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Theorem ovolctb 22977
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.
StepHypRef Expression
1 ensym 7863 . 2 (𝐴 ≈ ℕ → ℕ ≈ 𝐴)
2 bren 7822 . . . 4 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
3 simpll 785 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
4 f1of 6030 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
54adantl 480 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ⟶𝐴)
65ffvelrnda 6247 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ 𝐴)
73, 6sseldd 3563 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℝ)
87leidd 10438 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ≤ (𝑓𝑥))
9 df-br 4573 . . . . . . . . . . . . 13 ((𝑓𝑥) ≤ (𝑓𝑥) ↔ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
108, 9sylib 206 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
11 opelxpi 5057 . . . . . . . . . . . . 13 (((𝑓𝑥) ∈ ℝ ∧ (𝑓𝑥) ∈ ℝ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℝ × ℝ))
127, 7, 11syl2anc 690 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℝ × ℝ))
1310, 12elind 3754 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
14 df-ov 6525 . . . . . . . . . . . . 13 ((𝑓𝑥) I (𝑓𝑥)) = ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
15 opex 4848 . . . . . . . . . . . . . 14 ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V
16 fvi 6145 . . . . . . . . . . . . . 14 (⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V → ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
1715, 16ax-mp 5 . . . . . . . . . . . . 13 ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1814, 17eqtri 2626 . . . . . . . . . . . 12 ((𝑓𝑥) I (𝑓𝑥)) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1918mpteq2i 4658 . . . . . . . . . . 11 (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
2013, 19fmptd 6272 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
21 nnex 10868 . . . . . . . . . . . . 13 ℕ ∈ V
2221a1i 11 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ℕ ∈ V)
237recnd 9919 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℂ)
245feqmptd 6139 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓𝑥)))
2522, 23, 23, 24, 24offval2 6784 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))))
2625feq1d 5924 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((𝑓𝑓 I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ↔ (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ))))
2720, 26mpbird 245 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓𝑓 I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28 f1ofo 6037 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
2928adantl 480 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–onto𝐴)
30 forn 6011 . . . . . . . . . . . . . . 15 (𝑓:ℕ–onto𝐴 → ran 𝑓 = 𝐴)
3129, 30syl 17 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran 𝑓 = 𝐴)
3231eleq2d 2667 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦 ∈ ran 𝑓𝑦𝐴))
33 f1ofn 6031 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto𝐴𝑓 Fn ℕ)
3433adantl 480 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 Fn ℕ)
35 fvelrnb 6133 . . . . . . . . . . . . . 14 (𝑓 Fn ℕ → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦))
3634, 35syl 17 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦))
3732, 36bitr3d 268 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦𝐴 ↔ ∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦))
3825, 19syl6eq 2654 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩))
3938fveq1d 6085 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((𝑓𝑓 I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥))
40 eqid 2604 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4140fvmpt2 6180 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℕ ∧ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4215, 41mpan2 702 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4339, 42sylan9eq 2658 . . . . . . . . . . . . . . . . . 18 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑓 I 𝑓)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4443fveq2d 6087 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
45 fvex 6093 . . . . . . . . . . . . . . . . . 18 (𝑓𝑥) ∈ V
4645, 45op1st 7039 . . . . . . . . . . . . . . . . 17 (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
4744, 46syl6eq 2654 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (𝑓𝑥))
4847, 8eqbrtrd 4594 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥))
4943fveq2d 6087 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
5045, 45op2nd 7040 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
5149, 50syl6eq 2654 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (𝑓𝑥))
528, 51breqtrrd 4600 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))
5348, 52jca 552 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ∧ (𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))))
54 breq2 4576 . . . . . . . . . . . . . . 15 ((𝑓𝑥) = 𝑦 → ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ↔ (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦))
55 breq1 4575 . . . . . . . . . . . . . . 15 ((𝑓𝑥) = 𝑦 → ((𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)) ↔ 𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))))
5654, 55anbi12d 742 . . . . . . . . . . . . . 14 ((𝑓𝑥) = 𝑦 → (((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ∧ (𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))) ↔ ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))))
5753, 56syl5ibcom 233 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑥) = 𝑦 → ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))))
5857reximdva 2994 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦 → ∃𝑥 ∈ ℕ ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))))
5937, 58sylbid 228 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦𝐴 → ∃𝑥 ∈ ℕ ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))))
6059ralrimiv 2942 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ∀𝑦𝐴𝑥 ∈ ℕ ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))))
61 ovolficc 22956 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ (𝑓𝑓 I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ (𝑓𝑓 I 𝑓)) ↔ ∀𝑦𝐴𝑥 ∈ ℕ ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))))
6227, 61syldan 485 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝐴 ran ([,] ∘ (𝑓𝑓 I 𝑓)) ↔ ∀𝑦𝐴𝑥 ∈ ℕ ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))))
6360, 62mpbird 245 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝐴 ran ([,] ∘ (𝑓𝑓 I 𝑓)))
64 eqid 2604 . . . . . . . . . 10 seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)))
6564ovollb2 22976 . . . . . . . . 9 (((𝑓𝑓 I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ (𝑓𝑓 I 𝑓))) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ))
6627, 63, 65syl2anc 690 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ))
67 opelxpi 5057 . . . . . . . . . . . . . . . . . 18 (((𝑓𝑥) ∈ ℂ ∧ (𝑓𝑥) ∈ ℂ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℂ × ℂ))
6823, 23, 67syl2anc 690 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℂ × ℂ))
69 absf 13866 . . . . . . . . . . . . . . . . . . . 20 abs:ℂ⟶ℝ
70 subf 10129 . . . . . . . . . . . . . . . . . . . 20 − :(ℂ × ℂ)⟶ℂ
71 fco 5952 . . . . . . . . . . . . . . . . . . . 20 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7269, 70, 71mp2an 703 . . . . . . . . . . . . . . . . . . 19 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7372a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7473feqmptd 6139 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
75 fveq2 6083 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
76 df-ov 6525 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
7775, 76syl6eqr 2656 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)))
7868, 38, 74, 77fmptco 6283 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))))
79 cnmet 22312 . . . . . . . . . . . . . . . . . 18 (abs ∘ − ) ∈ (Met‘ℂ)
80 met0 21894 . . . . . . . . . . . . . . . . . 18 (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (𝑓𝑥) ∈ ℂ) → ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = 0)
8179, 23, 80sylancr 693 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = 0)
8281mpteq2dva 4661 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))) = (𝑥 ∈ ℕ ↦ 0))
8378, 82eqtrd 2638 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
84 fconstmpt 5070 . . . . . . . . . . . . . . 15 (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
8583, 84syl6eqr 2656 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)) = (ℕ × {0}))
8685seqeq3d 12621 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = seq1( + , (ℕ × {0})))
87 1z 11235 . . . . . . . . . . . . . 14 1 ∈ ℤ
88 nnuz 11550 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
8988ser0f 12666 . . . . . . . . . . . . . 14 (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
9087, 89ax-mp 5 . . . . . . . . . . . . 13 seq1( + , (ℕ × {0})) = (ℕ × {0})
9186, 90syl6eq 2654 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = (ℕ × {0}))
9291rneqd 5256 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = ran (ℕ × {0}))
93 1nn 10873 . . . . . . . . . . . 12 1 ∈ ℕ
94 ne0i 3874 . . . . . . . . . . . 12 (1 ∈ ℕ → ℕ ≠ ∅)
95 rnxp 5464 . . . . . . . . . . . 12 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
9693, 94, 95mp2b 10 . . . . . . . . . . 11 ran (ℕ × {0}) = {0}
9792, 96syl6eq 2654 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = {0})
9897supeq1d 8207 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
99 xrltso 11804 . . . . . . . . . 10 < Or ℝ*
100 0xr 9937 . . . . . . . . . 10 0 ∈ ℝ*
101 supsn 8233 . . . . . . . . . 10 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
10299, 100, 101mp2an 703 . . . . . . . . 9 sup({0}, ℝ*, < ) = 0
10398, 102syl6eq 2654 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ) = 0)
10466, 103breqtrd 4598 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ≤ 0)
105 ovolge0 22968 . . . . . . . 8 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
106105adantr 479 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 0 ≤ (vol*‘𝐴))
107 ovolcl 22965 . . . . . . . . 9 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
108107adantr 479 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ∈ ℝ*)
109 xrletri3 11815 . . . . . . . 8 (((vol*‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
110108, 100, 109sylancl 692 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
111104, 106, 110mpbir2and 958 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) = 0)
112111ex 448 . . . . 5 (𝐴 ⊆ ℝ → (𝑓:ℕ–1-1-onto𝐴 → (vol*‘𝐴) = 0))
113112exlimdv 1846 . . . 4 (𝐴 ⊆ ℝ → (∃𝑓 𝑓:ℕ–1-1-onto𝐴 → (vol*‘𝐴) = 0))
1142, 113syl5bi 230 . . 3 (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
115114imp 443 . 2 ((𝐴 ⊆ ℝ ∧ ℕ ≈ 𝐴) → (vol*‘𝐴) = 0)
1161, 115sylan2 489 1 ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1975  wne 2774  wral 2890  wrex 2891  Vcvv 3167  cin 3533  wss 3534  c0 3868  {csn 4119  cop 4125   cuni 4361   class class class wbr 4572  cmpt 4632   I cid 4933   Or wor 4943   × cxp 5021  ran crn 5024  ccom 5027   Fn wfn 5780  wf 5781  ontowfo 5783  1-1-ontowf1o 5784  cfv 5785  (class class class)co 6522  𝑓 cof 6765  1st c1st 7029  2nd c2nd 7030  cen 7810  supcsup 8201  cc 9785  cr 9786  0cc0 9787  1c1 9788   + caddc 9790  *cxr 9924   < clt 9925  cle 9926  cmin 10112  cn 10862  cz 11205  [,]cicc 12000  seqcseq 12613  abscabs 13763  Metcme 19494  vol*covol 22950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864  ax-pre-sup 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-of 6767  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423  df-er 7601  df-map 7718  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-sup 8203  df-inf 8204  df-oi 8270  df-card 8620  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-div 10529  df-nn 10863  df-2 10921  df-3 10922  df-n0 11135  df-z 11206  df-uz 11515  df-q 11616  df-rp 11660  df-xadd 11774  df-ioo 12001  df-ico 12003  df-icc 12004  df-fz 12148  df-fzo 12285  df-seq 12614  df-exp 12673  df-hash 12930  df-cj 13628  df-re 13629  df-im 13630  df-sqrt 13764  df-abs 13765  df-clim 14008  df-sum 14206  df-xmet 19501  df-met 19502  df-ovol 22952
This theorem is referenced by:  ovolq  22978  ovolctb2  22979  ovoliunnfl  32419
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