Theorem ovoliun 23799

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 23779, 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 10490	. . . . . 6 ⊢ -∞ ∈ ℝ*
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → -∞ ∈ ℝ*)
3		nnuz 12088	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
4		1zzd 11819	. . . . . . . . 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 6695	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
8	7	ffvelrnda 6670	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
9	3, 4, 8	serfre 13207	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
10		ovoliun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , 𝐺)
11	10	feq1i 6329	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
12	9, 11	sylibr 226	. . . . . . 7 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
13		1nn 11444	. . . . . . 7 ⊢ 1 ∈ ℕ
14		ffvelrn 6668	. . . . . . 7 ⊢ ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
15	12, 13, 14	sylancl 577	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ)
16	15	rexrd 10482	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ*)
17	12	frnd 6345	. . . . . . 7 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
18		ressxr 10476	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
19	17, 18	syl6ss 3866	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*)
20		supxrcl 12517	. . . . . 6 ⊢ (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
22	15	mnfltd 12329	. . . . 5 ⊢ (𝜑 → -∞ < (𝑇‘1))
23	12	ffnd 6339	. . . . . . 7 ⊢ (𝜑 → 𝑇 Fn ℕ)
24		fnfvelrn 6667	. . . . . . 7 ⊢ ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
25	23, 13, 24	sylancl 577	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26		supxrub 12526	. . . . . 6 ⊢ ((ran 𝑇 ⊆ ℝ* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
27	19, 25, 26	syl2anc 576	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
28	2, 16, 21, 22, 27	xrltletrd 12364	. . . 4 ⊢ (𝜑 → -∞ < sup(ran 𝑇, ℝ*, < ))
29		xrrebnd 12371	. . . . 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 694	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ*, < ) < +∞))
32		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
33		nfcsb1v 3800	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
34		csbeq1a 3791	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
35	32, 33, 34	cbviun 4825	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
36	35	fveq2i 6496	. . . . . . 7 ⊢ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) = (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴)
37		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
38		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
39	38, 33	nffv 6503	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
40	34	fveq2d 6497	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
41	37, 39, 40	cbvmpt 5021	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
42	6, 41	eqtri 2796	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
43		ovoliun.a	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
44	43	ralrimiva 3126	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
45		nfv 1873	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
46		nfcv 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
47	33, 46	nfss 3847	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
48	34	sseq1d 3884	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
49	45, 47, 48	cbvral 3373	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
50	44, 49	sylib 210	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
51	50	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	51	r19.21bi 3152	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	5	ralrimiva 3126	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
54	37	nfel1 2940	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
55	39	nfel1 2940	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
56	40	eleq1d 2844	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
57	54, 55, 56	cbvral 3373	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
58	53, 57	sylib 210	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
59	58	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	59	r19.21bi 3152	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61		simplr 756	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
62		simpr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
63	10, 42, 52, 60, 61, 62	ovoliunlem3 23798	. . . . . . 7 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
64	36, 63	syl5eqbr 4958	. . . . . 6 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
65	64	ralrimiva 3126	. . . . 5 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
66		iunss 4829	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
67	44, 66	sylibr 226	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68		ovolcl 23772	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
69	67, 68	syl 17	. . . . . 6 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
70		xralrple 12408	. . . . . 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 572	. . . . 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 249	. . . 4 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
73	72	ex 405	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
74	31, 73	sylbird 252	. 2 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) < +∞ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
75		nltpnft 12367	. . . 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 12335	. . . . 5 ⊢ ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
78	69, 77	syl 17	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
79		breq2 4927	. . . 4 ⊢ (sup(ran 𝑇, ℝ*, < ) = +∞ → ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ) ↔ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞))
80	78, 79	syl5ibrcom 239	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) = +∞ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
81	76, 80	sylbird 252	. 2 ⊢ (𝜑 → (¬ sup(ran 𝑇, ℝ*, < ) < +∞ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
82	74, 81	pm2.61d 172	1 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048  ∀wral 3082  ⦋csb 3782   ⊆ wss 3825  ∪ ciun 4786   class class class wbr 4923   ↦ cmpt 5002  ran crn 5401   Fn wfn 6177  ⟶wf 6178  ‘cfv 6182  (class class class)co 6970  supcsup 8691  ℝcr 10326  1c1 10328   + caddc 10330  +∞cpnf 10463  -∞cmnf 10464  ℝ^*cxr 10465   < clt 10466   ≤ cle 10467  ℕcn 11431  ℝ⁺crp 12197  seqcseq 13177  vol*covol 23756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8890  ax-cc 9647  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-pre-sup 10405

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-map 8200  df-pm 8201  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-sup 8693  df-inf 8694  df-oi 8761  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-2 11496  df-3 11497  df-n0 11701  df-z 11787  df-uz 12052  df-q 12156  df-rp 12198  df-ioo 12551  df-ico 12553  df-fz 12702  df-fzo 12843  df-fl 12970  df-seq 13178  df-exp 13238  df-hash 13499  df-cj 14309  df-re 14310  df-im 14311  df-sqrt 14445  df-abs 14446  df-clim 14696  df-rlim 14697  df-sum 14894  df-ovol 23758

This theorem is referenced by:  ovoliun2  23800  voliunlem2  23845  voliunlem3  23846  ex-ovoliunnfl  34324