Theorem ovoliun 24669

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 24649, 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 11032	. . . . . 6 ⊢ -∞ ∈ ℝ*
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → -∞ ∈ ℝ*)
3		nnuz 12621	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
4		1zzd 12351	. . . . . . . . 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 6988	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
8	7	ffvelrnda 6961	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
9	3, 4, 8	serfre 13752	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
10		ovoliun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , 𝐺)
11	10	feq1i 6591	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
12	9, 11	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
13		1nn 11984	. . . . . . 7 ⊢ 1 ∈ ℕ
14		ffvelrn 6959	. . . . . . 7 ⊢ ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
15	12, 13, 14	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ)
16	15	rexrd 11025	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ*)
17	12	frnd 6608	. . . . . . 7 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
18		ressxr 11019	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
19	17, 18	sstrdi 3933	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*)
20		supxrcl 13049	. . . . . 6 ⊢ (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
22	15	mnfltd 12860	. . . . 5 ⊢ (𝜑 → -∞ < (𝑇‘1))
23	12	ffnd 6601	. . . . . . 7 ⊢ (𝜑 → 𝑇 Fn ℕ)
24		fnfvelrn 6958	. . . . . . 7 ⊢ ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
25	23, 13, 24	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26		supxrub 13058	. . . . . 6 ⊢ ((ran 𝑇 ⊆ ℝ* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
27	19, 25, 26	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
28	2, 16, 21, 22, 27	xrltletrd 12895	. . . 4 ⊢ (𝜑 → -∞ < sup(ran 𝑇, ℝ*, < ))
29		xrrebnd 12902	. . . . 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 704	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ*, < ) < +∞))
32		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
33		nfcsb1v 3857	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
34		csbeq1a 3846	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
35	32, 33, 34	cbviun 4966	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
36	35	fveq2i 6777	. . . . . . 7 ⊢ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) = (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴)
37		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
38		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
39	38, 33	nffv 6784	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
40	34	fveq2d 6778	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
41	37, 39, 40	cbvmpt 5185	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
42	6, 41	eqtri 2766	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
43		ovoliun.a	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
44	43	ralrimiva 3103	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
45		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
46		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
47	33, 46	nfss 3913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
48	34	sseq1d 3952	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
49	45, 47, 48	cbvralw 3373	. . . . . . . . . . 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 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	51	r19.21bi 3134	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	5	ralrimiva 3103	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
54	37	nfel1 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
55	39	nfel1 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
56	40	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
57	54, 55, 56	cbvralw 3373	. . . . . . . . . . 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 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	59	r19.21bi 3134	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61		simplr 766	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
62		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
63	10, 42, 52, 60, 61, 62	ovoliunlem3 24668	. . . . . . 7 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
64	36, 63	eqbrtrid 5109	. . . . . 6 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
65	64	ralrimiva 3103	. . . . 5 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
66		iunss 4975	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
67	44, 66	sylibr 233	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68		ovolcl 24642	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
69	67, 68	syl 17	. . . . . 6 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
70		xralrple 12939	. . . . . 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 580	. . . . 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 413	. . 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 12898	. . . 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 12866	. . . . 5 ⊢ ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
78	69, 77	syl 17	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
79		breq2 5078	. . . 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*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⦋csb 3832   ⊆ wss 3887  ∪ ciun 4924   class class class wbr 5074   ↦ cmpt 5157  ran crn 5590   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  supcsup 9199  ℝcr 10870  1c1 10872   + caddc 10874  +∞cpnf 11006  -∞cmnf 11007  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010  ℕcn 11973  ℝ⁺crp 12730  seqcseq 13721  vol*covol 24626

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cc 10191  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-ioo 13083  df-ico 13085  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-ovol 24628

This theorem is referenced by:  ovoliun2  24670  voliunlem2  24715  voliunlem3  24716  ex-ovoliunnfl  35820