Theorem ovolctb 23065

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 7891	. 2 ⊢ (𝐴 ≈ ℕ → ℕ ≈ 𝐴)
2		bren 7850	. . . 4 ⊢ (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
3		simpll 786	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
4		f1of 6050	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ⟶𝐴)
5	4	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ⟶𝐴)
6	5	ffvelrnda 6267	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ 𝐴)
7	3, 6	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℝ)
8	7	leidd 10473	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ≤ (𝑓‘𝑥))
9		df-br 4584	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑥) ≤ (𝑓‘𝑥) ↔ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ≤ )
10	8, 9	sylib 207	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ≤ )
11		opelxpi 5072	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑥) ∈ ℝ ∧ (𝑓‘𝑥) ∈ ℝ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℝ × ℝ))
12	7, 7, 11	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℝ × ℝ))
13	10, 12	elind 3760	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
14		df-ov 6552	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ( I ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
15		opex 4859	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V
16		fvi 6165	. . . . . . . . . . . . . 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 2632	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) I (𝑓‘𝑥)) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩
19	18	mpteq2i 4669	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
20	13, 19	fmptd 6292	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
21		nnex 10903	. . . . . . . . . . . . 13 ⊢ ℕ ∈ V
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ℕ ∈ V)
23	7	recnd 9947	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ∈ ℂ)
24	5	feqmptd 6159	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓‘𝑥)))
25	22, 23, 23, 24, 24	offval2 6812	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥) I (𝑓‘𝑥))))
26	25	feq1d 5943	. . . . . . . . . 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 246	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘𝑓 I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28		f1ofo 6057	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:ℕ–onto→𝐴)
29	28	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓:ℕ–onto→𝐴)
30		forn 6031	. . . . . . . . . . . . . . 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 2673	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ 𝐴))
33		f1ofn 6051	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓 Fn ℕ)
34	33	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝑓 Fn ℕ)
35		fvelrnb 6153	. . . . . . . . . . . . . 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 269	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ℕ (𝑓‘𝑥) = 𝑦))
38	25, 19	syl6eq 2660	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑓 ∘𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
39	38	fveq1d 6105	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((𝑓 ∘𝑓 I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥))
40		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
41	40	fvmpt2 6200	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℕ ∧ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
42	15, 41	mpan2 703	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
43	39, 42	sylan9eq 2664	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓 ∘𝑓 I 𝑓)‘𝑥) = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
44	43	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
45		fvex 6113	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑥) ∈ V
46	45, 45	op1st 7067	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
47	44, 46	syl6eq 2660	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) = (𝑓‘𝑥))
48	47, 8	eqbrtrd 4605	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥))
49	43	fveq2d 6107	. . . . . . . . . . . . . . . . 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 7068	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩) = (𝑓‘𝑥)
51	49, 50	syl6eq 2660	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) = (𝑓‘𝑥))
52	8, 51	breqtrrd 4611	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)))
53	48, 52	jca 553	. . . . . . . . . . . . . 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 4587	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥) = 𝑦 → ((1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) ≤ (𝑓‘𝑥) ↔ (1st ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) ≤ 𝑦))
55		breq1 4586	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥) = 𝑦 → ((𝑓‘𝑥) ≤ (2nd ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥)) ↔ 𝑦 ≤ (2nd ‘((𝑓 ∘𝑓 I 𝑓)‘𝑥))))
56	54, 55	anbi12d 743	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 3000	. . . . . . . . . . . 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 229	. . . . . . . . . . 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 2948	. . . . . . . . . 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 23044	. . . . . . . . . . 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 486	. . . . . . . . . 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 246	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 𝐴 ⊆ ∪ ran ([,] ∘ (𝑓 ∘𝑓 I 𝑓)))
64		eqid 2610	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓)))
65	64	ovollb2 23064	. . . . . . . . 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 691	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))), ℝ*, < ))
67		opelxpi 5072	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓‘𝑥) ∈ ℂ ∧ (𝑓‘𝑥) ∈ ℂ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℂ × ℂ))
68	23, 23, 67	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ ∈ (ℂ × ℂ))
69		absf 13925	. . . . . . . . . . . . . . . . . . . 20 ⊢ abs:ℂ⟶ℝ
70		subf 10162	. . . . . . . . . . . . . . . . . . . 20 ⊢ − :(ℂ × ℂ)⟶ℂ
71		fco 5971	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
72	69, 70, 71	mp2an 704	. . . . . . . . . . . . . . . . . . 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 6159	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
75		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩))
76		df-ov 6552	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = ((abs ∘ − )‘⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩)
77	75, 76	syl6eqr 2662	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨(𝑓‘𝑥), (𝑓‘𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)))
78	68, 38, 74, 77	fmptco 6303	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))))
79		cnmet 22385	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
80		met0 21958	. . . . . . . . . . . . . . . . . 18 ⊢ (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (𝑓‘𝑥) ∈ ℂ) → ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = 0)
81	79, 23, 80	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥)) = 0)
82	81	mpteq2dva 4672	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓‘𝑥)(abs ∘ − )(𝑓‘𝑥))) = (𝑥 ∈ ℕ ↦ 0))
83	78, 82	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
84		fconstmpt 5085	. . . . . . . . . . . . . . 15 ⊢ (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
85	83, 84	syl6eqr 2662	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓)) = (ℕ × {0}))
86	85	seqeq3d 12671	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = seq1( + , (ℕ × {0})))
87		1z 11284	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
88		nnuz 11599	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
89	88	ser0f 12716	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
90	87, 89	ax-mp 5	. . . . . . . . . . . . 13 ⊢ seq1( + , (ℕ × {0})) = (ℕ × {0})
91	86, 90	syl6eq 2660	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = (ℕ × {0}))
92	91	rneqd 5274	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = ran (ℕ × {0}))
93		1nn 10908	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
94		ne0i 3880	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
95		rnxp 5483	. . . . . . . . . . . 12 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
96	93, 94, 95	mp2b 10	. . . . . . . . . . 11 ⊢ ran (ℕ × {0}) = {0}
97	92, 96	syl6eq 2660	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))) = {0})
98	97	supeq1d 8235	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
99		xrltso 11850	. . . . . . . . . 10 ⊢ < Or ℝ*
100		0xr 9965	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
101		supsn 8261	. . . . . . . . . 10 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
102	99, 100, 101	mp2an 704	. . . . . . . . 9 ⊢ sup({0}, ℝ*, < ) = 0
103	98, 102	syl6eq 2660	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓 ∘𝑓 I 𝑓))), ℝ*, < ) = 0)
104	66, 103	breqtrd 4609	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ≤ 0)
105		ovolge0 23056	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
106	105	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → 0 ≤ (vol*‘𝐴))
107		ovolcl 23053	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
108	107	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) ∈ ℝ*)
109		xrletri3 11861	. . . . . . . 8 ⊢ (((vol*‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
110	108, 100, 109	sylancl 693	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
111	104, 106, 110	mpbir2and 959	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) → (vol*‘𝐴) = 0)
112	111	ex 449	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (𝑓:ℕ–1-1-onto→𝐴 → (vol*‘𝐴) = 0))
113	112	exlimdv 1848	. . . 4 ⊢ (𝐴 ⊆ ℝ → (∃𝑓 𝑓:ℕ–1-1-onto→𝐴 → (vol*‘𝐴) = 0))
114	2, 113	syl5bi 231	. . 3 ⊢ (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
115	114	imp 444	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ ℕ ≈ 𝐴) → (vol*‘𝐴) = 0)
116	1, 115	sylan2 490	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   Or wor 4958   × cxp 5036  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  1^st c1st 7057  2^nd c2nd 7058   ≈ cen 7838  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℤcz 11254  [,]cicc 12049  seqcseq 12663  abscabs 13822  Metcme 19553  vol*covol 23038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xadd 11823  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-xmet 19560  df-met 19561  df-ovol 23040

This theorem is referenced by:  ovolq  23066  ovolctb2  23067  ovoliunnfl  32621