Theorem ovolctb 24936

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 8932	. . 3 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
2		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
3		f1of 6820	. . . . . . . . . . . . . . . 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	ffvelcdmda 7071	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ 𝐴)
6	2, 5	sseldd 3979	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℝ)
7	6	leidd 11762	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ≤ (𝑓‘𝑥))
8		df-br 5142	. . . . . . . . . . . 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 5707	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℝ × ℝ))
11	9, 10	elind 4190	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
12		df-ov 7396	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ( I ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
13		opex 5457	. . . . . . . . . . . . 13 ⊢ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V
14		fvi 6953	. . . . . . . . . . . . 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 2759	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩
17	16	mpteq2i 5246	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
18	11, 17	fmptd 7098	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19		nnex 12200	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ℕ ∈ V)
21	6	recnd 11224	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℂ)
22	4	feqmptd 6946	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓‘𝑥)))
23	20, 21, 21, 22, 22	offval2 7673	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘f I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))))
24	23	feq1d 6689	. . . . . . . . 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 6827	. . . . . . . . . . . . . . 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 6795	. . . . . . . . . . . . . 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 2818	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ 𝐴))
31		f1ofn 6821	. . . . . . . . . . . . . 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 6939	. . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘f I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
37	36	fveq1d 6880	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((𝑓 ∘f I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥))
38		eqid 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
39	38	fvmpt2 6995	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ ∧ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
40	13, 39	mpan2 689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
41	37, 40	sylan9eq 2791	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓 ∘f I 𝑓)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
42	41	fveq2d 6882	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
43		fvex 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑥) ∈ V
44	43, 43	op1st 7965	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
45	42, 44	eqtrdi 2787	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (𝑓‘𝑥))
46	45, 7	eqbrtrd 5163	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥))
47	41	fveq2d 6882	. . . . . . . . . . . . . . . 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 7966	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
49	47, 48	eqtrdi 2787	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (𝑓‘𝑥))
50	7, 49	breqtrrd 5169	. . . . . . . . . . . . . 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 5145	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥) = 𝑦 → ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥) ↔ (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦))
53		breq1 5144	. . . . . . . . . . . . . 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 3167	. . . . . . . . . . 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 3144	. . . . . . . . 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 24914	. . . . . . . . . 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 2731	. . . . . . . . 9 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)))
63	62	ovollb2 24935	. . . . . . . 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 5707	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℂ × ℂ))
66		absf 15266	. . . . . . . . . . . . . . . . . . 19 ⊢ abs:ℂ⟶ℝ
67		subf 11444	. . . . . . . . . . . . . . . . . . 19 ⊢ − :(ℂ × ℂ)⟶ℂ
68		fco 6728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
69	66, 67, 68	mp2an 690	. . . . . . . . . . . . . . . . . 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 6946	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
72		fveq2 6878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
73		df-ov 7396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
74	72, 73	eqtr4di 2789	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)))
75	65, 36, 71, 74	fmptco 7111	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))))
76		cnmet 24217	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
77		met0 23778	. . . . . . . . . . . . . . . . 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 5241	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ 0))
80	75, 79	eqtrd 2771	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
81		fconstmpt 5730	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
82	80, 81	eqtr4di 2789	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)) = (ℕ × {0}))
83	82	seqeq3d 13956	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , (ℕ × {0})))
84		1z 12574	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
85		nnuz 12847	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
86	85	ser0f 14003	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
87	84, 86	ax-mp 5	. . . . . . . . . . . 12 ⊢ seq1( + , (ℕ × {0})) = (ℕ × {0})
88	83, 87	eqtrdi 2787	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = (ℕ × {0}))
89	88	rneqd 5929	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = ran (ℕ × {0}))
90		1nn 12205	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
91		ne0i 4330	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
92		rnxp 6158	. . . . . . . . . . 11 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
93	90, 91, 92	mp2b 10	. . . . . . . . . 10 ⊢ ran (ℕ × {0}) = {0}
94	89, 93	eqtrdi 2787	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = {0})
95	94	supeq1d 9423	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
96		xrltso 13102	. . . . . . . . 9 ⊢ < Or ℝ*
97		0xr 11243	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
98		supsn 9449	. . . . . . . . 9 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
99	96, 97, 98	mp2an 690	. . . . . . . 8 ⊢ sup({0}, ℝ*, < ) = 0
100	95, 99	eqtrdi 2787	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ) = 0)
101	64, 100	breqtrd 5167	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ≤ 0)
102		ovolge0 24927	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
103	102	adantr 481	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 0 ≤ (vol*‘𝐴))
104		ovolcl 24924	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
105	104	adantr 481	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ∈ ℝ*)
106		xrletri3 13115	. . . . . . 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 711	. . . . 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	biimtrid 241	. 2 ⊢ (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
112		ensym 8982	. 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 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3473   ∩ cin 3943   ⊆ wss 3944  ∅c0 4318  {csn 4622  ⟨cop 4628  ∪ cuni 4901   class class class wbr 5141   ↦ cmpt 5224   I cid 5566   Or wor 5580   × cxp 5667  ran crn 5670   ∘ ccom 5673   Fn wfn 6527  ⟶wf 6528  –onto→wfo 6530  –1-1-onto→wf1o 6531  ‘cfv 6532  (class class class)co 7393   ∘_f cof 7651  1^st c1st 7955  2^nd c2nd 7956   ≈ cen 8919  supcsup 9417  ℂcc 11090  ℝcr 11091  0cc0 11092  1c1 11093   + caddc 11095  ℝ^*cxr 11229   < clt 11230   ≤ cle 11231   − cmin 11426  ℕcn 12194  ℤcz 12540  [,]cicc 13309  seqcseq 13948  abscabs 15163  Metcmet 20864  vol*covol 24908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-pre-sup 11170

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-of 7653  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-sup 9419  df-inf 9420  df-oi 9487  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-2 12257  df-3 12258  df-n0 12455  df-z 12541  df-uz 12805  df-q 12915  df-rp 12957  df-xadd 13075  df-ioo 13310  df-ico 13312  df-icc 13313  df-fz 13467  df-fzo 13610  df-seq 13949  df-exp 14010  df-hash 14273  df-cj 15028  df-re 15029  df-im 15030  df-sqrt 15164  df-abs 15165  df-clim 15414  df-sum 15615  df-xmet 20871  df-met 20872  df-ovol 24910

This theorem is referenced by:  ovolq  24937  ovolctb2  24938  ovoliunnfl  36334