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Theorem ovoliun 25254

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 25234, 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 11275	. . . . . 6 ⊢ -∞ ∈ ℝ^*
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → -∞ ∈ ℝ^*)
3		nnuz 12869	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
4		1zzd 12597	. . . . . . . . 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 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
9	3, 4, 8	serfre 14001	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
10		ovoliun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , 𝐺)
11	10	feq1i 6707	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
12	9, 11	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
13		1nn 12227	. . . . . . 7 ⊢ 1 ∈ ℕ
14		ffvelcdm 7082	. . . . . . 7 ⊢ ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
15	12, 13, 14	sylancl 584	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ)
16	15	rexrd 11268	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ^*)
17	12	frnd 6724	. . . . . . 7 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
18		ressxr 11262	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
19	17, 18	sstrdi 3993	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
20		supxrcl 13298	. . . . . 6 ⊢ (ran 𝑇 ⊆ ℝ^* → sup(ran 𝑇, ℝ^, < ) ∈ ℝ^)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ∈ ℝ^)
22	15	mnfltd 13108	. . . . 5 ⊢ (𝜑 → -∞ < (𝑇‘1))
23	12	ffnd 6717	. . . . . . 7 ⊢ (𝜑 → 𝑇 Fn ℕ)
24		fnfvelrn 7081	. . . . . . 7 ⊢ ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
25	23, 13, 24	sylancl 584	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26		supxrub 13307	. . . . . 6 ⊢ ((ran 𝑇 ⊆ ℝ^* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ^*, < ))
27	19, 25, 26	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ^*, < ))
28	2, 16, 21, 22, 27	xrltletrd 13144	. . . 4 ⊢ (𝜑 → -∞ < sup(ran 𝑇, ℝ^*, < ))
29		xrrebnd 13151	. . . . 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 703	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ^, < ) < +∞))
32		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
33		nfcsb1v 3917	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
34		csbeq1a 3906	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
35	32, 33, 34	cbviun 5038	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
36	35	fveq2i 6893	. . . . . . 7 ⊢ (vol‘∪ 𝑛 ∈ ℕ 𝐴) = (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴)
37		nfcv 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
38		nfcv 2901	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
39	38, 33	nffv 6900	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
40	34	fveq2d 6894	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol‘𝐴) = (vol‘⦋𝑚 / 𝑛⦌𝐴))
41	37, 39, 40	cbvmpt 5258	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol‘⦋𝑚 / 𝑛⦌𝐴))
42	6, 41	eqtri 2758	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
43		ovoliun.a	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
44	43	ralrimiva 3144	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
45		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
46		nfcv 2901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
47	33, 46	nfss 3973	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
48	34	sseq1d 4012	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
49	45, 47, 48	cbvralw 3301	. . . . . . . . . . 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 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	51	r19.21bi 3246	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	5	ralrimiva 3144	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
54	37	nfel1 2917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
55	39	nfel1 2917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
56	40	eleq1d 2816	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
57	54, 55, 56	cbvralw 3301	. . . . . . . . . . 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 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → ∀𝑚 ∈ ℕ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	59	r19.21bi 3246	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ ℕ) → (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61		simplr 765	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → sup(ran 𝑇, ℝ^, < ) ∈ ℝ)
62		simpr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
63	10, 42, 52, 60, 61, 62	ovoliunlem3 25253	. . . . . . 7 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥))
64	36, 63	eqbrtrid 5182	. . . . . 6 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥))
65	64	ralrimiva 3144	. . . . 5 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝑥))
66		iunss 5047	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
67	44, 66	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68		ovolcl 25227	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^)
69	67, 68	syl 17	. . . . . 6 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^)
70		xralrple 13188	. . . . . 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 578	. . . . 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 256	. . . 4 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ^, < ) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < ))
73	72	ex 411	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) ∈ ℝ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
74	31, 73	sylbird 259	. 2 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) < +∞ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ^*, < )))
75		nltpnft 13147	. . . 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 13114	. . . . 5 ⊢ ((vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
78	69, 77	syl 17	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
79		breq2 5151	. . . 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 259	. 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 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ⦋csb 3892 ⊆ wss 3947 ∪ ciun 4996 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 supcsup 9437 ℝcr 11111 1c1 11113 + caddc 11115 +∞cpnf 11249 -∞cmnf 11250 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 ℕcn 12216 ℝ⁺crp 12978 seqcseq 13970 vol*covol 25211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12979 df-ioo 13332 df-ico 13334 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-ovol 25213

This theorem is referenced by: ovoliun2 25255 voliunlem2 25300 voliunlem3 25301 ex-ovoliunnfl 36834

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