volfiniune - Mathbox for Thierry Arnoux

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Theorem volfiniune 32098

Description: The Lebesgue measure function is countably additive. This theorem is to volfiniun 24616 what voliune 32097 is to voliun 24623. (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 1189	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → 𝐴 ∈ Fin)
2		simpl2 1190	. . . . 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 3094	. . . . 5 ⊢ (∀𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ))
5	2, 3, 4	sylanbrc 582	. . . 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 1191	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → Disj 𝑛 ∈ 𝐴 𝐵)
7		volfiniun 24616	. . . 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 1369	. . 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 2906	. . . 4 ⊢ Ⅎ𝑛𝐴
10	9	nfel1 2922	. . . . . 6 ⊢ Ⅎ𝑛 𝐴 ∈ Fin
11		nfra1 3142	. . . . . 6 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol
12		nfdisj1 5049	. . . . . 6 ⊢ Ⅎ𝑛Disj 𝑛 ∈ 𝐴 𝐵
13	10, 11, 12	nf3an 1905	. . . . 5 ⊢ Ⅎ𝑛(𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵)
14		nfra1 3142	. . . . 5 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ
15	13, 14	nfan 1903	. . . 4 ⊢ Ⅎ𝑛((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
16	3	r19.21bi 3132	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ ℝ)
17		rspa 3130	. . . . . . . 8 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ 𝑛 ∈ 𝐴) → 𝐵 ∈ dom vol)
18		volf 24598	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
19	18	ffvelrni 6942	. . . . . . . 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 579	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,]+∞))
22		0xr 10953	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
23		pnfxr 10960	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
24		elicc1 13052	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((vol‘𝐵) ∈ (0[,]+∞) ↔ ((vol‘𝐵) ∈ ℝ^ ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) ≤ +∞)))
25	22, 23, 24	mp2an 688	. . . . . . 7 ⊢ ((vol‘𝐵) ∈ (0[,]+∞) ↔ ((vol‘𝐵) ∈ ℝ^* ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) ≤ +∞))
26	25	simp2bi 1144	. . . . . 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 12785	. . . . . 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 10908	. . . . . 6 ⊢ 0 ∈ ℝ
31		elico2 13072	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((vol‘𝐵) ∈ (0[,)+∞) ↔ ((vol‘𝐵) ∈ ℝ ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) < +∞)))
32	30, 23, 31	mp2an 688	. . . . 5 ⊢ ((vol‘𝐵) ∈ (0[,)+∞) ↔ ((vol‘𝐵) ∈ ℝ ∧ 0 ≤ (vol‘𝐵) ∧ (vol‘𝐵) < +∞))
33	16, 27, 29, 32	syl3anbrc 1341	. . . 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 31944	. . 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 2781	. 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 1918	. . . . . . . . 9 ⊢ Ⅎ𝑘(vol‘𝐵) = +∞
38		nfcv 2906	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol
39		nfcsb1v 3853	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵
40	38, 39	nffv 6766	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐵)
41	40	nfeq1 2921	. . . . . . . . 9 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞
42		csbeq1a 3842	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
43	42	fveqeq2d 6764	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((vol‘𝐵) = +∞ ↔ (vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞))
44	37, 41, 43	cbvrexw 3364	. . . . . . . 8 ⊢ (∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞ ↔ ∃𝑘 ∈ 𝐴 (vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞)
45	36, 44	sylib 217	. . . . . . 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 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol
47	42	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐵 ∈ dom vol ↔ ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol))
48	46, 47	rspc 3539	. . . . . . . . . . . . 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 710	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol)
51		finiunmbl 24613	. . . . . . . . . . . 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 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
54	9, 53, 39, 42	ssiun2sf 30800	. . . . . . . . . . . 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 24602	. . . . . . . . . . 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 1369	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
58	57	3adantl3 1166	. . . . . . . . 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 711	. . . . . . . 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 3107	. . . . . . 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 3184	. . . . . . 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 583	. . . . . 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 5073	. . . . . . . 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 3174	. . . . . 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 3167	. . . . . 6 ⊢ (∃𝑘 ∈ 𝐴 +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) → ∃𝑘+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
68		19.9v 1988	. . . . . 6 ⊢ (∃𝑘+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ↔ +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
69	67, 68	sylib 217	. . . . 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 13091	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
72	18	ffvelrni 6942	. . . . . . . . 9 ⊢ (∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ (0[,]+∞))
73	71, 72	sselid 3915	. . . . . . . 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 1130	. . . . . 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 12828	. . . . 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 231	. . 3 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) = +∞)
80		nfre1 3234	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞
81	13, 80	nfan 1903	. . . 4 ⊢ Ⅎ𝑛((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
82		simpl1 1189	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → 𝐴 ∈ Fin)
83	20	3ad2antl2 1184	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ (0[,]+∞))
84	83	adantlr 711	. . . 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 31941	. . 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 2781	. 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 891	. . . . 5 ⊢ (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ¬ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
88		rexnal 3165	. . . . . 6 ⊢ (∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ ↔ ¬ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
89	88	orbi2i 909	. . . . 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 230	. . . 4 ⊢ (∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ ∨ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ)
91		r19.29 3183	. . . . . . 7 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ) → ∃𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ ¬ (vol‘𝐵) ∈ ℝ))
92		xrge0nre 13114	. . . . . . . . 9 ⊢ (((vol‘𝐵) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐵) ∈ ℝ) → (vol‘𝐵) = +∞)
93	19, 92	sylan 579	. . . . . . . 8 ⊢ ((𝐵 ∈ dom vol ∧ ¬ (vol‘𝐵) ∈ ℝ) → (vol‘𝐵) = +∞)
94	93	reximi 3174	. . . . . . 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 963	. . . 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 1132	. 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 955	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 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ⦋csb 3828 ⊆ wss 3883 ∪ ciun 4921 Disj wdisj 5035 class class class wbr 5070 dom cdm 5580 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℝcr 10801 0cc0 10802 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 [,)cico 13010 [,]cicc 13011 Σcsu 15325 volcvol 24532 Σ^*cesum 31895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-ordt 17129 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-ps 18199 df-tsr 18200 df-plusf 18240 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-subrg 19937 df-abv 19992 df-lmod 20040 df-scaf 20041 df-sra 20349 df-rgmod 20350 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-tmd 23131 df-tgp 23132 df-tsms 23186 df-trg 23219 df-xms 23381 df-ms 23382 df-tms 23383 df-nm 23644 df-ngp 23645 df-nrg 23647 df-nlm 23648 df-ii 23946 df-cncf 23947 df-ovol 24533 df-vol 24534 df-limc 24935 df-dv 24936 df-log 25617 df-esum 31896

This theorem is referenced by: volmeas 32099

W3C validator