ovoliun - Metamath Proof Explorer

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Theorem ovoliun 25022

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 25002, 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		mnfxr 11271	. . . . . 6 ⊢ -∞ ∈ ℝ^*
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → -∞ ∈ ℝ^*)
3		nnuz 12865	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 12593	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
5		ovoliun.v	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
6		ovoliun.g	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
7	5, 6	fmptd 7114	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
8	7	ffvelcdmda 7087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
9	3, 4, 8	serfre 13997	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
10		ovoliun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , 𝐺)
11	10	feq1i 6709	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
12	9, 11	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
13		1nn 12223	. . . . . . 7 ⊢ 1 ∈ ℕ
14		ffvelcdm 7084	. . . . . . 7 ⊢ ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
15	12, 13, 14	sylancl 587	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ)
16	15	rexrd 11264	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ^*)
17	12	frnd 6726	. . . . . . 7 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
18		ressxr 11258	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
19	17, 18	sstrdi 3995	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
20		supxrcl 13294	. . . . . 6 ⊢ (ran 𝑇 ⊆ ℝ^* → sup(ran 𝑇, ℝ^, < ) ∈ ℝ^)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ∈ ℝ^)
22	15	mnfltd 13104	. . . . 5 ⊢ (𝜑 → -∞ < (𝑇‘1))
23	12	ffnd 6719	. . . . . . 7 ⊢ (𝜑 → 𝑇 Fn ℕ)
24		fnfvelrn 7083	. . . . . . 7 ⊢ ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
25	23, 13, 24	sylancl 587	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26		supxrub 13303	. . . . . 6 ⊢ ((ran 𝑇 ⊆ ℝ^* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ^*, < ))
27	19, 25, 26	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ^*, < ))
28	2, 16, 21, 22, 27	xrltletrd 13140	. . . 4 ⊢ (𝜑 → -∞ < sup(ran 𝑇, ℝ^*, < ))
29		xrrebnd 13147	. . . . 5 ⊢ (sup(ran 𝑇, ℝ^, < ) ∈ ℝ^ → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ^, < ) ∧ sup(ran 𝑇, ℝ^*, < ) < +∞)))
30	21, 29	syl 17	. . . 4 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ^, < ) ∧ sup(ran 𝑇, ℝ^*, < ) < +∞)))
31	28, 30	mpbirand 706	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ^, < ) < +∞))
32		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
33		nfcsb1v 3919	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
34		csbeq1a 3908	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
35	32, 33, 34	cbviun 5040	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
36	35	fveq2i 6895	. . . . . . 7 ⊢ (vol‘∪ 𝑛 ∈ ℕ 𝐴) = (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴)
37		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
38		nfcv 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
39	38, 33	nffv 6902	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
40	34	fveq2d 6896	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol‘𝐴) = (vol‘⦋𝑚 / 𝑛⦌𝐴))
41	37, 39, 40	cbvmpt 5260	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol‘⦋𝑚 / 𝑛⦌𝐴))
42	6, 41	eqtri 2761	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
43		ovoliun.a	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
44	43	ralrimiva 3147	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
45		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
46		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
47	33, 46	nfss 3975	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
48	34	sseq1d 4014	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
49	45, 47, 48	cbvralw 3304	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
50	44, 49	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
51	50	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	51	r19.21bi 3249	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	5	ralrimiva 3147	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
54	37	nfel1 2920	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
55	39	nfel1 2920	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
56	40	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
57	54, 55, 56	cbvralw 3304	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
58	53, 57	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
59	58	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → ∀𝑚 ∈ ℕ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	59	r19.21bi 3249	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ ℕ) → (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → sup(ran 𝑇, ℝ^, < ) ∈ ℝ)
62		simpr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
63	10, 42, 52, 60, 61, 62	ovoliunlem3 25021	. . . . . . 7 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥))
64	36, 63	eqbrtrid 5184	. . . . . 6 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥))
65	64	ralrimiva 3147	. . . . 5 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥))
66		iunss 5049	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
67	44, 66	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68		ovolcl 24995	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^)
69	67, 68	syl 17	. . . . . 6 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^)
70		xralrple 13184	. . . . . 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)))
71	69, 70	sylan 581	. . . . 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)))
72	65, 71	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < ))
73	72	ex 414	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
74	31, 73	sylbird 260	. 2 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) < +∞ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
75		nltpnft 13143	. . . 4 ⊢ (sup(ran 𝑇, ℝ^, < ) ∈ ℝ^ → (sup(ran 𝑇, ℝ^, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ^, < ) < +∞))
76	21, 75	syl 17	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ^, < ) < +∞))
77		pnfge 13110	. . . . 5 ⊢ ((vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
78	69, 77	syl 17	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
79		breq2 5153	. . . 4 ⊢ (sup(ran 𝑇, ℝ^, < ) = +∞ → ((vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^, < ) ↔ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞))
80	78, 79	syl5ibrcom 246	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) = +∞ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
81	76, 80	sylbird 260	. 2 ⊢ (𝜑 → (¬ sup(ran 𝑇, ℝ^, < ) < +∞ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
82	74, 81	pm2.61d 179	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⦋csb 3894 ⊆ wss 3949 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 ran crn 5678 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 supcsup 9435 ℝcr 11109 1c1 11111 + caddc 11113 +∞cpnf 11245 -∞cmnf 11246 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 ℕcn 12212 ℝ⁺crp 12974 seqcseq 13966 vol*covol 24979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-ioo 13328 df-ico 13330 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-ovol 24981

This theorem is referenced by: ovoliun2 25023 voliunlem2 25068 voliunlem3 25069 ex-ovoliunnfl 36531

W3C validator