Theorem ovolctb 24654

Description: The volume of a denumerable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)

Assertion

Ref	Expression
ovolctb	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)

Proof of Theorem ovolctb

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8743	. . 3 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
2		simpll 764	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
3		f1of 6716	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ⟶𝐴)
4	3	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ⟶𝐴)
5	4	ffvelrnda 6961	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ 𝐴)
6	2, 5	sseldd 3922	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℝ)
7	6	leidd 11541	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ≤ (𝑓‘𝑥))
8		df-br 5075	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ≤ (𝑓‘𝑥) ↔ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ≤ )
9	7, 8	sylib 217	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ≤ )
10	6, 6	opelxpd 5627	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℝ × ℝ))
11	9, 10	elind 4128	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
12		df-ov 7278	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ( I ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
13		opex 5379	. . . . . . . . . . . . 13 ⊢ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V
14		fvi 6844	. . . . . . . . . . . . 13 ⊢ (⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V → ( I ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
15	13, 14	ax-mp 5	. . . . . . . . . . . 12 ⊢ ( I ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩
16	12, 15	eqtri 2766	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩
17	16	mpteq2i 5179	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
18	11, 17	fmptd 6988	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19		nnex 11979	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ℕ ∈ V)
21	6	recnd 11003	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℂ)
22	4	feqmptd 6837	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓‘𝑥)))
23	20, 21, 21, 22, 22	offval2 7553	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘f I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))))
24	23	feq1d 6585	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((𝑓 ∘f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ↔ (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ))))
25	18, 24	mpbird 256	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
26		f1ofo 6723	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ–onto→𝐴)
27	26	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ–onto→𝐴)
28		forn 6691	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–onto→𝐴 → ran 𝑓 = 𝐴)
29	27, 28	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran 𝑓 = 𝐴)
30	29	eleq2d 2824	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ 𝐴))
31		f1ofn 6717	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓 Fn ℕ)
32	31	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓 Fn ℕ)
33		fvelrnb 6830	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn ℕ → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ ℕ (𝑓‘𝑥) = 𝑦))
34	32, 33	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ ℕ (𝑓‘𝑥) = 𝑦))
35	30, 34	bitr3d 280	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ℕ (𝑓‘𝑥) = 𝑦))
36	23, 17	eqtrdi 2794	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘f I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
37	36	fveq1d 6776	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((𝑓 ∘f I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥))
38		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
39	38	fvmpt2 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ ∧ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
40	13, 39	mpan2 688	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
41	37, 40	sylan9eq 2798	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓 ∘f I 𝑓)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
42	41	fveq2d 6778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
43		fvex 6787	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑥) ∈ V
44	43, 43	op1st 7839	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
45	42, 44	eqtrdi 2794	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (𝑓‘𝑥))
46	45, 7	eqbrtrd 5096	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥))
47	41	fveq2d 6778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (2nd ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
48	43, 43	op2nd 7840	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
49	47, 48	eqtrdi 2794	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (𝑓‘𝑥))
50	7, 49	breqtrrd 5102	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)))
51	46, 50	jca 512	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥))))
52		breq2 5078	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥) = 𝑦 → ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥) ↔ (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦))
53		breq1 5077	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥) = 𝑦 → ((𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)) ↔ 𝑦 ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥))))
54	52, 53	anbi12d 631	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑥) = 𝑦 → (((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥))) ↔ ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦 ∧ 𝑦 ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)))))
55	51, 54	syl5ibcom 244	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓‘𝑥) = 𝑦 → ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦 ∧ 𝑦 ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)))))
56	55	reximdva 3203	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (∃𝑥 ∈ ℕ (𝑓‘𝑥) = 𝑦 → ∃𝑥 ∈ ℕ ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦 ∧ 𝑦 ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)))))
57	35, 56	sylbid 239	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ 𝐴 → ∃𝑥 ∈ ℕ ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦 ∧ 𝑦 ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)))))
58	57	ralrimiv 3102	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ℕ ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦 ∧ 𝑦 ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥))))
59		ovolficc 24632	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑓 ∘f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ([,] ∘ (𝑓 ∘f I 𝑓)) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ℕ ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦 ∧ 𝑦 ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)))))
60	25, 59	syldan 591	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝐴 ⊆ ∪ ran ([,] ∘ (𝑓 ∘f I 𝑓)) ↔ ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ ℕ ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦 ∧ 𝑦 ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)))))
61	58, 60	mpbird 256	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝐴 ⊆ ∪ ran ([,] ∘ (𝑓 ∘f I 𝑓)))
62		eqid 2738	. . . . . . . . 9 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)))
63	62	ovollb2 24653	. . . . . . . 8 ⊢ (((𝑓 ∘f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ (𝑓 ∘f I 𝑓))) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ))
64	25, 61, 63	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ))
65	21, 21	opelxpd 5627	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℂ × ℂ))
66		absf 15049	. . . . . . . . . . . . . . . . . . 19 ⊢ abs:ℂ⟶ℝ
67		subf 11223	. . . . . . . . . . . . . . . . . . 19 ⊢ − :(ℂ × ℂ)⟶ℂ
68		fco 6624	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
69	66, 67, 68	mp2an 689	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
70	69	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
71	70	feqmptd 6837	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
72		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
73		df-ov 7278	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
74	72, 73	eqtr4di 2796	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)))
75	65, 36, 71, 74	fmptco 7001	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))))
76		cnmet 23935	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
77		met0 23496	. . . . . . . . . . . . . . . . 17 ⊢ (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (𝑓‘𝑥) ∈ ℂ) → ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = 0)
78	76, 21, 77	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = 0)
79	78	mpteq2dva 5174	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ 0))
80	75, 79	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
81		fconstmpt 5649	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
82	80, 81	eqtr4di 2796	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)) = (ℕ × {0}))
83	82	seqeq3d 13729	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , (ℕ × {0})))
84		1z 12350	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
85		nnuz 12621	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
86	85	ser0f 13776	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
87	84, 86	ax-mp 5	. . . . . . . . . . . 12 ⊢ seq1( + , (ℕ × {0})) = (ℕ × {0})
88	83, 87	eqtrdi 2794	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = (ℕ × {0}))
89	88	rneqd 5847	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = ran (ℕ × {0}))
90		1nn 11984	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
91		ne0i 4268	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
92		rnxp 6073	. . . . . . . . . . 11 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
93	90, 91, 92	mp2b 10	. . . . . . . . . 10 ⊢ ran (ℕ × {0}) = {0}
94	89, 93	eqtrdi 2794	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = {0})
95	94	supeq1d 9205	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
96		xrltso 12875	. . . . . . . . 9 ⊢ < Or ℝ*
97		0xr 11022	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
98		supsn 9231	. . . . . . . . 9 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
99	96, 97, 98	mp2an 689	. . . . . . . 8 ⊢ sup({0}, ℝ*, < ) = 0
100	95, 99	eqtrdi 2794	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ) = 0)
101	64, 100	breqtrd 5100	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ≤ 0)
102		ovolge0 24645	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
103	102	adantr 481	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 0 ≤ (vol*‘𝐴))
104		ovolcl 24642	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
105	104	adantr 481	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ∈ ℝ*)
106		xrletri3 12888	. . . . . . 7 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
107	105, 97, 106	sylancl 586	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
108	101, 103, 107	mpbir2and 710	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) = 0)
109	108	ex 413	. . . 4 ⊢ (𝐴 ⊆ ℝ → (𝑓:ℕ–1-1-onto→𝐴 → (vol*‘𝐴) = 0))
110	109	exlimdv 1936	. . 3 ⊢ (𝐴 ⊆ ℝ → (∃𝑓 𝑓:ℕ–1-1-onto→𝐴 → (vol*‘𝐴) = 0))
111	1, 110	syl5bi 241	. 2 ⊢ (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
112		ensym 8789	. 2 ⊢ (𝐴 ≈ ℕ → ℕ ≈ 𝐴)
113	111, 112	impel 506	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561  ⟨cop 4567  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157   I cid 5488   Or wor 5502   × cxp 5587  ran crn 5590   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  –onto→wfo 6431  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ∘_f cof 7531  1^st c1st 7829  2^nd c2nd 7830   ≈ cen 8730  supcsup 9199  ℂcc 10869  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010   − cmin 11205  ℕcn 11973  ℤcz 12319  [,]cicc 13082  seqcseq 13721  abscabs 14945  Metcmet 20583  vol*covol 24626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xadd 12849  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-xmet 20590  df-met 20591  df-ovol 24628

This theorem is referenced by:  ovolq  24655  ovolctb2  24656  ovoliunnfl  35819