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Theorem ovolctb 23239
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 7990 . 2 (𝐴 ≈ ℕ → ℕ ≈ 𝐴)
2 bren 7949 . . . 4 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
3 simpll 789 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
4 f1of 6124 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
54adantl 482 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ⟶𝐴)
65ffvelrnda 6345 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ 𝐴)
73, 6sseldd 3596 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℝ)
87leidd 10579 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ≤ (𝑓𝑥))
9 df-br 4645 . . . . . . . . . . . . 13 ((𝑓𝑥) ≤ (𝑓𝑥) ↔ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
108, 9sylib 208 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
11 opelxpi 5138 . . . . . . . . . . . . 13 (((𝑓𝑥) ∈ ℝ ∧ (𝑓𝑥) ∈ ℝ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℝ × ℝ))
127, 7, 11syl2anc 692 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℝ × ℝ))
1310, 12elind 3790 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
14 df-ov 6638 . . . . . . . . . . . . 13 ((𝑓𝑥) I (𝑓𝑥)) = ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
15 opex 4923 . . . . . . . . . . . . . 14 ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V
16 fvi 6242 . . . . . . . . . . . . . 14 (⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V → ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
1715, 16ax-mp 5 . . . . . . . . . . . . 13 ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1814, 17eqtri 2642 . . . . . . . . . . . 12 ((𝑓𝑥) I (𝑓𝑥)) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1918mpteq2i 4732 . . . . . . . . . . 11 (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
2013, 19fmptd 6371 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
21 nnex 11011 . . . . . . . . . . . . 13 ℕ ∈ V
2221a1i 11 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ℕ ∈ V)
237recnd 10053 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℂ)
245feqmptd 6236 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓𝑥)))
2522, 23, 23, 24, 24offval2 6899 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))))
2625feq1d 6017 . . . . . . . . . 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 6131 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
2928adantl 482 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–onto𝐴)
30 forn 6105 . . . . . . . . . . . . . . 15 (𝑓:ℕ–onto𝐴 → ran 𝑓 = 𝐴)
3129, 30syl 17 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran 𝑓 = 𝐴)
3231eleq2d 2685 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦 ∈ ran 𝑓𝑦𝐴))
33 f1ofn 6125 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto𝐴𝑓 Fn ℕ)
3433adantl 482 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 Fn ℕ)
35 fvelrnb 6230 . . . . . . . . . . . . . 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 2670 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓𝑓 I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩))
3938fveq1d 6180 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((𝑓𝑓 I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥))
40 eqid 2620 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4140fvmpt2 6278 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℕ ∧ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4215, 41mpan2 706 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4339, 42sylan9eq 2674 . . . . . . . . . . . . . . . . . 18 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑓 I 𝑓)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4443fveq2d 6182 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
45 fvex 6188 . . . . . . . . . . . . . . . . . 18 (𝑓𝑥) ∈ V
4645, 45op1st 7161 . . . . . . . . . . . . . . . . 17 (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
4744, 46syl6eq 2670 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (𝑓𝑥))
4847, 8eqbrtrd 4666 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥))
4943fveq2d 6182 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
5045, 45op2nd 7162 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
5149, 50syl6eq 2670 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)) = (𝑓𝑥))
528, 51breqtrrd 4672 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)))
5348, 52jca 554 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ∧ (𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))))
54 breq2 4648 . . . . . . . . . . . . . . 15 ((𝑓𝑥) = 𝑦 → ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ↔ (1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦))
55 breq1 4647 . . . . . . . . . . . . . . 15 ((𝑓𝑥) = 𝑦 → ((𝑓𝑥) ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥)) ↔ 𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))))
5654, 55anbi12d 746 . . . . . . . . . . . . . 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 3014 . . . . . . . . . . . 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 2962 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ∀𝑦𝐴𝑥 ∈ ℕ ((1st ‘((𝑓𝑓 I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓𝑓 I 𝑓)‘𝑥))))
61 ovolficc 23218 . . . . . . . . . . 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 487 . . . . . . . . . 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 2620 . . . . . . . . . 10 seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)))
6564ovollb2 23238 . . . . . . . . 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 692 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ))
67 opelxpi 5138 . . . . . . . . . . . . . . . . . 18 (((𝑓𝑥) ∈ ℂ ∧ (𝑓𝑥) ∈ ℂ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℂ × ℂ))
6823, 23, 67syl2anc 692 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℂ × ℂ))
69 absf 14058 . . . . . . . . . . . . . . . . . . . 20 abs:ℂ⟶ℝ
70 subf 10268 . . . . . . . . . . . . . . . . . . . 20 − :(ℂ × ℂ)⟶ℂ
71 fco 6045 . . . . . . . . . . . . . . . . . . . 20 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7269, 70, 71mp2an 707 . . . . . . . . . . . . . . . . . . 19 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7372a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7473feqmptd 6236 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
75 fveq2 6178 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
76 df-ov 6638 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
7775, 76syl6eqr 2672 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)))
7868, 38, 74, 77fmptco 6382 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))))
79 cnmet 22556 . . . . . . . . . . . . . . . . . 18 (abs ∘ − ) ∈ (Met‘ℂ)
80 met0 22129 . . . . . . . . . . . . . . . . . 18 (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (𝑓𝑥) ∈ ℂ) → ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = 0)
8179, 23, 80sylancr 694 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = 0)
8281mpteq2dva 4735 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))) = (𝑥 ∈ ℕ ↦ 0))
8378, 82eqtrd 2654 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
84 fconstmpt 5153 . . . . . . . . . . . . . . 15 (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
8583, 84syl6eqr 2672 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓)) = (ℕ × {0}))
8685seqeq3d 12792 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = seq1( + , (ℕ × {0})))
87 1z 11392 . . . . . . . . . . . . . 14 1 ∈ ℤ
88 nnuz 11708 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
8988ser0f 12837 . . . . . . . . . . . . . 14 (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
9087, 89ax-mp 5 . . . . . . . . . . . . 13 seq1( + , (ℕ × {0})) = (ℕ × {0})
9186, 90syl6eq 2670 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = (ℕ × {0}))
9291rneqd 5342 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = ran (ℕ × {0}))
93 1nn 11016 . . . . . . . . . . . 12 1 ∈ ℕ
94 ne0i 3913 . . . . . . . . . . . 12 (1 ∈ ℕ → ℕ ≠ ∅)
95 rnxp 5552 . . . . . . . . . . . 12 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
9693, 94, 95mp2b 10 . . . . . . . . . . 11 ran (ℕ × {0}) = {0}
9792, 96syl6eq 2670 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))) = {0})
9897supeq1d 8337 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
99 xrltso 11959 . . . . . . . . . 10 < Or ℝ*
100 0xr 10071 . . . . . . . . . 10 0 ∈ ℝ*
101 supsn 8363 . . . . . . . . . 10 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
10299, 100, 101mp2an 707 . . . . . . . . 9 sup({0}, ℝ*, < ) = 0
10398, 102syl6eq 2670 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓𝑓 I 𝑓))), ℝ*, < ) = 0)
10466, 103breqtrd 4670 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ≤ 0)
105 ovolge0 23230 . . . . . . . 8 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
106105adantr 481 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 0 ≤ (vol*‘𝐴))
107 ovolcl 23227 . . . . . . . . 9 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
108107adantr 481 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ∈ ℝ*)
109 xrletri3 11970 . . . . . . . 8 (((vol*‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
110108, 100, 109sylancl 693 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
111104, 106, 110mpbir2and 956 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) = 0)
112111ex 450 . . . . 5 (𝐴 ⊆ ℝ → (𝑓:ℕ–1-1-onto𝐴 → (vol*‘𝐴) = 0))
113112exlimdv 1859 . . . 4 (𝐴 ⊆ ℝ → (∃𝑓 𝑓:ℕ–1-1-onto𝐴 → (vol*‘𝐴) = 0))
1142, 113syl5bi 232 . . 3 (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
115114imp 445 . 2 ((𝐴 ⊆ ℝ ∧ ℕ ≈ 𝐴) → (vol*‘𝐴) = 0)
1161, 115sylan2 491 1 ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  wne 2791  wral 2909  wrex 2910  Vcvv 3195  cin 3566  wss 3567  c0 3907  {csn 4168  cop 4174   cuni 4427   class class class wbr 4644  cmpt 4720   I cid 5013   Or wor 5024   × cxp 5102  ran crn 5105  ccom 5108   Fn wfn 5871  wf 5872  ontowfo 5874  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  𝑓 cof 6880  1st c1st 7151  2nd c2nd 7152  cen 7937  supcsup 8331  cc 9919  cr 9920  0cc0 9921  1c1 9922   + caddc 9924  *cxr 10058   < clt 10059  cle 10060  cmin 10251  cn 11005  cz 11362  [,]cicc 12163  seqcseq 12784  abscabs 13955  Metcme 19713  vol*covol 23212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-q 11774  df-rp 11818  df-xadd 11932  df-ioo 12164  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398  df-xmet 19720  df-met 19721  df-ovol 23214
This theorem is referenced by:  ovolq  23240  ovolctb2  23241  ovoliunnfl  33422
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