91	83, 85, 86, 89, 90	elicod 13526	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,)+∞))
92	76, 81, 91	chvarfv 2277	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞))
93	79	cbvmptv 5209	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘(𝐸‘𝑚)))
94	92, 93	fmptd 7114	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))):ℕ⟶(0[,)+∞))
95		seqeq3 14149	. . . . . . 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 2784	. . . . 5 ⊢ 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))))
98	71, 72, 94, 97	sge0seq 47455	. . . 4 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = sup(ran 𝑆, ℝ^*, < ))
99	70, 98	eqtr4d 2799	. . 3 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
100	56, 99	syldan 603	. 2 ⊢ ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
101	41, 100	pm2.61dan 825	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 Vcvv 3451 ⊆ wss 3899 ∪ ciun 4951 Disj wdisj 5070 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 ran crn 5652 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 supcsup 9432 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 +∞cpnf 11340 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 ℕcn 12335 [,)cico 13478 [,]cicc 13479 seqcseq 14144 volcvol 25784 Σ^{^}csumge0 47371

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cc 10513 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-z 12694 df-uz 12966 df-q 13076 df-rp 13121 df-xadd 13242 df-ioo 13480 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-xmet 21671 df-met 21672 df-ovol 25785 df-vol 25786 df-sumge0 47372

This theorem is used by: voliunsge0 47482

W3C validator