volfiniune - Mathbox for Thierry Arnoux

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Theorem volfiniune 34387

Description: The Lebesgue measure function is countably additive. This theorem is to volfiniun 25504 what voliune 34386 is to voliun 25511. (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 1192	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → 𝐴 ∈ Fin)
2		simpl2 1193	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol)
3		simpr 484	. . . . 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 3096	. . . . 5 ⊢ (∀𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ))
5	2, 3, 4	sylanbrc 583	. . . 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 1194	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → Disj 𝑛 ∈ 𝐴 𝐵)
7		volfiniun 25504	. . . 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 1373	. . 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 2898	. . . 4 ⊢ Ⅎ𝑛𝐴
10	9	nfel1 2915	. . . . . 6 ⊢ Ⅎ𝑛 𝐴 ∈ Fin
11		nfra1 3260	. . . . . 6 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol
12		nfdisj1 5079	. . . . . 6 ⊢ Ⅎ𝑛Disj 𝑛 ∈ 𝐴 𝐵
13	10, 11, 12	nf3an 1902	. . . . 5 ⊢ Ⅎ𝑛(𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵)
14		nfra1 3260	. . . . 5 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ
15	13, 14	nfan 1900	. . . 4 ⊢ Ⅎ𝑛((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
16	3	r19.21bi 3228	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ ℝ)
17		rspa 3225	. . . . . . . 8 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ 𝑛 ∈ 𝐴) → 𝐵 ∈ dom vol)
18		volf 25486	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
19	18	ffvelcdmi 7028	. . . . . . . 8 ⊢ (𝐵 ∈ dom vol → (vol‘𝐵) ∈ (0[,]+∞))
20	17, 19	syl 17	. . . . . . 7 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,]+∞))
21	2, 20	sylan 580	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,]+∞))
22		0xr 11179	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
23		pnfxr 11186	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
24		elicc1 13305	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((vol‘𝐵) ∈ (0[,]+∞) ↔ ((vol‘𝐵) ∈ ℝ^ ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) ≤ +∞)))
25	22, 23, 24	mp2an 692	. . . . . . 7 ⊢ ((vol‘𝐵) ∈ (0[,]+∞) ↔ ((vol‘𝐵) ∈ ℝ^* ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) ≤ +∞))
26	25	simp2bi 1146	. . . . . 6 ⊢ ((vol‘𝐵) ∈ (0[,]+∞) → 0 ≤ (vol‘𝐵))
27	21, 26	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → 0 ≤ (vol‘𝐵))
28		ltpnf 13034	. . . . . 6 ⊢ ((vol‘𝐵) ∈ ℝ → (vol‘𝐵) < +∞)
29	16, 28	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) < +∞)
30		0re 11134	. . . . . 6 ⊢ 0 ∈ ℝ
31		elico2 13326	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((vol‘𝐵) ∈ (0[,)+∞) ↔ ((vol‘𝐵) ∈ ℝ ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) < +∞)))
32	30, 23, 31	mp2an 692	. . . . 5 ⊢ ((vol‘𝐵) ∈ (0[,)+∞) ↔ ((vol‘𝐵) ∈ ℝ ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) < +∞))
33	16, 27, 29, 32	syl3anbrc 1344	. . . 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 34233	. . 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 2774	. 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 484	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
37		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘(vol‘𝐵) = +∞
38		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol
39		nfcsb1v 3873	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵
40	38, 39	nffv 6844	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐵)
41	40	nfeq1 2914	. . . . . . . . 9 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞
42		csbeq1a 3863	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
43	42	fveqeq2d 6842	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((vol‘𝐵) = +∞ ↔ (vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞))
44	37, 41, 43	cbvrexw 3279	. . . . . . . 8 ⊢ (∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞ ↔ ∃𝑘 ∈ 𝐴 (vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞)
45	36, 44	sylib 218	. . . . . . 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 2915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol
47	42	eleq1d 2821	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐵 ∈ dom vol ↔ ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol))
48	46, 47	rspc 3564	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐴 → (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol → ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol))
49	48	impcom 407	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol)
50	49	adantll 714	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol)
51		finiunmbl 25501	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) → ∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol)
52	51	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → ∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol)
53		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
54	9, 53, 39, 42	ssiun2sf 32634	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐴 → ⦋𝑘 / 𝑛⦌𝐵 ⊆ ∪ 𝑛 ∈ 𝐴 𝐵)
55	54	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑛⦌𝐵 ⊆ ∪ 𝑛 ∈ 𝐴 𝐵)
56		volss 25490	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
58	57	3adantl3 1169	. . . . . . . . 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 715	. . . . . . . 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 3128	. . . . . . 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 3100	. . . . . . 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 584	. . . . . 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 5101	. . . . . . . 8 ⊢ ((vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞ → ((vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ↔ +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵)))
64	63	biimpa 476	. . . . . . 7 ⊢ (((vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵)) → +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
65	64	reximi 3074	. . . . . 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 17	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → ∃𝑘 ∈ 𝐴 +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
67		rexex 3066	. . . . . 6 ⊢ (∃𝑘 ∈ 𝐴 +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) → ∃𝑘+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
68		19.9v 1985	. . . . . 6 ⊢ (∃𝑘+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ↔ +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
69	67, 68	sylib 218	. . . . 5 ⊢ (∃𝑘 ∈ 𝐴 +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) → +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
70	66, 69	syl 17	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
71		iccssxr 13346	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
72	18	ffvelcdmi 7028	. . . . . . . . 9 ⊢ (∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ (0[,]+∞))
73	71, 72	sselid 3931	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^*)
74	51, 73	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^*)
75	74	3adant3 1132	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^*)
76	75	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^*)
77		xgepnf 13080	. . . . 5 ⊢ ((vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ ℝ^* → (+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ↔ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = +∞))
78	76, 77	syl 17	. . . 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 232	. . 3 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = +∞)
80		nfre1 3261	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞
81	13, 80	nfan 1900	. . . 4 ⊢ Ⅎ𝑛((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
82		simpl1 1192	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → 𝐴 ∈ Fin)
83	20	3ad2antl2 1187	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,]+∞))
84	83	adantlr 715	. . . 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 34230	. . 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 2774	. 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 894	. . . . 5 ⊢ (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ¬ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
88		rexnal 3088	. . . . . 6 ⊢ (∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ ↔ ¬ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
89	88	orbi2i 912	. . . . 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 231	. . . 4 ⊢ (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ)
91		r19.29 3099	. . . . . . 7 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ) → ∃𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ ¬ (vol‘𝐵) ∈ ℝ))
92		xrge0nre 13369	. . . . . . . . 9 ⊢ (((vol‘𝐵) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐵) ∈ ℝ) → (vol‘𝐵) = +∞)
93	19, 92	sylan 580	. . . . . . . 8 ⊢ ((𝐵 ∈ dom vol ∧ ¬ (vol‘𝐵) ∈ ℝ) → (vol‘𝐵) = +∞)
94	93	reximi 3074	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ ¬ (vol‘𝐵) ∈ ℝ) → ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
95	91, 94	syl 17	. . . . . 6 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ) → ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
96	95	ex 412	. . . . 5 ⊢ (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ → ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞))
97	96	orim2d 968	. . . 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 20	. . 3 ⊢ (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞))
99	98	3ad2ant2 1134	. 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 960	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 ⦋csb 3849 ⊆ wss 3901 ∪ ciun 4946 Disj wdisj 5065 class class class wbr 5098 dom cdm 5624 ‘cfv 6492 (class class class)co 7358 Fincfn 8883 ℝcr 11025 0cc0 11026 +∞cpnf 11163 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 [,)cico 13263 [,]cicc 13264 Σcsu 15609 volcvol 25420 Σ^*cesum 34184

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-ordt 17422 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-ps 18489 df-tsr 18490 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-subrng 20479 df-subrg 20503 df-abv 20742 df-lmod 20813 df-scaf 20814 df-sra 21125 df-rgmod 21126 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-tmd 24016 df-tgp 24017 df-tsms 24071 df-trg 24104 df-xms 24264 df-ms 24265 df-tms 24266 df-nm 24526 df-ngp 24527 df-nrg 24529 df-nlm 24530 df-ii 24826 df-cncf 24827 df-ovol 25421 df-vol 25422 df-limc 25823 df-dv 25824 df-log 26521 df-esum 34185

This theorem is referenced by: volmeas 34388

W3C validator