Theorem ovoliun2 25560

Description: The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 25559.) (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)
ovoliun2.t	⊢ (𝜑 → 𝑇 ∈ dom ⇝ )

Assertion

Ref	Expression
ovoliun2	⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))

Distinct variable group:   𝜑,𝑛

Allowed substitution hints:   𝐴(𝑛)   𝑇(𝑛)   𝐺(𝑛)

Proof of Theorem ovoliun2

Dummy variables 𝑘 𝑚 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovoliun.t	. . 3 ⊢ 𝑇 = seq1( + , 𝐺)
2		ovoliun.g	. . 3 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
3		ovoliun.a	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
4		ovoliun.v	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
5	1, 2, 3, 4	ovoliun 25559	. 2 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
6		nnuz 12946	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
7		1zzd 12674	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
8		fvex 6933	. . . . . . . . . . 11 ⊢ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V
9		nfcv 2908	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(vol*‘𝐴)
10		nfcv 2908	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
11		nfcsb1v 3946	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
12	10, 11	nffv 6930	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
13		csbeq1a 3935	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
14	13	fveq2d 6924	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
15	9, 12, 14	cbvmpt 5277	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
16	2, 15	eqtri 2768	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
17	16	fvmpt2 7040	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
18	8, 17	mpan2 690	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
19	18	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
20	4	ralrimiva 3152	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
21	9	nfel1 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
22	12	nfel1 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
23	14	eleq1d 2829	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
24	21, 22, 23	cbvralw 3312	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
25	20, 24	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
26	25	r19.21bi 3257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
27	19, 26	eqeltrd 2844	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
28	6, 7, 27	serfre 14082	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
29	1	feq1i 6738	. . . . . . 7 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
30	28, 29	sylibr 234	. . . . . 6 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
31	30	frnd 6755	. . . . 5 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
32		1nn 12304	. . . . . . . 8 ⊢ 1 ∈ ℕ
33	30	fdmd 6757	. . . . . . . 8 ⊢ (𝜑 → dom 𝑇 = ℕ)
34	32, 33	eleqtrrid 2851	. . . . . . 7 ⊢ (𝜑 → 1 ∈ dom 𝑇)
35	34	ne0d 4365	. . . . . 6 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
36		dm0rn0 5949	. . . . . . 7 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
37	36	necon3bii 2999	. . . . . 6 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
38	35, 37	sylib 218	. . . . 5 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
39		ovoliun2.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ dom ⇝ )
40	1, 39	eqeltrrid 2849	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
41	6, 7, 19, 26, 40	isumrecl 15813	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
42		elfznn 13613	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
43	42	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → 𝑚 ∈ ℕ)
44	43, 18	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
45		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
46	45, 6	eleqtrdi 2854	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
47		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
48	47, 42, 26	syl2an 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
49	48	recnd 11318	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℂ)
50	44, 46, 49	fsumser 15778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (seq1( + , 𝐺)‘𝑘))
51	1	fveq1i 6921	. . . . . . . . . 10 ⊢ (𝑇‘𝑘) = (seq1( + , 𝐺)‘𝑘)
52	50, 51	eqtr4di 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (𝑇‘𝑘))
53		fzfid 14024	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ∈ Fin)
54		fz1ssnn 13615	. . . . . . . . . . . 12 ⊢ (1...𝑘) ⊆ ℕ
55	54	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ⊆ ℕ)
56	3	ralrimiva 3152	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
57		nfv 1913	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
58		nfcv 2908	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛ℝ
59	11, 58	nfss 4001	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
60	13	sseq1d 4040	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
61	57, 59, 60	cbvralw 3312	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
62	56, 61	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
63	62	r19.21bi 3257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
64		ovolge0 25535	. . . . . . . . . . . 12 ⊢ (⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ → 0 ≤ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
65	63, 64	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0 ≤ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
66	6, 7, 53, 55, 19, 26, 65, 40	isumless 15893	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
67	66	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
68	52, 67	eqbrtrrd 5190	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
69	68	ralrimiva 3152	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
70		brralrspcev 5226	. . . . . . 7 ⊢ ((Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥)
71	41, 69, 70	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥)
72	30	ffnd 6748	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Fn ℕ)
73		breq1 5169	. . . . . . . . 9 ⊢ (𝑧 = (𝑇‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑇‘𝑘) ≤ 𝑥))
74	73	ralrn 7122	. . . . . . . 8 ⊢ (𝑇 Fn ℕ → (∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥))
75	72, 74	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥))
76	75	rexbidv 3185	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥))
77	71, 76	mpbird 257	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥)
78		supxrre 13389	. . . . 5 ⊢ ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
79	31, 38, 77, 78	syl3anc 1371	. . . 4 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
80	6, 1, 7, 19, 26, 65, 71	isumsup 15895	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) = sup(ran 𝑇, ℝ, < ))
81	79, 80	eqtr4d 2783	. . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
82	14, 9, 12	cbvsum 15743	. . 3 ⊢ Σ𝑛 ∈ ℕ (vol*‘𝐴) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)
83	81, 82	eqtr4di 2798	. 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑛 ∈ ℕ (vol*‘𝐴))
84	5, 83	breqtrd 5192	1 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  Vcvv 3488  ⦋csb 3921   ⊆ wss 3976  ∅c0 4352  ∪ ciun 5015   class class class wbr 5166   ↦ cmpt 5249  dom cdm 5700  ran crn 5701   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  supcsup 9509  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  ℕcn 12293  ℤ_≥cuz 12903  ...cfz 13567  seqcseq 14052   ⇝ cli 15530  Σcsu 15734  vol*covol 25516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cc 10504  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-ioo 13411  df-ico 13413  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-sum 15735  df-ovol 25518

This theorem is referenced by:  ovoliunnul  25561  vitalilem5  25666