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Theorem ovolctb 23477
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 8157 . 2 (𝐴 ≈ ℕ → ℕ ≈ 𝐴)
2 bren 8117 . . . 4 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
3 simpll 742 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
4 f1of 6278 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
54adantl 467 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ⟶𝐴)
65ffvelrnda 6502 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ 𝐴)
73, 6sseldd 3751 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℝ)
87leidd 10795 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ≤ (𝑓𝑥))
9 df-br 4785 . . . . . . . . . . . . 13 ((𝑓𝑥) ≤ (𝑓𝑥) ↔ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
108, 9sylib 208 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
11 opelxpi 5288 . . . . . . . . . . . . 13 (((𝑓𝑥) ∈ ℝ ∧ (𝑓𝑥) ∈ ℝ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℝ × ℝ))
127, 7, 11syl2anc 565 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℝ × ℝ))
1310, 12elind 3947 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
14 df-ov 6795 . . . . . . . . . . . . 13 ((𝑓𝑥) I (𝑓𝑥)) = ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
15 opex 5060 . . . . . . . . . . . . . 14 ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V
16 fvi 6397 . . . . . . . . . . . . . 14 (⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V → ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
1715, 16ax-mp 5 . . . . . . . . . . . . 13 ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1814, 17eqtri 2792 . . . . . . . . . . . 12 ((𝑓𝑥) I (𝑓𝑥)) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1918mpteq2i 4873 . . . . . . . . . . 11 (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
2013, 19fmptd 6527 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
21 nnex 11227 . . . . . . . . . . . . 13 ℕ ∈ V
2221a1i 11 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ℕ ∈ V)
237recnd 10269 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℂ)
245feqmptd 6391 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓𝑥)))
2522, 23, 23, 24, 24offval2 7060 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))))
2625feq1d 6170 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((𝑓𝑓 I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ↔ (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ))))
2720, 26mpbird 247 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓𝑓 I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28 f1ofo 6285 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
2928adantl 467 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–onto𝐴)
30 forn 6259 . . . . . . . . . . . . . . 15 (𝑓:ℕ–onto𝐴 → ran 𝑓 = 𝐴)
3129, 30syl 17 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran 𝑓 = 𝐴)
3231eleq2d 2835 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦 ∈ ran 𝑓𝑦𝐴))
33 f1ofn 6279 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto𝐴𝑓 Fn ℕ)
3433adantl 467 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 Fn ℕ)
35 fvelrnb 6385 . . . . . . . . . . . . . 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 270 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦𝐴 ↔ ∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦))
3825, 19syl6eq 2820 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩))
3938fveq1d 6334 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((𝑓𝑓 I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥))
40 eqid 2770 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4140fvmpt2 6433 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℕ ∧ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4215, 41mpan2 663 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4339, 42sylan9eq 2824 . . . . . . . . . . . . . . . . . 18 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑓 I 𝑓)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4443fveq2d 6336 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
45 fvex 6342 . . . . . . . . . . . . . . . . . 18 (𝑓𝑥) ∈ V
4645, 45op1st 7322 . . . . . . . . . . . . . . . . 17 (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
4744, 46syl6eq 2820 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (𝑓𝑥))
4847, 8eqbrtrd 4806 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥))
4943fveq2d 6336 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
5045, 45op2nd 7323 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
5149, 50syl6eq 2820 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (𝑓𝑥))
528, 51breqtrrd 4812 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))
5348, 52jca 495 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ∧ (𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))))
54 breq2 4788 . . . . . . . . . . . . . . 15 ((𝑓𝑥) = 𝑦 → ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ↔ (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦))
55 breq1 4787 . . . . . . . . . . . . . . 15 ((𝑓𝑥) = 𝑦 → ((𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)) ↔ 𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))))
5654, 55anbi12d 608 . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑥) = 𝑦 → ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))))
5857reximdva 3164 . . . . . . . . . . . 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 230 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦𝐴 → ∃𝑥 ∈ ℕ ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))))
6059ralrimiv 3113 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ∀𝑦𝐴𝑥 ∈ ℕ ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))))
61 ovolficc 23455 . . . . . . . . . . 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 571 . . . . . . . . . 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 247 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝐴 ran ([,] ∘ (𝑓𝑓 I 𝑓)))
64 eqid 2770 . . . . . . . . . 10 seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)))
6564ovollb2 23476 . . . . . . . . 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 565 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ))
67 opelxpi 5288 . . . . . . . . . . . . . . . . . 18 (((𝑓𝑥) ∈ ℂ ∧ (𝑓𝑥) ∈ ℂ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℂ × ℂ))
6823, 23, 67syl2anc 565 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℂ × ℂ))
69 absf 14284 . . . . . . . . . . . . . . . . . . . 20 abs:ℂ⟶ℝ
70 subf 10484 . . . . . . . . . . . . . . . . . . . 20 − :(ℂ × ℂ)⟶ℂ
71 fco 6198 . . . . . . . . . . . . . . . . . . . 20 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7269, 70, 71mp2an 664 . . . . . . . . . . . . . . . . . . 19 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7372a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7473feqmptd 6391 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
75 fveq2 6332 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
76 df-ov 6795 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
7775, 76syl6eqr 2822 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)))
7868, 38, 74, 77fmptco 6538 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))))
79 cnmet 22794 . . . . . . . . . . . . . . . . . 18 (abs ∘ − ) ∈ (Met‘ℂ)
80 met0 22367 . . . . . . . . . . . . . . . . . 18 (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (𝑓𝑥) ∈ ℂ) → ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = 0)
8179, 23, 80sylancr 567 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = 0)
8281mpteq2dva 4876 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))) = (𝑥 ∈ ℕ ↦ 0))
8378, 82eqtrd 2804 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
84 fconstmpt 5303 . . . . . . . . . . . . . . 15 (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
8583, 84syl6eqr 2822 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)) = (ℕ × {0}))
8685seqeq3d 13015 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = seq1( + , (ℕ × {0})))
87 1z 11608 . . . . . . . . . . . . . 14 1 ∈ ℤ
88 nnuz 11924 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
8988ser0f 13060 . . . . . . . . . . . . . 14 (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
9087, 89ax-mp 5 . . . . . . . . . . . . 13 seq1( + , (ℕ × {0})) = (ℕ × {0})
9186, 90syl6eq 2820 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = (ℕ × {0}))
9291rneqd 5491 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = ran (ℕ × {0}))
93 1nn 11232 . . . . . . . . . . . 12 1 ∈ ℕ
94 ne0i 4067 . . . . . . . . . . . 12 (1 ∈ ℕ → ℕ ≠ ∅)
95 rnxp 5705 . . . . . . . . . . . 12 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
9693, 94, 95mp2b 10 . . . . . . . . . . 11 ran (ℕ × {0}) = {0}
9792, 96syl6eq 2820 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = {0})
9897supeq1d 8507 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
99 xrltso 12178 . . . . . . . . . 10 < Or ℝ*
100 0xr 10287 . . . . . . . . . 10 0 ∈ ℝ*
101 supsn 8533 . . . . . . . . . 10 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
10299, 100, 101mp2an 664 . . . . . . . . 9 sup({0}, ℝ*, < ) = 0
10398, 102syl6eq 2820 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ) = 0)
10466, 103breqtrd 4810 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ≤ 0)
105 ovolge0 23468 . . . . . . . 8 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
106105adantr 466 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 0 ≤ (vol*‘𝐴))
107 ovolcl 23465 . . . . . . . . 9 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
108107adantr 466 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ∈ ℝ*)
109 xrletri3 12189 . . . . . . . 8 (((vol*‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
110108, 100, 109sylancl 566 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
111104, 106, 110mpbir2and 684 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) = 0)
112111ex 397 . . . . 5 (𝐴 ⊆ ℝ → (𝑓:ℕ–1-1-onto𝐴 → (vol*‘𝐴) = 0))
113112exlimdv 2012 . . . 4 (𝐴 ⊆ ℝ → (∃𝑓 𝑓:ℕ–1-1-onto𝐴 → (vol*‘𝐴) = 0))
1142, 113syl5bi 232 . . 3 (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
115114imp 393 . 2 ((𝐴 ⊆ ℝ ∧ ℕ ≈ 𝐴) → (vol*‘𝐴) = 0)
1161, 115sylan2 572 1 ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wex 1851  wcel 2144  wne 2942  wral 3060  wrex 3061  Vcvv 3349  cin 3720  wss 3721  c0 4061  {csn 4314  cop 4320   cuni 4572   class class class wbr 4784  cmpt 4861   I cid 5156   Or wor 5169   × cxp 5247  ran crn 5250  ccom 5253   Fn wfn 6026  wf 6027  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  𝑓 cof 7041  1st c1st 7312  2nd c2nd 7313  cen 8105  supcsup 8501  cc 10135  cr 10136  0cc0 10137  1c1 10138   + caddc 10140  *cxr 10274   < clt 10275  cle 10276  cmin 10467  cn 11221  cz 11578  [,]cicc 12382  seqcseq 13007  abscabs 14181  Metcme 19946  vol*covol 23449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-inf 8504  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-q 11991  df-rp 12035  df-xadd 12151  df-ioo 12383  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-sum 14624  df-xmet 19953  df-met 19954  df-ovol 23451
This theorem is referenced by:  ovolq  23478  ovolctb2  23479  ovoliunnfl  33777
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