volfiniune - Mathbox for Thierry Arnoux

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Theorem volfiniune 34620

Description: The Lebesgue measure function is countably additive. This theorem is to volfiniun 25706 what voliune 34619 is to voliun 25713. (Contributed by Thierry Arnoux, 16-Oct-2017.)

**Assertion**
Ref	Expression
volfiniune	⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = Σ^*𝑛 ∈ 𝐴(vol‘𝐵))

Distinct variable group: 𝐴,𝑛 Allowed substitution hint: 𝐵(𝑛)

Proof of Theorem volfiniune Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1210	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → 𝐴 ∈ Fin)
2		simpl2 1211	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol)
3		simpr 489	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
4		r19.26 3125	. . . . 5 ⊢ (∀𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ))
5	2, 3, 4	sylanbrc 594	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → ∀𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ))
6		simpl3 1212	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → Disj 𝑛 ∈ 𝐴 𝐵)
7		volfiniun 25706	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑛 ∈ 𝐴 𝐵) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = Σ𝑛 ∈ 𝐴 (vol‘𝐵))
8	1, 5, 6, 7	syl3anc 1398	. . 3 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = Σ𝑛 ∈ 𝐴 (vol‘𝐵))
9		nfcv 2925	. . . 4 ⊢ Ⅎ𝑛𝐴
10	9	nfel1 2941	. . . . . 6 ⊢ Ⅎ𝑛 𝐴 ∈ Fin
11		nfra1 3289	. . . . . 6 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol
12		nfdisj1 5090	. . . . . 6 ⊢ Ⅎ𝑛Disj 𝑛 ∈ 𝐴 𝐵
13	10, 11, 12	nf3an 1931	. . . . 5 ⊢ Ⅎ𝑛(𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵)
14		nfra1 3289	. . . . 5 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ
15	13, 14	nfan 1929	. . . 4 ⊢ Ⅎ𝑛((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
16	3	r19.21bi 3257	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ ℝ)
17		rspa 3254	. . . . . . . 8 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ 𝑛 ∈ 𝐴) → 𝐵 ∈ dom vol)
18		volf 25688	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
19	18	ffvelcdmi 7078	. . . . . . . 8 ⊢ (𝐵 ∈ dom vol → (vol‘𝐵) ∈ (0[,]+∞))
20	17, 19	syl 18	. . . . . . 7 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,]+∞))
21	2, 20	sylan 591	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,]+∞))
22		0xr 11251	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
23		pnfxr 11258	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
24		elicc1 13411	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((vol‘𝐵) ∈ (0[,]+∞) ↔ ((vol‘𝐵) ∈ ℝ^ ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) ≤ +∞)))
25	22, 23, 24	mp2an 704	. . . . . . 7 ⊢ ((vol‘𝐵) ∈ (0[,]+∞) ↔ ((vol‘𝐵) ∈ ℝ^* ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) ≤ +∞))
26	25	simp2bi 1164	. . . . . 6 ⊢ ((vol‘𝐵) ∈ (0[,]+∞) → 0 ≤ (vol‘𝐵))
27	21, 26	syl 18	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → 0 ≤ (vol‘𝐵))
28		ltpnf 13140	. . . . . 6 ⊢ ((vol‘𝐵) ∈ ℝ → (vol‘𝐵) < +∞)
29	16, 28	syl 18	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) < +∞)
30		0re 11205	. . . . . 6 ⊢ 0 ∈ ℝ
31		elico2 13432	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((vol‘𝐵) ∈ (0[,)+∞) ↔ ((vol‘𝐵) ∈ ℝ ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) < +∞)))
32	30, 23, 31	mp2an 704	. . . . 5 ⊢ ((vol‘𝐵) ∈ (0[,)+∞) ↔ ((vol‘𝐵) ∈ ℝ ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) < +∞))
33	16, 27, 29, 32	syl3anbrc 1362	. . . 4 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,)+∞))
34	9, 15, 1, 33	esumpfinvalf 34466	. . 3 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → Σ^*𝑛 ∈ 𝐴(vol‘𝐵) = Σ𝑛 ∈ 𝐴 (vol‘𝐵))
35	8, 34	eqtr4d 2801	. 2 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = Σ^*𝑛 ∈ 𝐴(vol‘𝐵))
36		simpr 489	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
37		nfv 1944	. . . . . . . . 9 ⊢ Ⅎ𝑘(vol‘𝐵) = +∞
38		nfcv 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol
39		nfcsb1v 3877	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵
40	38, 39	nffv 6891	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐵)
41	40	nfeq1 2940	. . . . . . . . 9 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞
42		csbeq1a 3867	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
43	42	fveqeq2d 6889	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((vol‘𝐵) = +∞ ↔ (vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞))
44	37, 41, 43	cbvrexw 3308	. . . . . . . 8 ⊢ (∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞ ↔ ∃𝑘 ∈ 𝐴 (vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞)
45	36, 44	sylib 221	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → ∃𝑘 ∈ 𝐴 (vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞)
46	39	nfel1 2941	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol
47	42	eleq1d 2848	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐵 ∈ dom vol ↔ ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol))
48	46, 47	rspc 3569	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐴 → (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol → ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol))
49	48	impcom 412	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol)
50	49	adantll 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol)
51		finiunmbl 25703	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) → ∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol)
52	51	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → ∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol)
53		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
54	9, 53, 39, 42	ssiun2sf 32904	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐴 → ⦋𝑘 / 𝑛⦌𝐵 ⊆ ∪ 𝑛 ∈ 𝐴 𝐵)
55	54	adantl 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑛⦌𝐵 ⊆ ∪ 𝑛 ∈ 𝐴 𝐵)
56		volss 25692	. . . . . . . . . . 11 ⊢ ((⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol ∧ ∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ⦋𝑘 / 𝑛⦌𝐵 ⊆ ∪ 𝑛 ∈ 𝐴 𝐵) → (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
57	50, 52, 55, 56	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
58	57	3adantl3 1187	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ 𝑘 ∈ 𝐴) → (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
59	58	adantlr 727	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) ∧ 𝑘 ∈ 𝐴) → (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
60	59	ralrimiva 3157	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → ∀𝑘 ∈ 𝐴 (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
61		r19.29r 3129	. . . . . . 7 ⊢ ((∃𝑘 ∈ 𝐴 (vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞ ∧ ∀𝑘 ∈ 𝐴 (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵)) → ∃𝑘 ∈ 𝐴 ((vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵)))
62	45, 60, 61	syl2anc 595	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → ∃𝑘 ∈ 𝐴 ((vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵)))
63		breq1 5112	. . . . . . . 8 ⊢ ((vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞ → ((vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ↔ +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵)))
64	63	biimpa 481	. . . . . . 7 ⊢ (((vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵)) → +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
65	64	reximi 3103	. . . . . 6 ⊢ (∃𝑘 ∈ 𝐴 ((vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵)) → ∃𝑘 ∈ 𝐴 +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
66	62, 65	syl 18	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → ∃𝑘 ∈ 𝐴 +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
67		rexex 3095	. . . . . 6 ⊢ (∃𝑘 ∈ 𝐴 +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) → ∃𝑘+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
68		19.9v 2014	. . . . . 6 ⊢ (∃𝑘+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ↔ +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
69	67, 68	sylib 221	. . . . 5 ⊢ (∃𝑘 ∈ 𝐴 +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) → +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
70	66, 69	syl 18	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
71		iccssxr 13452	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
72	18	ffvelcdmi 7078	. . . . . . . . 9 ⊢ (∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ (0[,]+∞))
73	71, 72	sselid 3935	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^*)
74	51, 73	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^*)
75	74	3adant3 1150	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^*)
76	75	adantr 485	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^*)
77		xgepnf 13186	. . . . 5 ⊢ ((vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^* → (+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ↔ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = +∞))
78	76, 77	syl 18	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → (+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ↔ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = +∞))
79	70, 78	mpbid 235	. . 3 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = +∞)
80		nfre1 3290	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞
81	13, 80	nfan 1929	. . . 4 ⊢ Ⅎ𝑛((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
82		simpl1 1210	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → 𝐴 ∈ Fin)
83	20	3ad2antl2 1205	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,]+∞))
84	83	adantlr 727	. . . 4 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,]+∞))
85	81, 82, 84, 36	esumpinfval 34463	. . 3 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → Σ^*𝑛 ∈ 𝐴(vol‘𝐵) = +∞)
86	79, 85	eqtr4d 2801	. 2 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = Σ^*𝑛 ∈ 𝐴(vol‘𝐵))
87		exmid 907	. . . . 5 ⊢ (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ¬ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
88		rexnal 3117	. . . . . 6 ⊢ (∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ ↔ ¬ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
89	88	orbi2i 925	. . . . 5 ⊢ ((∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ) ↔ (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ¬ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ))
90	87, 89	mpbir 234	. . . 4 ⊢ (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ)
91		r19.29 3128	. . . . . . 7 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ) → ∃𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ ¬ (vol‘𝐵) ∈ ℝ))
92		xrge0nre 13475	. . . . . . . . 9 ⊢ (((vol‘𝐵) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐵) ∈ ℝ) → (vol‘𝐵) = +∞)
93	19, 92	sylan 591	. . . . . . . 8 ⊢ ((𝐵 ∈ dom vol ∧ ¬ (vol‘𝐵) ∈ ℝ) → (vol‘𝐵) = +∞)
94	93	reximi 3103	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ ¬ (vol‘𝐵) ∈ ℝ) → ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
95	91, 94	syl 18	. . . . . 6 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ) → ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
96	95	ex 417	. . . . 5 ⊢ (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ → ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞))
97	96	orim2d 982	. . . 4 ⊢ (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol → ((∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ) → (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)))
98	90, 97	mpi 21	. . 3 ⊢ (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞))
99	98	3ad2ant2 1152	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) → (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞))
100	35, 86, 99	mpjaodan 973	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = Σ^*𝑛 ∈ 𝐴(vol‘𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ⦋csb 3853 ⊆ wss 3905 ∪ ciun 4956 Disj wdisj 5076 class class class wbr 5109 dom cdm 5661 ‘cfv 6536 (class class class)co 7410 Fincfn 8939 ℝcr 11094 0cc0 11095 +∞cpnf 11235 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 [,)cico 13369 [,]cicc 13370 Σcsu 15733 volcvol 25622 Σ^*cesum 34417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5077 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-ef 16116 df-sin 16118 df-cos 16119 df-pi 16121 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-ordt 17550 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-ps 18617 df-tsr 18618 df-plusf 18692 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-mhm 18836 df-submnd 18837 df-grp 18998 df-minusg 18999 df-sbg 19000 df-mulg 19129 df-subg 19184 df-cntz 19382 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-cring 20313 df-subrng 20645 df-subrg 20669 df-abv 20912 df-lmod 20983 df-scaf 20984 df-sra 21294 df-rgmod 21295 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-haus 23472 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-tmd 24229 df-tgp 24230 df-tsms 24284 df-trg 24317 df-xms 24477 df-ms 24478 df-tms 24479 df-nm 24739 df-ngp 24740 df-nrg 24742 df-nlm 24743 df-ii 25036 df-cncf 25037 df-ovol 25623 df-vol 25624 df-limc 26025 df-dv 26026 df-log 26721 df-esum 34418

This theorem is referenced by: volmeas 34621

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