Theorem ovoliun2 25461

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 25460.) (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 25460	. 2 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
6		nnuz 12788	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
7		1zzd 12520	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
8		fvex 6845	. . . . . . . . . . 11 ⊢ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V
9		nfcv 2896	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(vol*‘𝐴)
10		nfcv 2896	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
11		nfcsb1v 3871	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
12	10, 11	nffv 6842	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
13		csbeq1a 3861	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
14	13	fveq2d 6836	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
15	9, 12, 14	cbvmpt 5198	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
16	2, 15	eqtri 2757	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
17	16	fvmpt2 6950	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
18	8, 17	mpan2 691	. . . . . . . . . 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 3126	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
21	9	nfel1 2913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
22	12	nfel1 2913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
23	14	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
24	21, 22, 23	cbvralw 3276	. . . . . . . . . . 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 3226	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
27	19, 26	eqeltrd 2834	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
28	6, 7, 27	serfre 13952	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
29	1	feq1i 6651	. . . . . . 7 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
30	28, 29	sylibr 234	. . . . . 6 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
31	30	frnd 6668	. . . . 5 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
32		1nn 12154	. . . . . . . 8 ⊢ 1 ∈ ℕ
33	30	fdmd 6670	. . . . . . . 8 ⊢ (𝜑 → dom 𝑇 = ℕ)
34	32, 33	eleqtrrid 2841	. . . . . . 7 ⊢ (𝜑 → 1 ∈ dom 𝑇)
35	34	ne0d 4292	. . . . . 6 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
36		dm0rn0 5871	. . . . . . 7 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
37	36	necon3bii 2982	. . . . . 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 2839	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
41	6, 7, 19, 26, 40	isumrecl 15686	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
42		elfznn 13467	. . . . . . . . . . . . 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 2844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
47		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
48	47, 42, 26	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
49	48	recnd 11158	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℂ)
50	44, 46, 49	fsumser 15651	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (seq1( + , 𝐺)‘𝑘))
51	1	fveq1i 6833	. . . . . . . . . 10 ⊢ (𝑇‘𝑘) = (seq1( + , 𝐺)‘𝑘)
52	50, 51	eqtr4di 2787	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (𝑇‘𝑘))
53		fzfid 13894	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ∈ Fin)
54		fz1ssnn 13469	. . . . . . . . . . . 12 ⊢ (1...𝑘) ⊆ ℕ
55	54	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ⊆ ℕ)
56	3	ralrimiva 3126	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
57		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
58		nfcv 2896	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛ℝ
59	11, 58	nfss 3924	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
60	13	sseq1d 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
61	57, 59, 60	cbvralw 3276	. . . . . . . . . . . . . 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 3226	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
64		ovolge0 25436	. . . . . . . . . . . 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 15766	. . . . . . . . . 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 5120	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
69	68	ralrimiva 3126	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
70		brralrspcev 5156	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥)
72	30	ffnd 6661	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Fn ℕ)
73		breq1 5099	. . . . . . . . 9 ⊢ (𝑧 = (𝑇‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑇‘𝑘) ≤ 𝑥))
74	73	ralrn 7031	. . . . . . . 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 3158	. . . . . 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 13240	. . . . 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 1373	. . . 4 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
80	6, 1, 7, 19, 26, 65, 71	isumsup 15768	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) = sup(ran 𝑇, ℝ, < ))
81	79, 80	eqtr4d 2772	. . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
82	14, 9, 12	cbvsum 15616	. . 3 ⊢ Σ𝑛 ∈ ℕ (vol*‘𝐴) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)
83	81, 82	eqtr4di 2787	. 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑛 ∈ ℕ (vol*‘𝐴))
84	5, 83	breqtrd 5122	1 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  Vcvv 3438  ⦋csb 3847   ⊆ wss 3899  ∅c0 4283  ∪ ciun 4944   class class class wbr 5096   ↦ cmpt 5177  dom cdm 5622  ran crn 5623   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  supcsup 9341  ℝcr 11023  0cc0 11024  1c1 11025   + caddc 11027  ℝ^*cxr 11163   < clt 11164   ≤ cle 11165  ℕcn 12143  ℤ_≥cuz 12749  ...cfz 13421  seqcseq 13922   ⇝ cli 15405  Σcsu 15607  vol*covol 25417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-cc 10343  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-pm 8764  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-inf 9344  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-q 12860  df-rp 12904  df-ioo 13263  df-ico 13265  df-fz 13422  df-fzo 13569  df-fl 13710  df-seq 13923  df-exp 13983  df-hash 14252  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-clim 15409  df-rlim 15410  df-sum 15608  df-ovol 25419

This theorem is referenced by:  ovoliunnul  25462  vitalilem5  25567