Theorem ovoliun 25413

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 25393, 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 11238	. . . . . 6 ⊢ -∞ ∈ ℝ*
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → -∞ ∈ ℝ*)
3		nnuz 12843	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
4		1zzd 12571	. . . . . . . . 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 7089	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
8	7	ffvelcdmda 7059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
9	3, 4, 8	serfre 14003	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
10		ovoliun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , 𝐺)
11	10	feq1i 6682	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
12	9, 11	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
13		1nn 12204	. . . . . . 7 ⊢ 1 ∈ ℕ
14		ffvelcdm 7056	. . . . . . 7 ⊢ ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
15	12, 13, 14	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ)
16	15	rexrd 11231	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ*)
17	12	frnd 6699	. . . . . . 7 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
18		ressxr 11225	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
19	17, 18	sstrdi 3962	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*)
20		supxrcl 13282	. . . . . 6 ⊢ (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
22	15	mnfltd 13091	. . . . 5 ⊢ (𝜑 → -∞ < (𝑇‘1))
23	12	ffnd 6692	. . . . . . 7 ⊢ (𝜑 → 𝑇 Fn ℕ)
24		fnfvelrn 7055	. . . . . . 7 ⊢ ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
25	23, 13, 24	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26		supxrub 13291	. . . . . 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 13128	. . . 4 ⊢ (𝜑 → -∞ < sup(ran 𝑇, ℝ*, < ))
29		xrrebnd 13135	. . . . 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 707	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ*, < ) < +∞))
32		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
33		nfcsb1v 3889	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
34		csbeq1a 3879	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
35	32, 33, 34	cbviun 5003	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
36	35	fveq2i 6864	. . . . . . 7 ⊢ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) = (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴)
37		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
38		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
39	38, 33	nffv 6871	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
40	34	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
41	37, 39, 40	cbvmpt 5212	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
42	6, 41	eqtri 2753	. . . . . . . 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 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
46		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
47	33, 46	nfss 3942	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
48	34	sseq1d 3981	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
49	45, 47, 48	cbvralw 3282	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
50	44, 49	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
51	50	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	51	r19.21bi 3230	. . . . . . . 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 2909	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
55	39	nfel1 2909	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
56	40	eleq1d 2814	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
57	54, 55, 56	cbvralw 3282	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
58	53, 57	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
59	58	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	59	r19.21bi 3230	. . . . . . . 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 484	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
63	10, 42, 52, 60, 61, 62	ovoliunlem3 25412	. . . . . . 7 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
64	36, 63	eqbrtrid 5145	. . . . . 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 5012	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
67	44, 66	sylibr 234	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68		ovolcl 25386	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
69	67, 68	syl 17	. . . . . 6 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
70		xralrple 13172	. . . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
73	72	ex 412	. . 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 13131	. . . 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 13097	. . . . 5 ⊢ ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
78	69, 77	syl 17	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
79		breq2 5114	. . . 4 ⊢ (sup(ran 𝑇, ℝ*, < ) = +∞ → ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ) ↔ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞))
80	78, 79	syl5ibrcom 247	. . 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*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ⦋csb 3865   ⊆ wss 3917  ∪ ciun 4958   class class class wbr 5110   ↦ cmpt 5191  ran crn 5642   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  supcsup 9398  ℝcr 11074  1c1 11076   + caddc 11078  +∞cpnf 11212  -∞cmnf 11213  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216  ℕcn 12193  ℝ⁺crp 12958  seqcseq 13973  vol*covol 25370

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cc 10395  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-ioo 13317  df-ico 13319  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-rlim 15462  df-sum 15660  df-ovol 25372

This theorem is referenced by:  ovoliun2  25414  voliunlem2  25459  voliunlem3  25460  ex-ovoliunnfl  37664