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

Theorem ovolctb 24936
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 8932 . . 3 (ℕ ≈ 𝐴 ↔ ∃𝑓 𝑓:ℕ–1-1-onto𝐴)
2 simpll 765 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → 𝐴 ⊆ ℝ)
3 f1of 6820 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ⟶𝐴)
43adantl 482 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ⟶𝐴)
54ffvelcdmda 7071 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ 𝐴)
62, 5sseldd 3979 . . . . . . . . . . . . 13 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℝ)
76leidd 11762 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ≤ (𝑓𝑥))
8 df-br 5142 . . . . . . . . . . . 12 ((𝑓𝑥) ≤ (𝑓𝑥) ↔ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
97, 8sylib 217 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ≤ )
106, 6opelxpd 5707 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℝ × ℝ))
119, 10elind 4190 . . . . . . . . . 10 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
12 df-ov 7396 . . . . . . . . . . . 12 ((𝑓𝑥) I (𝑓𝑥)) = ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
13 opex 5457 . . . . . . . . . . . . 13 ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V
14 fvi 6953 . . . . . . . . . . . . 13 (⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V → ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
1513, 14ax-mp 5 . . . . . . . . . . . 12 ( I ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1612, 15eqtri 2759 . . . . . . . . . . 11 ((𝑓𝑥) I (𝑓𝑥)) = ⟨(𝑓𝑥), (𝑓𝑥)⟩
1716mpteq2i 5246 . . . . . . . . . 10 (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
1811, 17fmptd 7098 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19 nnex 12200 . . . . . . . . . . . 12 ℕ ∈ V
2019a1i 11 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ℕ ∈ V)
216recnd 11224 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (𝑓𝑥) ∈ ℂ)
224feqmptd 6946 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 = (𝑥 ∈ ℕ ↦ (𝑓𝑥)))
2320, 21, 21, 22, 22offval2 7673 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓f I 𝑓) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))))
2423feq1d 6689 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((𝑓f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ↔ (𝑥 ∈ ℕ ↦ ((𝑓𝑥) I (𝑓𝑥))):ℕ⟶( ≤ ∩ (ℝ × ℝ))))
2518, 24mpbird 256 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓f I 𝑓):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
26 f1ofo 6827 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto𝐴𝑓:ℕ–onto𝐴)
2726adantl 482 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓:ℕ–onto𝐴)
28 forn 6795 . . . . . . . . . . . . . 14 (𝑓:ℕ–onto𝐴 → ran 𝑓 = 𝐴)
2927, 28syl 17 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran 𝑓 = 𝐴)
3029eleq2d 2818 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦 ∈ ran 𝑓𝑦𝐴))
31 f1ofn 6821 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-onto𝐴𝑓 Fn ℕ)
3231adantl 482 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝑓 Fn ℕ)
33 fvelrnb 6939 . . . . . . . . . . . . 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 280 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦𝐴 ↔ ∃𝑥 ∈ ℕ (𝑓𝑥) = 𝑦))
3623, 17eqtrdi 2787 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑓f I 𝑓) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩))
3736fveq1d 6880 . . . . . . . . . . . . . . . . . 18 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((𝑓f I 𝑓)‘𝑥) = ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥))
38 eqid 2731 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)
3938fvmpt2 6995 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ ∧ ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ V) → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4013, 39mpan2 689 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℕ → ((𝑥 ∈ ℕ ↦ ⟨(𝑓𝑥), (𝑓𝑥)⟩)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4137, 40sylan9eq 2791 . . . . . . . . . . . . . . . . 17 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓f I 𝑓)‘𝑥) = ⟨(𝑓𝑥), (𝑓𝑥)⟩)
4241fveq2d 6882 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓f I 𝑓)‘𝑥)) = (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
43 fvex 6891 . . . . . . . . . . . . . . . . 17 (𝑓𝑥) ∈ V
4443, 43op1st 7965 . . . . . . . . . . . . . . . 16 (1st ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
4542, 44eqtrdi 2787 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓f I 𝑓)‘𝑥)) = (𝑓𝑥))
4645, 7eqbrtrd 5163 . . . . . . . . . . . . . 14 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ (𝑓𝑥))
4741fveq2d 6882 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓f I 𝑓)‘𝑥)) = (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
4843, 43op2nd 7966 . . . . . . . . . . . . . . . 16 (2nd ‘⟨(𝑓𝑥), (𝑓𝑥)⟩) = (𝑓𝑥)
4947, 48eqtrdi 2787 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → (2nd ‘((𝑓f I 𝑓)‘𝑥)) = (𝑓𝑥))
507, 49breqtrrd 5169 . . . . . . . . . . . . . 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 5145 . . . . . . . . . . . . . 14 ((𝑓𝑥) = 𝑦 → ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ (𝑓𝑥) ↔ (1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ 𝑦))
53 breq1 5144 . . . . . . . . . . . . . 14 ((𝑓𝑥) = 𝑦 → ((𝑓𝑥) ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥)) ↔ 𝑦 ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥))))
5452, 53anbi12d 631 . . . . . . . . . . . . 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 244 . . . . . . . . . . . 12 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ((𝑓𝑥) = 𝑦 → ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥)))))
5655reximdva 3167 . . . . . . . . . . 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 239 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑦𝐴 → ∃𝑥 ∈ ℕ ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥)))))
5857ralrimiv 3144 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ∀𝑦𝐴𝑥 ∈ ℕ ((1st ‘((𝑓f I 𝑓)‘𝑥)) ≤ 𝑦𝑦 ≤ (2nd ‘((𝑓f I 𝑓)‘𝑥))))
59 ovolficc 24914 . . . . . . . . . 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 256 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 𝐴 ran ([,] ∘ (𝑓f I 𝑓)))
62 eqid 2731 . . . . . . . . 9 seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓)))
6362ovollb2 24935 . . . . . . . 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 5707 . . . . . . . . . . . . . . . 16 (((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) ∧ 𝑥 ∈ ℕ) → ⟨(𝑓𝑥), (𝑓𝑥)⟩ ∈ (ℂ × ℂ))
66 absf 15266 . . . . . . . . . . . . . . . . . . 19 abs:ℂ⟶ℝ
67 subf 11444 . . . . . . . . . . . . . . . . . . 19 − :(ℂ × ℂ)⟶ℂ
68 fco 6728 . . . . . . . . . . . . . . . . . . 19 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
6966, 67, 68mp2an 690 . . . . . . . . . . . . . . . . . 18 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7069a1i 11 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7170feqmptd 6946 . . . . . . . . . . . . . . . 16 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (abs ∘ − ) = (𝑦 ∈ (ℂ × ℂ) ↦ ((abs ∘ − )‘𝑦)))
72 fveq2 6878 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩))
73 df-ov 7396 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)) = ((abs ∘ − )‘⟨(𝑓𝑥), (𝑓𝑥)⟩)
7472, 73eqtr4di 2789 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨(𝑓𝑥), (𝑓𝑥)⟩ → ((abs ∘ − )‘𝑦) = ((𝑓𝑥)(abs ∘ − )(𝑓𝑥)))
7565, 36, 71, 74fmptco 7111 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓f I 𝑓)) = (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))))
76 cnmet 24217 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) ∈ (Met‘ℂ)
77 met0 23778 . . . . . . . . . . . . . . . . 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 5241 . . . . . . . . . . . . . . 15 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (𝑥 ∈ ℕ ↦ ((𝑓𝑥)(abs ∘ − )(𝑓𝑥))) = (𝑥 ∈ ℕ ↦ 0))
8075, 79eqtrd 2771 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓f I 𝑓)) = (𝑥 ∈ ℕ ↦ 0))
81 fconstmpt 5730 . . . . . . . . . . . . . 14 (ℕ × {0}) = (𝑥 ∈ ℕ ↦ 0)
8280, 81eqtr4di 2789 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ((abs ∘ − ) ∘ (𝑓f I 𝑓)) = (ℕ × {0}))
8382seqeq3d 13956 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = seq1( + , (ℕ × {0})))
84 1z 12574 . . . . . . . . . . . . 13 1 ∈ ℤ
85 nnuz 12847 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
8685ser0f 14003 . . . . . . . . . . . . 13 (1 ∈ ℤ → seq1( + , (ℕ × {0})) = (ℕ × {0}))
8784, 86ax-mp 5 . . . . . . . . . . . 12 seq1( + , (ℕ × {0})) = (ℕ × {0})
8883, 87eqtrdi 2787 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = (ℕ × {0}))
8988rneqd 5929 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = ran (ℕ × {0}))
90 1nn 12205 . . . . . . . . . . 11 1 ∈ ℕ
91 ne0i 4330 . . . . . . . . . . 11 (1 ∈ ℕ → ℕ ≠ ∅)
92 rnxp 6158 . . . . . . . . . . 11 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
9390, 91, 92mp2b 10 . . . . . . . . . 10 ran (ℕ × {0}) = {0}
9489, 93eqtrdi 2787 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))) = {0})
9594supeq1d 9423 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
96 xrltso 13102 . . . . . . . . 9 < Or ℝ*
97 0xr 11243 . . . . . . . . 9 0 ∈ ℝ*
98 supsn 9449 . . . . . . . . 9 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
9996, 97, 98mp2an 690 . . . . . . . 8 sup({0}, ℝ*, < ) = 0
10095, 99eqtrdi 2787 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑓f I 𝑓))), ℝ*, < ) = 0)
10164, 100breqtrd 5167 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ≤ 0)
102 ovolge0 24927 . . . . . . 7 (𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝐴))
103102adantr 481 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → 0 ≤ (vol*‘𝐴))
104 ovolcl 24924 . . . . . . . 8 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
105104adantr 481 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑓:ℕ–1-1-onto𝐴) → (vol*‘𝐴) ∈ ℝ*)
106 xrletri3 13115 . . . . . . 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 711 . . . . 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 1936 . . 3 (𝐴 ⊆ ℝ → (∃𝑓 𝑓:ℕ–1-1-onto𝐴 → (vol*‘𝐴) = 0))
1111, 110biimtrid 241 . 2 (𝐴 ⊆ ℝ → (ℕ ≈ 𝐴 → (vol*‘𝐴) = 0))
112 ensym 8982 . 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 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wne 2939  wral 3060  wrex 3069  Vcvv 3473  cin 3943  wss 3944  c0 4318  {csn 4622  cop 4628   cuni 4901   class class class wbr 5141  cmpt 5224   I cid 5566   Or wor 5580   × cxp 5667  ran crn 5670  ccom 5673   Fn wfn 6527  wf 6528  ontowfo 6530  1-1-ontowf1o 6531  cfv 6532  (class class class)co 7393  f cof 7651  1st c1st 7955  2nd c2nd 7956  cen 8919  supcsup 9417  cc 11090  cr 11091  0cc0 11092  1c1 11093   + caddc 11095  *cxr 11229   < clt 11230  cle 11231  cmin 11426  cn 12194  cz 12540  [,]cicc 13309  seqcseq 13948  abscabs 15163  Metcmet 20864  vol*covol 24908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-pre-sup 11170
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-of 7653  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-sup 9419  df-inf 9420  df-oi 9487  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-2 12257  df-3 12258  df-n0 12455  df-z 12541  df-uz 12805  df-q 12915  df-rp 12957  df-xadd 13075  df-ioo 13310  df-ico 13312  df-icc 13313  df-fz 13467  df-fzo 13610  df-seq 13949  df-exp 14010  df-hash 14273  df-cj 15028  df-re 15029  df-im 15030  df-sqrt 15164  df-abs 15165  df-clim 15414  df-sum 15615  df-xmet 20871  df-met 20872  df-ovol 24910
This theorem is referenced by:  ovolq  24937  ovolctb2  24938  ovoliunnfl  36334
  Copyright terms: Public domain W3C validator