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Theorem ovolctb 24999
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 βŠ† ℝ ∧ 𝐴 β‰ˆ β„•) β†’ (vol*β€˜π΄) = 0)
Proof of Theorem ovolctb
Dummy variables 𝑓 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bren 8946 . . 3 (β„• β‰ˆ 𝐴 ↔ βˆƒπ‘“ 𝑓:ℕ–1-1-onto→𝐴)
2 simpll 766 . . . . . . . . . . . . . 14 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ 𝐴 βŠ† ℝ)
3 f1of 6831 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto→𝐴 β†’ 𝑓:β„•βŸΆπ΄)
43adantl 483 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝑓:β„•βŸΆπ΄)
54ffvelcdmda 7084 . . . . . . . . . . . . . 14 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (π‘“β€˜π‘₯) ∈ 𝐴)
62, 5sseldd 3983 . . . . . . . . . . . . 13 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (π‘“β€˜π‘₯) ∈ ℝ)
76leidd 11777 . . . . . . . . . . . 12 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘₯))
8 df-br 5149 . . . . . . . . . . . 12 ((π‘“β€˜π‘₯) ≀ (π‘“β€˜π‘₯) ↔ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ ∈ ≀ )
97, 8sylib 217 . . . . . . . . . . 11 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ ∈ ≀ )
106, 6opelxpd 5714 . . . . . . . . . . 11 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ ∈ (ℝ Γ— ℝ))
119, 10elind 4194 . . . . . . . . . 10 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
12 df-ov 7409 . . . . . . . . . . . 12 ((π‘“β€˜π‘₯) I (π‘“β€˜π‘₯)) = ( I β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
13 opex 5464 . . . . . . . . . . . . 13 ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ ∈ V
14 fvi 6965 . . . . . . . . . . . . 13 (⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ ∈ V β†’ ( I β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩) = ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
1513, 14ax-mp 5 . . . . . . . . . . . 12 ( I β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩) = ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩
1612, 15eqtri 2761 . . . . . . . . . . 11 ((π‘“β€˜π‘₯) I (π‘“β€˜π‘₯)) = ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩
1716mpteq2i 5253 . . . . . . . . . 10 (π‘₯ ∈ β„• ↦ ((π‘“β€˜π‘₯) I (π‘“β€˜π‘₯))) = (π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
1811, 17fmptd 7111 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (π‘₯ ∈ β„• ↦ ((π‘“β€˜π‘₯) I (π‘“β€˜π‘₯))):β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
19 nnex 12215 . . . . . . . . . . . 12 β„• ∈ V
2019a1i 11 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ β„• ∈ V)
216recnd 11239 . . . . . . . . . . 11 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (π‘“β€˜π‘₯) ∈ β„‚)
224feqmptd 6958 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝑓 = (π‘₯ ∈ β„• ↦ (π‘“β€˜π‘₯)))
2320, 21, 21, 22, 22offval2 7687 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑓 ∘f I 𝑓) = (π‘₯ ∈ β„• ↦ ((π‘“β€˜π‘₯) I (π‘“β€˜π‘₯))))
2423feq1d 6700 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ((𝑓 ∘f I 𝑓):β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ↔ (π‘₯ ∈ β„• ↦ ((π‘“β€˜π‘₯) I (π‘“β€˜π‘₯))):β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))))
2518, 24mpbird 257 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑓 ∘f I 𝑓):β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
26 f1ofo 6838 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto→𝐴 β†’ 𝑓:ℕ–onto→𝐴)
2726adantl 483 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝑓:ℕ–onto→𝐴)
28 forn 6806 . . . . . . . . . . . . . 14 (𝑓:ℕ–onto→𝐴 β†’ ran 𝑓 = 𝐴)
2927, 28syl 17 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ran 𝑓 = 𝐴)
3029eleq2d 2820 . . . . . . . . . . . 12 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ 𝐴))
31 f1ofn 6832 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-onto→𝐴 β†’ 𝑓 Fn β„•)
3231adantl 483 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝑓 Fn β„•)
33 fvelrnb 6950 . . . . . . . . . . . . 13 (𝑓 Fn β„• β†’ (𝑦 ∈ ran 𝑓 ↔ βˆƒπ‘₯ ∈ β„• (π‘“β€˜π‘₯) = 𝑦))
3432, 33syl 17 . . . . . . . . . . . 12 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑦 ∈ ran 𝑓 ↔ βˆƒπ‘₯ ∈ β„• (π‘“β€˜π‘₯) = 𝑦))
3530, 34bitr3d 281 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑦 ∈ 𝐴 ↔ βˆƒπ‘₯ ∈ β„• (π‘“β€˜π‘₯) = 𝑦))
3623, 17eqtrdi 2789 . . . . . . . . . . . . . . . . . . 19 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑓 ∘f I 𝑓) = (π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩))
3736fveq1d 6891 . . . . . . . . . . . . . . . . . 18 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ((𝑓 ∘f I 𝑓)β€˜π‘₯) = ((π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)β€˜π‘₯))
38 eqid 2733 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩) = (π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
3938fvmpt2 7007 . . . . . . . . . . . . . . . . . . 19 ((π‘₯ ∈ β„• ∧ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ ∈ V) β†’ ((π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)β€˜π‘₯) = ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
4013, 39mpan2 690 . . . . . . . . . . . . . . . . . 18 (π‘₯ ∈ β„• β†’ ((π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)β€˜π‘₯) = ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
4137, 40sylan9eq 2793 . . . . . . . . . . . . . . . . 17 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ ((𝑓 ∘f I 𝑓)β€˜π‘₯) = ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
4241fveq2d 6893 . . . . . . . . . . . . . . . 16 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) = (1st β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩))
43 fvex 6902 . . . . . . . . . . . . . . . . 17 (π‘“β€˜π‘₯) ∈ V
4443, 43op1st 7980 . . . . . . . . . . . . . . . 16 (1st β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩) = (π‘“β€˜π‘₯)
4542, 44eqtrdi 2789 . . . . . . . . . . . . . . 15 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) = (π‘“β€˜π‘₯))
4645, 7eqbrtrd 5170 . . . . . . . . . . . . . 14 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ (π‘“β€˜π‘₯))
4741fveq2d 6893 . . . . . . . . . . . . . . . 16 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) = (2nd β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩))
4843, 43op2nd 7981 . . . . . . . . . . . . . . . 16 (2nd β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩) = (π‘“β€˜π‘₯)
4947, 48eqtrdi 2789 . . . . . . . . . . . . . . 15 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) = (π‘“β€˜π‘₯))
507, 49breqtrrd 5176 . . . . . . . . . . . . . 14 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (π‘“β€˜π‘₯) ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)))
5146, 50jca 513 . . . . . . . . . . . . 13 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ ((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ (π‘“β€˜π‘₯) ∧ (π‘“β€˜π‘₯) ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯))))
52 breq2 5152 . . . . . . . . . . . . . 14 ((π‘“β€˜π‘₯) = 𝑦 β†’ ((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ (π‘“β€˜π‘₯) ↔ (1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ 𝑦))
53 breq1 5151 . . . . . . . . . . . . . 14 ((π‘“β€˜π‘₯) = 𝑦 β†’ ((π‘“β€˜π‘₯) ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ↔ 𝑦 ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯))))
5452, 53anbi12d 632 . . . . . . . . . . . . 13 ((π‘“β€˜π‘₯) = 𝑦 β†’ (((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ (π‘“β€˜π‘₯) ∧ (π‘“β€˜π‘₯) ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯))) ↔ ((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ 𝑦 ∧ 𝑦 ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)))))
5551, 54syl5ibcom 244 . . . . . . . . . . . 12 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ ((π‘“β€˜π‘₯) = 𝑦 β†’ ((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ 𝑦 ∧ 𝑦 ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)))))
5655reximdva 3169 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (βˆƒπ‘₯ ∈ β„• (π‘“β€˜π‘₯) = 𝑦 β†’ βˆƒπ‘₯ ∈ β„• ((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ 𝑦 ∧ 𝑦 ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)))))
5735, 56sylbid 239 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑦 ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ β„• ((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ 𝑦 ∧ 𝑦 ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)))))
5857ralrimiv 3146 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ βˆ€π‘¦ ∈ 𝐴 βˆƒπ‘₯ ∈ β„• ((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ 𝑦 ∧ 𝑦 ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯))))
59 ovolficc 24977 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ (𝑓 ∘f I 𝑓):β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (𝐴 βŠ† βˆͺ ran ([,] ∘ (𝑓 ∘f I 𝑓)) ↔ βˆ€π‘¦ ∈ 𝐴 βˆƒπ‘₯ ∈ β„• ((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ 𝑦 ∧ 𝑦 ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)))))
6025, 59syldan 592 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝐴 βŠ† βˆͺ ran ([,] ∘ (𝑓 ∘f I 𝑓)) ↔ βˆ€π‘¦ ∈ 𝐴 βˆƒπ‘₯ ∈ β„• ((1st β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)) ≀ 𝑦 ∧ 𝑦 ≀ (2nd β€˜((𝑓 ∘f I 𝑓)β€˜π‘₯)))))
6158, 60mpbird 257 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝐴 βŠ† βˆͺ ran ([,] ∘ (𝑓 ∘f I 𝑓)))
62 eqid 2733 . . . . . . . . 9 seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓)))
6362ovollb2 24998 . . . . . . . 8 (((𝑓 ∘f I 𝑓):β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝐴 βŠ† βˆͺ ran ([,] ∘ (𝑓 ∘f I 𝑓))) β†’ (vol*β€˜π΄) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ))
6425, 61, 63syl2anc 585 . . . . . . 7 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (vol*β€˜π΄) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ))
6521, 21opelxpd 5714 . . . . . . . . . . . . . . . 16 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ ∈ (β„‚ Γ— β„‚))
66 absf 15281 . . . . . . . . . . . . . . . . . . 19 abs:β„‚βŸΆβ„
67 subf 11459 . . . . . . . . . . . . . . . . . . 19 βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚
68 fco 6739 . . . . . . . . . . . . . . . . . . 19 ((abs:β„‚βŸΆβ„ ∧ βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚) β†’ (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„)
6966, 67, 68mp2an 691 . . . . . . . . . . . . . . . . . 18 (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„
7069a1i 11 . . . . . . . . . . . . . . . . 17 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„)
7170feqmptd 6958 . . . . . . . . . . . . . . . 16 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (abs ∘ βˆ’ ) = (𝑦 ∈ (β„‚ Γ— β„‚) ↦ ((abs ∘ βˆ’ )β€˜π‘¦)))
72 fveq2 6889 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ β†’ ((abs ∘ βˆ’ )β€˜π‘¦) = ((abs ∘ βˆ’ )β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩))
73 df-ov 7409 . . . . . . . . . . . . . . . . 17 ((π‘“β€˜π‘₯)(abs ∘ βˆ’ )(π‘“β€˜π‘₯)) = ((abs ∘ βˆ’ )β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
7472, 73eqtr4di 2791 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ β†’ ((abs ∘ βˆ’ )β€˜π‘¦) = ((π‘“β€˜π‘₯)(abs ∘ βˆ’ )(π‘“β€˜π‘₯)))
7565, 36, 71, 74fmptco 7124 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓)) = (π‘₯ ∈ β„• ↦ ((π‘“β€˜π‘₯)(abs ∘ βˆ’ )(π‘“β€˜π‘₯))))
76 cnmet 24280 . . . . . . . . . . . . . . . . 17 (abs ∘ βˆ’ ) ∈ (Metβ€˜β„‚)
77 met0 23841 . . . . . . . . . . . . . . . . 17 (((abs ∘ βˆ’ ) ∈ (Metβ€˜β„‚) ∧ (π‘“β€˜π‘₯) ∈ β„‚) β†’ ((π‘“β€˜π‘₯)(abs ∘ βˆ’ )(π‘“β€˜π‘₯)) = 0)
7876, 21, 77sylancr 588 . . . . . . . . . . . . . . . 16 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ ((π‘“β€˜π‘₯)(abs ∘ βˆ’ )(π‘“β€˜π‘₯)) = 0)
7978mpteq2dva 5248 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (π‘₯ ∈ β„• ↦ ((π‘“β€˜π‘₯)(abs ∘ βˆ’ )(π‘“β€˜π‘₯))) = (π‘₯ ∈ β„• ↦ 0))
8075, 79eqtrd 2773 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓)) = (π‘₯ ∈ β„• ↦ 0))
81 fconstmpt 5737 . . . . . . . . . . . . . 14 (β„• Γ— {0}) = (π‘₯ ∈ β„• ↦ 0)
8280, 81eqtr4di 2791 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓)) = (β„• Γ— {0}))
8382seqeq3d 13971 . . . . . . . . . . . 12 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , (β„• Γ— {0})))
84 1z 12589 . . . . . . . . . . . . 13 1 ∈ β„€
85 nnuz 12862 . . . . . . . . . . . . . 14 β„• = (β„€β‰₯β€˜1)
8685ser0f 14018 . . . . . . . . . . . . 13 (1 ∈ β„€ β†’ seq1( + , (β„• Γ— {0})) = (β„• Γ— {0}))
8784, 86ax-mp 5 . . . . . . . . . . . 12 seq1( + , (β„• Γ— {0})) = (β„• Γ— {0})
8883, 87eqtrdi 2789 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))) = (β„• Γ— {0}))
8988rneqd 5936 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))) = ran (β„• Γ— {0}))
90 1nn 12220 . . . . . . . . . . 11 1 ∈ β„•
91 ne0i 4334 . . . . . . . . . . 11 (1 ∈ β„• β†’ β„• β‰  βˆ…)
92 rnxp 6167 . . . . . . . . . . 11 (β„• β‰  βˆ… β†’ ran (β„• Γ— {0}) = {0})
9390, 91, 92mp2b 10 . . . . . . . . . 10 ran (β„• Γ— {0}) = {0}
9489, 93eqtrdi 2789 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))) = {0})
9594supeq1d 9438 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ) = sup({0}, ℝ*, < ))
96 xrltso 13117 . . . . . . . . 9 < Or ℝ*
97 0xr 11258 . . . . . . . . 9 0 ∈ ℝ*
98 supsn 9464 . . . . . . . . 9 (( < Or ℝ* ∧ 0 ∈ ℝ*) β†’ sup({0}, ℝ*, < ) = 0)
9996, 97, 98mp2an 691 . . . . . . . 8 sup({0}, ℝ*, < ) = 0
10095, 99eqtrdi 2789 . . . . . . 7 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))), ℝ*, < ) = 0)
10164, 100breqtrd 5174 . . . . . 6 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (vol*β€˜π΄) ≀ 0)
102 ovolge0 24990 . . . . . . 7 (𝐴 βŠ† ℝ β†’ 0 ≀ (vol*β€˜π΄))
103102adantr 482 . . . . . 6 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 0 ≀ (vol*β€˜π΄))
104 ovolcl 24987 . . . . . . . 8 (𝐴 βŠ† ℝ β†’ (vol*β€˜π΄) ∈ ℝ*)
105104adantr 482 . . . . . . 7 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (vol*β€˜π΄) ∈ ℝ*)
106 xrletri3 13130 . . . . . . 7 (((vol*β€˜π΄) ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ ((vol*β€˜π΄) = 0 ↔ ((vol*β€˜π΄) ≀ 0 ∧ 0 ≀ (vol*β€˜π΄))))
107105, 97, 106sylancl 587 . . . . . 6 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ((vol*β€˜π΄) = 0 ↔ ((vol*β€˜π΄) ≀ 0 ∧ 0 ≀ (vol*β€˜π΄))))
108101, 103, 107mpbir2and 712 . . . . 5 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (vol*β€˜π΄) = 0)
109108ex 414 . . . 4 (𝐴 βŠ† ℝ β†’ (𝑓:ℕ–1-1-onto→𝐴 β†’ (vol*β€˜π΄) = 0))
110109exlimdv 1937 . . 3 (𝐴 βŠ† ℝ β†’ (βˆƒπ‘“ 𝑓:ℕ–1-1-onto→𝐴 β†’ (vol*β€˜π΄) = 0))
1111, 110biimtrid 241 . 2 (𝐴 βŠ† ℝ β†’ (β„• β‰ˆ 𝐴 β†’ (vol*β€˜π΄) = 0))
112 ensym 8996 . 2 (𝐴 β‰ˆ β„• β†’ β„• β‰ˆ 𝐴)
113111, 112impel 507 1 ((𝐴 βŠ† ℝ ∧ 𝐴 β‰ˆ β„•) β†’ (vol*β€˜π΄) = 0)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βŸ¨cop 4634  βˆͺ cuni 4908   class class class wbr 5148   ↦ cmpt 5231   I cid 5573   Or wor 5587   Γ— cxp 5674  ran crn 5677   ∘ ccom 5680   Fn wfn 6536  βŸΆwf 6537  β€“ontoβ†’wfo 6539  β€“1-1-ontoβ†’wf1o 6540  β€˜cfv 6541  (class class class)co 7406   ∘f cof 7665  1st c1st 7970  2nd c2nd 7971   β‰ˆ cen 8933  supcsup 9432  β„‚cc 11105  β„cr 11106  0cc0 11107  1c1 11108   + caddc 11110  β„*cxr 11244   < clt 11245   ≀ cle 11246   βˆ’ cmin 11441  β„•cn 12209  β„€cz 12555  [,]cicc 13324  seqcseq 13963  abscabs 15178  Metcmet 20923  vol*covol 24971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xadd 13090  df-ioo 13325  df-ico 13327  df-icc 13328  df-fz 13482  df-fzo 13625  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-sum 15630  df-xmet 20930  df-met 20931  df-ovol 24973
This theorem is referenced by:  ovolq  25000  ovolctb2  25001  ovoliunnfl  36519
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