voliunsge0lem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > voliunsge0lem		Structured version Visualization version GIF version

Theorem voliunsge0lem 46580

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 1915	. . . . 5 ⊢ Ⅎ𝑛𝜑
2		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑛vol
3		nfiu1 4975	. . . . . . 7 ⊢ Ⅎ𝑛∪ 𝑛 ∈ ℕ (𝐸‘𝑛)
4	2, 3	nffv 6832	. . . . . 6 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
5	4	nfeq1 2910	. . . . 5 ⊢ Ⅎ𝑛(vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = +∞
6		iccssxr 13330	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ^*
7		volf 25457	. . . . . . . . . . . 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 7017	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom vol)
11	10	ralrimiva 3124	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐸‘𝑛) ∈ dom vol)
12		iunmbl 25481	. . . . . . . . . . . 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 7018	. . . . . . . . . 10 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ (0[,]+∞))
15	6, 14	sselid 3927	. . . . . . . . 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 2737	. . . . . . . . 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 4994	. . . . . . . . . . 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 25461	. . . . . . . . . 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 5111	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ (vol‘(𝐸‘𝑛)) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
28	17, 27	xrgepnfd 45440	. . . . . 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 3239	. . . 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 3257	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞
33	1, 32	nfan 1900	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞)
34		nnex 12131	. . . . 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 7018	. . . . 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 46553	. . 3 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = +∞)
41	31, 40	eqtr4d 2769	. 2 ⊢ ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))))
42		ralnex 3058	. . . . . 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 2954	. . . . . . . . . 10 ⊢ (¬ (vol‘(𝐸‘𝑛)) = +∞ → (vol‘(𝐸‘𝑛)) ≠ +∞)
47	46	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸‘𝑛)) = +∞) → (vol‘(𝐸‘𝑛)) ≠ +∞)
48		ge0xrre 45641	. . . . . . . . 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 11160	. . . . . . . . 9 ⊢ ((vol‘(𝐸‘𝑛)) ∈ ℝ → (vol‘(𝐸‘𝑛)) ≠ +∞)
52	51	neneqd 2933	. . . . . . . 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 3153	. . . . 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 3256	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ
59	1, 58	nfan 1900	. . . . . 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 3221	. . . . . . . . 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 3230	. . . . 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 25482	. . . . 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 12503	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → 1 ∈ ℤ)
73		nnuz 12775	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
74		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
75	59, 74	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑛((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)
76		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑛(vol‘(𝐸‘𝑚)) ∈ (0[,)+∞)
77	75, 76	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑛(((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞))
78		eleq1w 2814	. . . . . . . . 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 2816	. . . . . . . 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 11159	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
84	83	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
85		pnfxr 11166	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
86	85	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
87	62	rexrd 11162	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ ℝ^*)
88		volge0 46069	. . . . . . . . . 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 13020	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) < +∞)
92	84, 86, 87, 90, 91	elicod 13295	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸‘𝑛)) ∈ (0[,)+∞))
93	77, 82, 92	chvarfv 2243	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸‘𝑚)) ∈ (0[,)+∞))
94	80	cbvmptv 5193	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘(𝐸‘𝑚)))
95	93, 94	fmptd 7047	. . . . 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 2754	. . . . 5 ⊢ 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛))))
99	72, 73, 95, 98	sge0seq 46554	. . . 4 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸‘𝑛)) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸‘𝑛)))) = sup(ran 𝑆, ℝ^*, < ))
100	71, 99	eqtr4d 2769	. . 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 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 Vcvv 3436 ⊆ wss 3897 ∪ ciun 4939 Disj wdisj 5056 class class class wbr 5089 ↦ cmpt 5170 dom cdm 5614 ran crn 5615 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 supcsup 9324 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 +∞cpnf 11143 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 ℕcn 12125 [,)cico 13247 [,]cicc 13248 seqcseq 13908 volcvol 25391 Σ^{^}csumge0 46470

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cc 10326 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-xadd 13012 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 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 21284 df-met 21285 df-ovol 25392 df-vol 25393 df-sumge0 46471

This theorem is referenced by: voliunsge0 46581

W3C validator