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Theorem ovoliun 23080

Description: The Lebesgue outer measure function is countably sub-additive. (Many books allow +∞ as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 23060, so we need not consider this case here, although we do allow the sum itself to be infinite.) (Contributed by Mario Carneiro, 12-Jun-2014.)

**Hypotheses**
Ref	Expression
ovoliun.t	⊢ 𝑇 = seq1( + , 𝐺)
ovoliun.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)

**Assertion**
Ref	Expression
ovoliun	⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^, < ))

Distinct variable group: 𝜑,𝑛 Allowed substitution hints: 𝐴(𝑛) 𝑇(𝑛) 𝐺(𝑛)

Proof of Theorem ovoliun Dummy variables 𝑘 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11599	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 11285	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
3		ovoliun.v	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
4		ovoliun.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
5	3, 4	fmptd 6292	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
6	5	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
7	1, 2, 6	serfre 12692	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
8		ovoliun.t	. . . . . . . . . 10 ⊢ 𝑇 = seq1( + , 𝐺)
9	8	feq1i 5949	. . . . . . . . 9 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
10	7, 9	sylibr 223	. . . . . . . 8 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
11		frn 5966	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ → ran 𝑇 ⊆ ℝ)
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
13		ressxr 9962	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
14	12, 13	syl6ss 3580	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
15		supxrcl 12017	. . . . . 6 ⊢ (ran 𝑇 ⊆ ℝ^* → sup(ran 𝑇, ℝ^, < ) ∈ ℝ^)
16	14, 15	syl 17	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ∈ ℝ^)
17		xrrebnd 11873	. . . . 5 ⊢ (sup(ran 𝑇, ℝ^, < ) ∈ ℝ^ → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ^, < ) ∧ sup(ran 𝑇, ℝ^*, < ) < +∞)))
18	16, 17	syl 17	. . . 4 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ^, < ) ∧ sup(ran 𝑇, ℝ^*, < ) < +∞)))
19		mnfxr 9975	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
20	19	a1i 11	. . . . . 6 ⊢ (𝜑 → -∞ ∈ ℝ^*)
21		1nn 10908	. . . . . . . 8 ⊢ 1 ∈ ℕ
22		ffvelrn 6265	. . . . . . . 8 ⊢ ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
23	10, 21, 22	sylancl 693	. . . . . . 7 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ)
24	23	rexrd 9968	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ^*)
25		mnflt 11833	. . . . . . 7 ⊢ ((𝑇‘1) ∈ ℝ → -∞ < (𝑇‘1))
26	23, 25	syl 17	. . . . . 6 ⊢ (𝜑 → -∞ < (𝑇‘1))
27		ffn 5958	. . . . . . . . 9 ⊢ (𝑇:ℕ⟶ℝ → 𝑇 Fn ℕ)
28	10, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Fn ℕ)
29		fnfvelrn 6264	. . . . . . . 8 ⊢ ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
30	28, 21, 29	sylancl 693	. . . . . . 7 ⊢ (𝜑 → (𝑇‘1) ∈ ran 𝑇)
31		supxrub 12026	. . . . . . 7 ⊢ ((ran 𝑇 ⊆ ℝ^* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ^*, < ))
32	14, 30, 31	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ^*, < ))
33	20, 24, 16, 26, 32	xrltletrd 11868	. . . . 5 ⊢ (𝜑 → -∞ < sup(ran 𝑇, ℝ^*, < ))
34	33	biantrurd 528	. . . 4 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) < +∞ ↔ (-∞ < sup(ran 𝑇, ℝ^, < ) ∧ sup(ran 𝑇, ℝ^*, < ) < +∞)))
35	18, 34	bitr4d 270	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ^, < ) < +∞))
36		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
37		nfcsb1v 3515	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
38		csbeq1a 3508	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
39	36, 37, 38	cbviun 4493	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
40	39	fveq2i 6106	. . . . . . 7 ⊢ (vol‘∪ 𝑛 ∈ ℕ 𝐴) = (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴)
41		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
42		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
43	42, 37	nffv 6110	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
44	38	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol‘𝐴) = (vol‘⦋𝑚 / 𝑛⦌𝐴))
45	41, 43, 44	cbvmpt 4677	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol‘⦋𝑚 / 𝑛⦌𝐴))
46	4, 45	eqtri 2632	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
47		ovoliun.a	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
48	47	ralrimiva 2949	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
49		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
50		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
51	37, 50	nfss 3561	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
52	38	sseq1d 3595	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
53	49, 51, 52	cbvral 3143	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
54	48, 53	sylib 207	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
55	54	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
56	55	r19.21bi 2916	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
57	3	ralrimiva 2949	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
58	41	nfel1 2765	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
59	43	nfel1 2765	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
60	44	eleq1d 2672	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
61	58, 59, 60	cbvral 3143	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
62	57, 61	sylib 207	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
63	62	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → ∀𝑚 ∈ ℕ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
64	63	r19.21bi 2916	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ ℕ) → (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
65		simplr 788	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → sup(ran 𝑇, ℝ^, < ) ∈ ℝ)
66		simpr 476	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
67	8, 46, 56, 64, 65, 66	ovoliunlem3 23079	. . . . . . 7 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥))
68	40, 67	syl5eqbr 4618	. . . . . 6 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥))
69	68	ralrimiva 2949	. . . . 5 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥))
70		iunss 4497	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
71	48, 70	sylibr 223	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
72		ovolcl 23053	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^)
73	71, 72	syl 17	. . . . . 6 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^)
74		xralrple 11910	. . . . . 6 ⊢ (((vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^ ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) → ((vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^, < ) ↔ ∀𝑥 ∈ ℝ⁺ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥)))
75	73, 74	sylan 487	. . . . 5 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) → ((vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^, < ) ↔ ∀𝑥 ∈ ℝ⁺ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥)))
76	69, 75	mpbird 246	. . . 4 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < ))
77	76	ex 449	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
78	35, 77	sylbird 249	. 2 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) < +∞ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
79		nltpnft 11871	. . . 4 ⊢ (sup(ran 𝑇, ℝ^, < ) ∈ ℝ^ → (sup(ran 𝑇, ℝ^, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ^, < ) < +∞))
80	16, 79	syl 17	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ^, < ) < +∞))
81		pnfge 11840	. . . . 5 ⊢ ((vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
82	73, 81	syl 17	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
83		breq2 4587	. . . 4 ⊢ (sup(ran 𝑇, ℝ^, < ) = +∞ → ((vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^, < ) ↔ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞))
84	82, 83	syl5ibrcom 236	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) = +∞ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
85	80, 84	sylbird 249	. 2 ⊢ (𝜑 → (¬ sup(ran 𝑇, ℝ^, < ) < +∞ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
86	78, 85	pm2.61d 169	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⦋csb 3499 ⊆ wss 3540 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℝcr 9814 1c1 9816 + caddc 9818 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 ℕcn 10897 ℝ⁺crp 11708 seqcseq 12663 vol*covol 23038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-ioo 12050 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-ovol 23040

This theorem is referenced by: ovoliun2 23081 voliunlem2 23126 voliunlem3 23127 ex-ovoliunnfl 32622

W3C validator