Theorem ovolctb 25538

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 8993	. . 3 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
2		simpll 767	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
3		f1of 6848	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ⟶𝐴)
4	3	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ⟶𝐴)
5	4	ffvelcdmda 7103	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ 𝐴)
6	2, 5	sseldd 3995	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℝ)
7	6	leidd 11826	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ≤ (𝑓‘𝑥))
8		df-br 5148	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ≤ (𝑓‘𝑥) ↔ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ≤ )
9	7, 8	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ≤ )
10	6, 6	opelxpd 5727	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℝ × ℝ))
11	9, 10	elind 4209	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
12		df-ov 7433	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ( I ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
13		opex 5474	. . . . . . . . . . . . 13 ⊢ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V
14		fvi 6984	. . . . . . . . . . . . 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 2762	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩
17	16	mpteq2i 5252	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
18	11, 17	fmptd 7133	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19		nnex 12269	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ℕ ∈ V)
21	6	recnd 11286	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℂ)
22	4	feqmptd 6976	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓‘𝑥)))
23	20, 21, 21, 22, 22	offval2 7716	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘f I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))))
24	23	feq1d 6720	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
26		f1ofo 6855	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ–onto→𝐴)
27	26	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ–onto→𝐴)
28		forn 6823	. . . . . . . . . . . . . 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 6849	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓 Fn ℕ)
32	31	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓 Fn ℕ)
33		fvelrnb 6968	. . . . . . . . . . . . 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 281	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ℕ (𝑓‘𝑥) = 𝑦))
36	23, 17	eqtrdi 2790	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘f I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
37	36	fveq1d 6908	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((𝑓 ∘f I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥))
38		eqid 2734	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
39	38	fvmpt2 7026	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ ∧ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
40	13, 39	mpan2 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
41	37, 40	sylan9eq 2794	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓 ∘f I 𝑓)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
42	41	fveq2d 6910	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
43		fvex 6919	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑥) ∈ V
44	43, 43	op1st 8020	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
45	42, 44	eqtrdi 2790	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (𝑓‘𝑥))
46	45, 7	eqbrtrd 5169	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥))
47	41	fveq2d 6910	. . . . . . . . . . . . . . . 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 8021	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
49	47, 48	eqtrdi 2790	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)) = (𝑓‘𝑥))
50	7, 49	breqtrrd 5175	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)))
51	46, 50	jca 511	. . . . . . . . . . . . 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 5151	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥) = 𝑦 → ((1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥) ↔ (1st ‘((𝑓 ∘f I 𝑓)‘𝑥)) ≤ 𝑦))
53		breq1 5150	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑥) = 𝑦 → ((𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥)) ↔ 𝑦 ≤ (2nd ‘((𝑓 ∘f I 𝑓)‘𝑥))))
54	52, 53	anbi12d 632	. . . . . . . . . . . . 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 245	. . . . . . . . . . . 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 3165	. . . . . . . . . . 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 240	. . . . . . . . . 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 3142	. . . . . . . . 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 25516	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝐴 ⊆ ∪ ran ([,] ∘ (𝑓 ∘f I 𝑓)))
62		eqid 2734	. . . . . . . . 9 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)))
63	62	ovollb2 25537	. . . . . . . 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 5727	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℂ × ℂ))
66		absf 15372	. . . . . . . . . . . . . . . . . . 19 ⊢ abs:ℂ⟶ℝ
67		subf 11507	. . . . . . . . . . . . . . . . . . 19 ⊢ − :(ℂ × ℂ)⟶ℂ
68		fco 6760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
69	66, 67, 68	mp2an 692	. . . . . . . . . . . . . . . . . 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 6976	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
72		fveq2 6906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
73		df-ov 7433	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
74	72, 73	eqtr4di 2792	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)))
75	65, 36, 71, 74	fmptco 7148	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))))
76		cnmet 24807	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
77		met0 24368	. . . . . . . . . . . . . . . . 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 5247	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ 0))
80	75, 79	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
81		fconstmpt 5750	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
82	80, 81	eqtr4di 2792	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓)) = (ℕ × {0}))
83	82	seqeq3d 14046	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , (ℕ × {0})))
84		1z 12644	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
85		nnuz 12918	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ≥‘1)
86	85	ser0f 14092	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
87	84, 86	ax-mp 5	. . . . . . . . . . . 12 ⊢ seq1( + , (ℕ × {0})) = (ℕ × {0})
88	83, 87	eqtrdi 2790	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = (ℕ × {0}))
89	88	rneqd 5951	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = ran (ℕ × {0}))
90		1nn 12274	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
91		ne0i 4346	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
92		rnxp 6191	. . . . . . . . . . 11 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
93	90, 91, 92	mp2b 10	. . . . . . . . . 10 ⊢ ran (ℕ × {0}) = {0}
94	89, 93	eqtrdi 2790	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))) = {0})
95	94	supeq1d 9483	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
96		xrltso 13179	. . . . . . . . 9 ⊢ < Or ℝ*
97		0xr 11305	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
98		supsn 9509	. . . . . . . . 9 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
99	96, 97, 98	mp2an 692	. . . . . . . 8 ⊢ sup({0}, ℝ*, < ) = 0
100	95, 99	eqtrdi 2790	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ) = 0)
101	64, 100	breqtrd 5173	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ≤ 0)
102		ovolge0 25529	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
103	102	adantr 480	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 0 ≤ (vol*‘𝐴))
104		ovolcl 25526	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
105	104	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ∈ ℝ*)
106		xrletri3 13192	. . . . . . 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 713	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) = 0)
109	108	ex 412	. . . 4 ⊢ (𝐴 ⊆ ℝ → (𝑓:ℕ–1-1-onto→𝐴 → (vol*‘𝐴) = 0))
110	109	exlimdv 1930	. . 3 ⊢ (𝐴 ⊆ ℝ → (∃𝑓 𝑓:ℕ–1-1-onto→𝐴 → (vol*‘𝐴) = 0))
111	1, 110	biimtrid 242	. 2 ⊢ (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
112		ensym 9041	. 2 ⊢ (𝐴 ≈ ℕ → ℕ ≈ 𝐴)
113	111, 112	impel 505	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3477   ∩ cin 3961   ⊆ wss 3962  ∅c0 4338  {csn 4630  ⟨cop 4636  ∪ cuni 4911   class class class wbr 5147   ↦ cmpt 5230   I cid 5581   Or wor 5595   × cxp 5686  ran crn 5689   ∘ ccom 5692   Fn wfn 6557  ⟶wf 6558  –onto→wfo 6560  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430   ∘_f cof 7694  1^st c1st 8010  2^nd c2nd 8011   ≈ cen 8980  supcsup 9477  ℂcc 11150  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155  ℝ^*cxr 11291   < clt 11292   ≤ cle 11293   − cmin 11489  ℕcn 12263  ℤcz 12610  [,]cicc 13386  seqcseq 14038  abscabs 15269  Metcmet 21367  vol*covol 25510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xadd 13152  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-xmet 21374  df-met 21375  df-ovol 25512

This theorem is referenced by:  ovolq  25539  ovolctb2  25540  ovoliunnfl  37648