MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolctb Structured version   Visualization version   GIF version

Theorem ovolctb 24006
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 bren 8507 . . 3 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
2 simpll 763 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
3 f1of 6612 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
43adantl 482 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ⟶𝐴)
54ffvelrnda 6847 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ 𝐴)
62, 5sseldd 3972 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℝ)
76leidd 11195 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ≤ (𝑓𝑥))
8 df-br 5064 . . . . . . . . . . . 12 ((𝑓𝑥) ≤ (𝑓𝑥) ↔ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
97, 8sylib 219 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
106, 6opelxpd 5592 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℝ × ℝ))
119, 10elind 4175 . . . . . . . . . 10 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
12 df-ov 7151 . . . . . . . . . . . 12 ((𝑓𝑥) I (𝑓𝑥)) = ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
13 opex 5353 . . . . . . . . . . . . 13 ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V
14 fvi 6737 . . . . . . . . . . . . 13 (⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V → ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
1513, 14ax-mp 5 . . . . . . . . . . . 12 ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1612, 15eqtri 2849 . . . . . . . . . . 11 ((𝑓𝑥) I (𝑓𝑥)) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1716mpteq2i 5155 . . . . . . . . . 10 (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
1811, 17fmptd 6874 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19 nnex 11633 . . . . . . . . . . . 12 ℕ ∈ V
2019a1i 11 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ℕ ∈ V)
216recnd 10658 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℂ)
224feqmptd 6730 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓𝑥)))
2320, 21, 21, 22, 22offval2 7416 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓f I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))))
2423feq1d 6496 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((𝑓f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ↔ (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ))))
2518, 24mpbird 258 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
26 f1ofo 6619 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
2726adantl 482 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–onto𝐴)
28 forn 6590 . . . . . . . . . . . . . 14 (𝑓:ℕ–onto𝐴 → ran 𝑓 = 𝐴)
2927, 28syl 17 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran 𝑓 = 𝐴)
3029eleq2d 2903 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦 ∈ ran 𝑓𝑦𝐴))
31 f1ofn 6613 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-onto𝐴𝑓 Fn ℕ)
3231adantl 482 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 Fn ℕ)
33 fvelrnb 6723 . . . . . . . . . . . . 13 (𝑓 Fn ℕ → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦))
3432, 33syl 17 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦 ∈ ran 𝑓 ↔ ∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦))
3530, 34bitr3d 282 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦𝐴 ↔ ∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦))
3623, 17syl6eq 2877 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓f I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩))
3736fveq1d 6669 . . . . . . . . . . . . . . . . . 18 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((𝑓f I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥))
38 eqid 2826 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
3938fvmpt2 6775 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ ∧ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4013, 39mpan2 687 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4137, 40sylan9eq 2881 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓f I 𝑓)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4241fveq2d 6671 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓f I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
43 fvex 6680 . . . . . . . . . . . . . . . . 17 (𝑓𝑥) ∈ V
4443, 43op1st 7688 . . . . . . . . . . . . . . . 16 (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
4542, 44syl6eq 2877 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓f I 𝑓)‘𝑥)) = (𝑓𝑥))
4645, 7eqbrtrd 5085 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ (𝑓𝑥))
4741fveq2d 6671 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓f I 𝑓)‘𝑥)) = (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
4843, 43op2nd 7689 . . . . . . . . . . . . . . . 16 (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
4947, 48syl6eq 2877 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓f I 𝑓)‘𝑥)) = (𝑓𝑥))
507, 49breqtrrd 5091 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥)))
5146, 50jca 512 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ∧ (𝑓𝑥) ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥))))
52 breq2 5067 . . . . . . . . . . . . . 14 ((𝑓𝑥) = 𝑦 → ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ↔ (1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ 𝑦))
53 breq1 5066 . . . . . . . . . . . . . 14 ((𝑓𝑥) = 𝑦 → ((𝑓𝑥) ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥)) ↔ 𝑦 ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥))))
5452, 53anbi12d 630 . . . . . . . . . . . . 13 ((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)))))
5551, 54syl5ibcom 246 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑥) = 𝑦 → ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥)))))
5655reximdva 3279 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦 → ∃𝑥 ∈ ℕ ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥)))))
5735, 56sylbid 241 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦𝐴 → ∃𝑥 ∈ ℕ ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥)))))
5857ralrimiv 3186 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ∀𝑦𝐴𝑥 ∈ ℕ ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥))))
59 ovolficc 23984 . . . . . . . . . 10 ((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)))))
6025, 59syldan 591 . . . . . . . . 9 ((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)))))
6158, 60mpbird 258 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝐴 ran ([,] ∘ (𝑓f I 𝑓)))
62 eqid 2826 . . . . . . . . 9 seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓)))
6362ovollb2 24005 . . . . . . . 8 (((𝑓f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ran ([,] ∘ (𝑓f I 𝑓))) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))), ℝ*, < ))
6425, 61, 63syl2anc 584 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))), ℝ*, < ))
6521, 21opelxpd 5592 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℂ × ℂ))
66 absf 14687 . . . . . . . . . . . . . . . . . . 19 abs:ℂ⟶ℝ
67 subf 10877 . . . . . . . . . . . . . . . . . . 19 − :(ℂ × ℂ)⟶ℂ
68 fco 6528 . . . . . . . . . . . . . . . . . . 19 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
6966, 67, 68mp2an 688 . . . . . . . . . . . . . . . . . 18 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7069a1i 11 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7170feqmptd 6730 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
72 fveq2 6667 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
73 df-ov 7151 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
7472, 73syl6eqr 2879 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)))
7565, 36, 71, 74fmptco 6887 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓f I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))))
76 cnmet 23295 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) ∈ (Met‘ℂ)
77 met0 22868 . . . . . . . . . . . . . . . . 17 (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (𝑓𝑥) ∈ ℂ) → ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = 0)
7876, 21, 77sylancr 587 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = 0)
7978mpteq2dva 5158 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))) = (𝑥 ∈ ℕ ↦ 0))
8075, 79eqtrd 2861 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓f I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
81 fconstmpt 5613 . . . . . . . . . . . . . 14 (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
8280, 81syl6eqr 2879 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓f I 𝑓)) = (ℕ × {0}))
8382seqeq3d 13367 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = seq1( + , (ℕ × {0})))
84 1z 12001 . . . . . . . . . . . . 13 1 ∈ ℤ
85 nnuz 12270 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
8685ser0f 13413 . . . . . . . . . . . . 13 (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
8784, 86ax-mp 5 . . . . . . . . . . . 12 seq1( + , (ℕ × {0})) = (ℕ × {0})
8883, 87syl6eq 2877 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = (ℕ × {0}))
8988rneqd 5807 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = ran (ℕ × {0}))
90 1nn 11638 . . . . . . . . . . 11 1 ∈ ℕ
91 ne0i 4304 . . . . . . . . . . 11 (1 ∈ ℕ → ℕ ≠ ∅)
92 rnxp 6025 . . . . . . . . . . 11 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
9390, 91, 92mp2b 10 . . . . . . . . . 10 ran (ℕ × {0}) = {0}
9489, 93syl6eq 2877 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = {0})
9594supeq1d 8899 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
96 xrltso 12524 . . . . . . . . 9 < Or ℝ*
97 0xr 10677 . . . . . . . . 9 0 ∈ ℝ*
98 supsn 8925 . . . . . . . . 9 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
9996, 97, 98mp2an 688 . . . . . . . 8 sup({0}, ℝ*, < ) = 0
10095, 99syl6eq 2877 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))), ℝ*, < ) = 0)
10164, 100breqtrd 5089 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ≤ 0)
102 ovolge0 23997 . . . . . . 7 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
103102adantr 481 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 0 ≤ (vol*‘𝐴))
104 ovolcl 23994 . . . . . . . 8 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
105104adantr 481 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ∈ ℝ*)
106 xrletri3 12537 . . . . . . 7 (((vol*‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
107105, 97, 106sylancl 586 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((vol*‘𝐴) = 0 ↔ ((vol*‘𝐴) ≤ 0 ∧ 0 ≤ (vol*‘𝐴))))
108101, 103, 107mpbir2and 709 . . . . 5 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) = 0)
109108ex 413 . . . 4 (𝐴 ⊆ ℝ → (𝑓:ℕ–1-1-onto𝐴 → (vol*‘𝐴) = 0))
110109exlimdv 1927 . . 3 (𝐴 ⊆ ℝ → (∃𝑓 𝑓:ℕ–1-1-onto𝐴 → (vol*‘𝐴) = 0))
1111, 110syl5bi 243 . 2 (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
112 ensym 8547 . 2 (𝐴 ≈ ℕ → ℕ ≈ 𝐴)
113111, 112impel 506 1 ((𝐴 ⊆ ℝ ∧ 𝐴 ≈ ℕ) → (vol*‘𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wne 3021  wral 3143  wrex 3144  Vcvv 3500  cin 3939  wss 3940  c0 4295  {csn 4564  cop 4570   cuni 4837   class class class wbr 5063  cmpt 5143   I cid 5458   Or wor 5472   × cxp 5552  ran crn 5555  ccom 5558   Fn wfn 6347  wf 6348  ontowfo 6350  1-1-ontowf1o 6351  cfv 6352  (class class class)co 7148  f cof 7397  1st c1st 7678  2nd c2nd 7679  cen 8495  supcsup 8893  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529  *cxr 10663   < clt 10664  cle 10665  cmin 10859  cn 11627  cz 11970  [,]cicc 12731  seqcseq 13359  abscabs 14583  Metcmet 20447  vol*covol 23978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7399  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-q 12338  df-rp 12380  df-xadd 12498  df-ioo 12732  df-ico 12734  df-icc 12735  df-fz 12883  df-fzo 13024  df-seq 13360  df-exp 13420  df-hash 13681  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-clim 14835  df-sum 15033  df-xmet 20454  df-met 20455  df-ovol 23980
This theorem is referenced by:  ovolq  24007  ovolctb2  24008  ovoliunnfl  34801
  Copyright terms: Public domain W3C validator