Theorem ovoliun2 25255

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 25254.) (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 25254	. 2 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
6		nnuz 12869	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
7		1zzd 12597	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
8		fvex 6903	. . . . . . . . . . 11 ⊢ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V
9		nfcv 2901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(vol*‘𝐴)
10		nfcv 2901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
11		nfcsb1v 3917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
12	10, 11	nffv 6900	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
13		csbeq1a 3906	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
14	13	fveq2d 6894	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
15	9, 12, 14	cbvmpt 5258	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
16	2, 15	eqtri 2758	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
17	16	fvmpt2 7008	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
18	8, 17	mpan2 687	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
19	18	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
20	4	ralrimiva 3144	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
21	9	nfel1 2917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
22	12	nfel1 2917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
23	14	eleq1d 2816	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
24	21, 22, 23	cbvralw 3301	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
25	20, 24	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
26	25	r19.21bi 3246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
27	19, 26	eqeltrd 2831	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
28	6, 7, 27	serfre 14001	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
29	1	feq1i 6707	. . . . . . 7 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
30	28, 29	sylibr 233	. . . . . 6 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
31	30	frnd 6724	. . . . 5 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
32		1nn 12227	. . . . . . . 8 ⊢ 1 ∈ ℕ
33	30	fdmd 6727	. . . . . . . 8 ⊢ (𝜑 → dom 𝑇 = ℕ)
34	32, 33	eleqtrrid 2838	. . . . . . 7 ⊢ (𝜑 → 1 ∈ dom 𝑇)
35	34	ne0d 4334	. . . . . 6 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
36		dm0rn0 5923	. . . . . . 7 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
37	36	necon3bii 2991	. . . . . 6 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
38	35, 37	sylib 217	. . . . 5 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
39		ovoliun2.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ dom ⇝ )
40	1, 39	eqeltrrid 2836	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
41	6, 7, 19, 26, 40	isumrecl 15715	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
42		elfznn 13534	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
43	42	adantl 480	. . . . . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
46	45, 6	eleqtrdi 2841	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
47		simpl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
48	47, 42, 26	syl2an 594	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
49	48	recnd 11246	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℂ)
50	44, 46, 49	fsumser 15680	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (seq1( + , 𝐺)‘𝑘))
51	1	fveq1i 6891	. . . . . . . . . 10 ⊢ (𝑇‘𝑘) = (seq1( + , 𝐺)‘𝑘)
52	50, 51	eqtr4di 2788	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (𝑇‘𝑘))
53		fzfid 13942	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ∈ Fin)
54		fz1ssnn 13536	. . . . . . . . . . . 12 ⊢ (1...𝑘) ⊆ ℕ
55	54	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ⊆ ℕ)
56	3	ralrimiva 3144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
57		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
58		nfcv 2901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛ℝ
59	11, 58	nfss 3973	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
60	13	sseq1d 4012	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
61	57, 59, 60	cbvralw 3301	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
62	56, 61	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
63	62	r19.21bi 3246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
64		ovolge0 25230	. . . . . . . . . . . 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 15795	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
67	66	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
68	52, 67	eqbrtrrd 5171	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
69	68	ralrimiva 3144	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
70		brralrspcev 5207	. . . . . . 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 582	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥)
72	30	ffnd 6717	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Fn ℕ)
73		breq1 5150	. . . . . . . . 9 ⊢ (𝑧 = (𝑇‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑇‘𝑘) ≤ 𝑥))
74	73	ralrn 7088	. . . . . . . 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 3176	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥))
77	71, 76	mpbird 256	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥)
78		supxrre 13310	. . . . 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 1369	. . . 4 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
80	6, 1, 7, 19, 26, 65, 71	isumsup 15797	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) = sup(ran 𝑇, ℝ, < ))
81	79, 80	eqtr4d 2773	. . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
82	9, 12, 14	cbvsumi 15647	. . 3 ⊢ Σ𝑛 ∈ ℕ (vol*‘𝐴) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)
83	81, 82	eqtr4di 2788	. 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑛 ∈ ℕ (vol*‘𝐴))
84	5, 83	breqtrd 5173	1 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3472  ⦋csb 3892   ⊆ wss 3947  ∅c0 4321  ∪ ciun 4996   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675  ran crn 5676   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411  supcsup 9437  ℝcr 11111  0cc0 11112  1c1 11113   + caddc 11115  ℝ^*cxr 11251   < clt 11252   ≤ cle 11253  ℕcn 12216  ℤ_≥cuz 12826  ...cfz 13488  seqcseq 13970   ⇝ cli 15432  Σcsu 15636  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:  ovoliunnul  25256  vitalilem5  25361