voliunsge0lem - Mathbox for Glauco Siliprandi

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Theorem voliunsge0lem 47303

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 2922	. . . . . . 7 ⊢ Ⅎ𝑛vol
3		nfiu1 4986	. . . . . . 7 ⊢ Ⅎ𝑛∪ 𝑛 ∈ ℕ (𝐸‘𝑛)
4	2, 3	nffv 6889	. . . . . 6 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
5	4	nfeq1 2937	. . . . 5 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞
6		iccssxr 13486	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ^*
7		volf 25760	. . . . . . . . . . . 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 7078	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom vol)
11	10	ralrimiva 3154	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
12		iunmbl 25784	. . . . . . . . . . . 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 7079	. . . . . . . . . 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 2766	. . . . . . . . 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 25764	. . . . . . . . . 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 46164	. . . . . 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 3269	. . . 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 3287	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞
33	1, 32	nfan 1932	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
34		nnex 12266	. . . . 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 7079	. . . . 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 47276	. . 3 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = +∞)
41	31, 40	eqtr4d 2798	. 2 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
42		ralnex 3088	. . . . 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 2981	. . . . . . . . . 10 ⊢ (¬ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) ≠ +∞)
46	45	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ≠ +∞)
47		ge0xrre 46364	. . . . . . . . 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 11284	. . . . . . . . 9 ⊢ ((vol‘(𝐸‘𝑛)) ∈ ℝ → (vol‘(𝐸‘𝑛)) ≠ +∞)
51	50	neneqd 2960	. . . . . . . 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 3183	. . . . 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 3286	. . . . . . 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 3251	. . . . . . . . 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 3260	. . . . 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 25785	. . . . 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 12652	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → 1 ∈ ℤ)
72		nnuz 12929	. . . . 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 2843	. . . . . . . . 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 6884	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (vol‘(𝐸‘𝑛)) = (vol‘(𝐸‘𝑚)))
80	79	eleq1d 2845	. . . . . . . 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 11283	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
83	82	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
84		pnfxr 11290	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
85	84	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
86	61	rexrd 11286	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ^*)
87		volge0 46792	. . . . . . . . . 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 13175	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) < +∞)
91	83, 85, 86, 89, 90	elicod 13451	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,)+∞))
92	76, 81, 91	chvarfv 2276	. . . . . 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 7108	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))):ℕ⟶(0[,)+∞))
95		seqeq3 14073	. . . . . . 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 2783	. . . . 5 ⊢ 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))))
98	71, 72, 94, 97	sge0seq 47277	. . . 4 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = sup(ran 𝑆, ℝ^*, < ))
99	70, 98	eqtr4d 2798	. . 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 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3899 ∪ ciun 4951 Disj wdisj 5070 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ran crn 5656 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 supcsup 9413 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 +∞cpnf 11267 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 ℕcn 12260 [,)cico 13403 [,]cicc 13404 seqcseq 14068 volcvol 25694 Σ^{^}csumge0 47193

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cc 10440 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-oi 9485 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-z 12619 df-uz 12891 df-q 13001 df-rp 13046 df-xadd 13167 df-ioo 13405 df-ico 13407 df-icc 13408 df-fz 13565 df-fzo 13713 df-fl 13856 df-seq 14069 df-exp 14129 df-hash 14398 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-clim 15578 df-rlim 15579 df-sum 15777 df-xmet 21581 df-met 21582 df-ovol 25695 df-vol 25696 df-sumge0 47194

This theorem is used by: voliunsge0 47304

W3C validator