volfiniune - Mathbox for Thierry Arnoux

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Theorem volfiniune 34590

Description: The Lebesgue measure function is countably additive. This theorem is to volfiniun 25689 what voliune 34589 is to voliun 25696. (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 1208	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → 𝐴 ∈ Fin)
2		simpl2 1209	. . . . 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 3132	. . . . 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 1210	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) → Disj 𝑛 ∈ 𝐴 𝐵)
7		volfiniun 25689	. . . 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 1396	. . 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 2932	. . . 4 ⊢ Ⅎ𝑛𝐴
10	9	nfel1 2948	. . . . . 6 ⊢ Ⅎ𝑛 𝐴 ∈ Fin
11		nfra1 3296	. . . . . 6 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol
12		nfdisj1 5095	. . . . . 6 ⊢ Ⅎ𝑛Disj 𝑛 ∈ 𝐴 𝐵
13	10, 11, 12	nf3an 1929	. . . . 5 ⊢ Ⅎ𝑛(𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵)
14		nfra1 3296	. . . . 5 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ
15	13, 14	nfan 1927	. . . 4 ⊢ Ⅎ𝑛((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ)
16	3	r19.21bi 3264	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∀𝑛 ∈ 𝐴 (vol‘𝐵) ∈ ℝ) ∧ 𝑛 ∈ 𝐴) → (vol‘𝐵) ∈ ℝ)
17		rspa 3261	. . . . . . . 8 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ 𝑛 ∈ 𝐴) → 𝐵 ∈ dom vol)
18		volf 25671	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
19	18	ffvelcdmi 7082	. . . . . . . 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 11259	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
23		pnfxr 11266	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
24		elicc1 13419	. . . . . . . 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 1162	. . . . . 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 13148	. . . . . 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 11213	. . . . . 6 ⊢ 0 ∈ ℝ
31		elico2 13440	. . . . . 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 1360	. . . 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 34436	. . 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 2808	. 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 1942	. . . . . . . . 9 ⊢ Ⅎ𝑘(vol‘𝐵) = +∞
38		nfcv 2932	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol
39		nfcsb1v 3885	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵
40	38, 39	nffv 6895	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐵)
41	40	nfeq1 2947	. . . . . . . . 9 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞
42		csbeq1a 3875	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
43	42	fveqeq2d 6893	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((vol‘𝐵) = +∞ ↔ (vol‘⦋𝑘 / 𝑛⦌𝐵) = +∞))
44	37, 41, 43	cbvrexw 3315	. . . . . . . 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 2948	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol
47	42	eleq1d 2855	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐵 ∈ dom vol ↔ ⦋𝑘 / 𝑛⦌𝐵 ∈ dom vol))
48	46, 47	rspc 3577	. . . . . . . . . . . . 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 25686	. . . . . . . . . . . 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 2932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
54	9, 53, 39, 42	ssiun2sf 32874	. . . . . . . . . . . 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 25675	. . . . . . . . . . 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 1396	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol) ∧ 𝑘 ∈ 𝐴) → (vol‘⦋𝑘 / 𝑛⦌𝐵) ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
58	57	3adantl3 1185	. . . . . . . . 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 3164	. . . . . . 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 3136	. . . . . . 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 5117	. . . . . . . 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 3110	. . . . . 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 3102	. . . . . 6 ⊢ (∃𝑘 ∈ 𝐴 +∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵) → ∃𝑘+∞ ≤ (vol‘∪ 𝑛 ∈ 𝐴 𝐵))
68		19.9v 2012	. . . . . 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 13460	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
72	18	ffvelcdmi 7082	. . . . . . . . 9 ⊢ (∪ 𝑛 ∈ 𝐴 𝐵 ∈ dom vol → (vol‘∪ 𝑛 ∈ 𝐴 𝐵) ∈ (0[,]+∞))
73	71, 72	sselid 3943	. . . . . . . 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 1148	. . . . . 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 13194	. . . . 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 3297	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞
81	13, 80	nfan 1927	. . . 4 ⊢ Ⅎ𝑛((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞)
82		simpl1 1208	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ Disj 𝑛 ∈ 𝐴 𝐵) ∧ ∃𝑛 ∈ 𝐴 (vol‘𝐵) = +∞) → 𝐴 ∈ Fin)
83	20	3ad2antl2 1203	. . . . 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 34433	. . 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 2808	. 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 3124	. . . . . 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 3135	. . . . . . 7 ⊢ ((∀𝑛 ∈ 𝐴 𝐵 ∈ dom vol ∧ ∃𝑛 ∈ 𝐴 ¬ (vol‘𝐵) ∈ ℝ) → ∃𝑛 ∈ 𝐴 (𝐵 ∈ dom vol ∧ ¬ (vol‘𝐵) ∈ ℝ))
92		xrge0nre 13483	. . . . . . . . 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 3110	. . . . . . 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 1150	. 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 1101 = wceq 1568 ∃wex 1807 ∈ wcel 2150 ∀wral 3086 ∃wrex 3096 ⦋csb 3861 ⊆ wss 3913 ∪ ciun 4961 Disj wdisj 5081 class class class wbr 5114 dom cdm 5665 ‘cfv 6540 (class class class)co 7414 Fincfn 8946 ℝcr 11102 0cc0 11103 +∞cpnf 11243 ℝ^*cxr 11245 < clt 11246 ≤ cle 11247 [,)cico 13377 [,]cicc 13378 Σcsu 15740 volcvol 25605 Σ^*cesum 34387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736 ax-inf2 9613 ax-cnex 11159 ax-resscn 11160 ax-1cn 11161 ax-icn 11162 ax-addcl 11163 ax-addrcl 11164 ax-mulcl 11165 ax-mulrcl 11166 ax-mulcom 11167 ax-addass 11168 ax-mulass 11169 ax-distr 11170 ax-i2m1 11171 ax-1ne0 11172 ax-1rid 11173 ax-rnegex 11174 ax-rrecex 11175 ax-cnre 11176 ax-pre-lttri 11177 ax-pre-lttrn 11178 ax-pre-ltadd 11179 ax-pre-mulgt0 11180 ax-pre-sup 11181 ax-addf 11182 ax-mulf 11183

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-nel 3072 df-ral 3087 df-rex 3097 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-disj 5082 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-se 5619 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7678 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-oadd 8460 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8899 df-en 8947 df-dom 8948 df-sdom 8949 df-fin 8950 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-dju 9890 df-card 9928 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11446 df-neg 11447 df-div 11875 df-nn 12237 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12508 df-xnn0 12581 df-z 12595 df-dec 12715 df-uz 12866 df-q 12976 df-rp 13020 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13379 df-ioc 13380 df-ico 13381 df-icc 13382 df-fz 13539 df-fzo 13686 df-fl 13828 df-mod 13906 df-seq 14041 df-exp 14101 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15107 df-cj 15153 df-re 15154 df-im 15155 df-sqrt 15289 df-abs 15290 df-limsup 15525 df-clim 15542 df-rlim 15543 df-sum 15741 df-ef 16124 df-sin 16126 df-cos 16127 df-pi 16129 df-struct 17210 df-sets 17227 df-slot 17245 df-ndx 17257 df-base 17273 df-ress 17294 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-rest 17478 df-topn 17479 df-0g 17497 df-gsum 17498 df-topgen 17499 df-pt 17500 df-prds 17503 df-ordt 17558 df-xrs 17559 df-qtop 17564 df-imas 17565 df-xps 17567 df-mre 17641 df-mrc 17642 df-acs 17644 df-ps 18625 df-tsr 18626 df-plusf 18700 df-mgm 18701 df-sgrp 18780 df-mnd 18796 df-mhm 18844 df-submnd 18845 df-grp 19006 df-minusg 19007 df-sbg 19008 df-mulg 19137 df-subg 19192 df-cntz 19390 df-cmn 19855 df-abl 19856 df-mgp 20220 df-rng 20234 df-ur 20267 df-ring 20320 df-cring 20321 df-subrng 20634 df-subrg 20658 df-abv 20895 df-lmod 20966 df-scaf 20967 df-sra 21277 df-rgmod 21278 df-psmet 21497 df-xmet 21498 df-met 21499 df-bl 21500 df-mopn 21501 df-fbas 21502 df-fg 21503 df-cnfld 21506 df-top 23034 df-topon 23051 df-topsp 23073 df-bases 23086 df-cld 23159 df-ntr 23160 df-cls 23161 df-nei 23238 df-lp 23276 df-perf 23277 df-cn 23367 df-cnp 23368 df-haus 23455 df-tx 23702 df-hmeo 23895 df-fil 23986 df-fm 24078 df-flim 24079 df-flf 24080 df-tmd 24212 df-tgp 24213 df-tsms 24267 df-trg 24300 df-xms 24460 df-ms 24461 df-tms 24462 df-nm 24722 df-ngp 24723 df-nrg 24725 df-nlm 24726 df-ii 25019 df-cncf 25020 df-ovol 25606 df-vol 25607 df-limc 26008 df-dv 26009 df-log 26701 df-esum 34388

This theorem is referenced by: volmeas 34591

W3C validator