Theorem ovolctb 23258

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		ensym 8005	. 2 ⊢ (𝐴 ≈ ℕ → ℕ ≈ 𝐴)
2		bren 7964	. . . 4 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
3		simpll 790	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
4		f1of 6137	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ⟶𝐴)
5	4	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ⟶𝐴)
6	5	ffvelrnda 6359	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ 𝐴)
7	3, 6	sseldd 3604	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℝ)
8	7	leidd 10594	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ≤ (𝑓‘𝑥))
9		df-br 4654	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑥) ≤ (𝑓‘𝑥) ↔ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ≤ )
10	8, 9	sylib 208	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ≤ )
11		opelxpi 5148	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑥) ∈ ℝ ∧ (𝑓‘𝑥) ∈ ℝ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℝ × ℝ))
12	7, 7, 11	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℝ × ℝ))
13	10, 12	elind 3798	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
14		df-ov 6653	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ( I ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
15		opex 4932	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V
16		fvi 6255	. . . . . . . . . . . . . 14 ⊢ (⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V → ( I ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ( I ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩
18	14, 17	eqtri 2644	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩
19	18	mpteq2i 4741	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
20	13, 19	fmptd 6385	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
21		nnex 11026	. . . . . . . . . . . . 13 ⊢ ℕ ∈ V
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ℕ ∈ V)
23	7	recnd 10068	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℂ)
24	5	feqmptd 6249	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓‘𝑥)))
25	22, 23, 23, 24, 24	offval2 6914	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))))
26	25	feq1d 6030	. . . . . . . . . 10 ⊢ ((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))))
27	20, 26	mpbird 247	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘𝑓 I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28		f1ofo 6144	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ–onto→𝐴)
29	28	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ–onto→𝐴)
30		forn 6118	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝐴 → ran 𝑓 = 𝐴)
31	29, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran 𝑓 = 𝐴)
32	31	eleq2d 2687	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ 𝐴))
33		f1ofn 6138	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓 Fn ℕ)
34	33	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓 Fn ℕ)
35		fvelrnb 6243	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn ℕ → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ ℕ (𝑓‘𝑥) = 𝑦))
36	34, 35	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ ℕ (𝑓‘𝑥) = 𝑦))
37	32, 36	bitr3d 270	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ℕ (𝑓‘𝑥) = 𝑦))
38	25, 19	syl6eq 2672	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
39	38	fveq1d 6193	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((𝑓 ∘𝑓 I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥))
40		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
41	40	fvmpt2 6291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
42	15, 41	mpan2 707	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
43	39, 42	sylan9eq 2676	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓 ∘𝑓 I 𝑓)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
44	43	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
45		fvex 6201	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑥) ∈ V
46	45, 45	op1st 7176	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
47	44, 46	syl6eq 2672	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) = (𝑓‘𝑥))
48	47, 8	eqbrtrd 4675	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥))
49	43	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) = (2nd ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
50	45, 45	op2nd 7177	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
51	49, 50	syl6eq 2672	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) = (𝑓‘𝑥))
52	8, 51	breqtrrd 4681	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)))
53	48, 52	jca 554	. . . . . . . . . . . . . 14 ⊢ (((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))))
54		breq2 4657	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥) = 𝑦 → ((1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥) ↔ (1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) ≤ 𝑦))
55		breq1 4656	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥) = 𝑦 → ((𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) ↔ 𝑦 ≤ (2nd ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥))))
56	54, 55	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ ((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)))))
57	53, 56	syl5ibcom 235	. . . . . . . . . . . . 13 ⊢ (((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)))))
58	57	reximdva 3017	. . . . . . . . . . . 12 ⊢ ((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)))))
59	37, 58	sylbid 230	. . . . . . . . . . 11 ⊢ ((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)))))
60	59	ralrimiv 2965	. . . . . . . . . 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))))
61		ovolficc 23237	. . . . . . . . . . 11 ⊢ ((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)))))
62	27, 61	syldan 487	. . . . . . . . . 10 ⊢ ((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)))))
63	60, 62	mpbird 247	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝐴 ⊆ ∪ ran ([,] ∘ (𝑓 ∘𝑓 I 𝑓)))
64		eqid 2622	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓)))
65	64	ovollb2 23257	. . . . . . . . 9 ⊢ (((𝑓 ∘𝑓 I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ (𝑓 ∘𝑓 I 𝑓))) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))), ℝ*, < ))
66	27, 63, 65	syl2anc 693	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))), ℝ*, < ))
67		opelxpi 5148	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓‘𝑥) ∈ ℂ ∧ (𝑓‘𝑥) ∈ ℂ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℂ × ℂ))
68	23, 23, 67	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℂ × ℂ))
69		absf 14077	. . . . . . . . . . . . . . . . . . . 20 ⊢ abs:ℂ⟶ℝ
70		subf 10283	. . . . . . . . . . . . . . . . . . . 20 ⊢ − :(ℂ × ℂ)⟶ℂ
71		fco 6058	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
72	69, 70, 71	mp2an 708	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
73	72	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
74	73	feqmptd 6249	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
75		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
76		df-ov 6653	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
77	75, 76	syl6eqr 2674	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)))
78	68, 38, 74, 77	fmptco 6396	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))))
79		cnmet 22575	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
80		met0 22148	. . . . . . . . . . . . . . . . . 18 ⊢ (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (𝑓‘𝑥) ∈ ℂ) → ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = 0)
81	79, 23, 80	sylancr 695	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = 0)
82	81	mpteq2dva 4744	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ 0))
83	78, 82	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
84		fconstmpt 5163	. . . . . . . . . . . . . . 15 ⊢ (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
85	83, 84	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓)) = (ℕ × {0}))
86	85	seqeq3d 12809	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = seq1( + , (ℕ × {0})))
87		1z 11407	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
88		nnuz 11723	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
89	88	ser0f 12854	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
90	87, 89	ax-mp 5	. . . . . . . . . . . . 13 ⊢ seq1( + , (ℕ × {0})) = (ℕ × {0})
91	86, 90	syl6eq 2672	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = (ℕ × {0}))
92	91	rneqd 5353	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = ran (ℕ × {0}))
93		1nn 11031	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
94		ne0i 3921	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
95		rnxp 5564	. . . . . . . . . . . 12 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
96	93, 94, 95	mp2b 10	. . . . . . . . . . 11 ⊢ ran (ℕ × {0}) = {0}
97	92, 96	syl6eq 2672	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = {0})
98	97	supeq1d 8352	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
99		xrltso 11974	. . . . . . . . . 10 ⊢ < Or ℝ*
100		0xr 10086	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
101		supsn 8378	. . . . . . . . . 10 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
102	99, 100, 101	mp2an 708	. . . . . . . . 9 ⊢ sup({0}, ℝ*, < ) = 0
103	98, 102	syl6eq 2672	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))), ℝ*, < ) = 0)
104	66, 103	breqtrd 4679	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ≤ 0)
105		ovolge0 23249	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
106	105	adantr 481	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 0 ≤ (vol*‘𝐴))
107		ovolcl 23246	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
108	107	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ∈ ℝ*)
109		xrletri3 11985	. . . . . . . 8 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
110	108, 100, 109	sylancl 694	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
111	104, 106, 110	mpbir2and 957	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) = 0)
112	111	ex 450	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (𝑓:ℕ–1-1-onto→𝐴 → (vol*‘𝐴) = 0))
113	112	exlimdv 1861	. . . 4 ⊢ (𝐴 ⊆ ℝ → (∃𝑓 𝑓:ℕ–1-1-onto→𝐴 → (vol*‘𝐴) = 0))
114	2, 113	syl5bi 232	. . 3 ⊢ (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
115	114	imp 445	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ ℕ ≈ 𝐴) → (vol*‘𝐴) = 0)
116	1, 115	sylan2 491	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   I cid 5023   Or wor 5034   × cxp 5112  ran crn 5115   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  1^st c1st 7166  2^nd c2nd 7167   ≈ cen 7952  supcsup 8346  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℤcz 11377  [,]cicc 12178  seqcseq 12801  abscabs 13974  Metcme 19732  vol*covol 23231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233

This theorem is referenced by:  ovolq  23259  ovolctb2  23260  ovoliunnfl  33451