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Theorem ovolctb 25231
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 8951 . . 3 (β„• β‰ˆ 𝐴 ↔ βˆƒπ‘“ 𝑓:ℕ–1-1-onto→𝐴)
2 simpll 765 . . . . . . . . . . . . . 14 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ 𝐴 βŠ† ℝ)
3 f1of 6833 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto→𝐴 β†’ 𝑓:β„•βŸΆπ΄)
43adantl 482 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝑓:β„•βŸΆπ΄)
54ffvelcdmda 7086 . . . . . . . . . . . . . 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 11784 . . . . . . . . . . . 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 5715 . . . . . . . . . . 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 7414 . . . . . . . . . . . 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 6967 . . . . . . . . . . . . 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 2760 . . . . . . . . . . 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 7115 . . . . . . . . 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 12222 . . . . . . . . . . . 12 β„• ∈ V
2019a1i 11 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ β„• ∈ V)
216recnd 11246 . . . . . . . . . . 11 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ (π‘“β€˜π‘₯) ∈ β„‚)
224feqmptd 6960 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝑓 = (π‘₯ ∈ β„• ↦ (π‘“β€˜π‘₯)))
2320, 21, 21, 22, 22offval2 7692 . . . . . . . . . 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 6702 . . . . . . . . 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 256 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑓 ∘f I 𝑓):β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
26 f1ofo 6840 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto→𝐴 β†’ 𝑓:ℕ–onto→𝐴)
2726adantl 482 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝑓:ℕ–onto→𝐴)
28 forn 6808 . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . 12 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ 𝐴))
31 f1ofn 6834 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-onto→𝐴 β†’ 𝑓 Fn β„•)
3231adantl 482 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝑓 Fn β„•)
33 fvelrnb 6952 . . . . . . . . . . . . 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 280 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑦 ∈ 𝐴 ↔ βˆƒπ‘₯ ∈ β„• (π‘“β€˜π‘₯) = 𝑦))
3623, 17eqtrdi 2788 . . . . . . . . . . . . . . . . . . 19 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (𝑓 ∘f I 𝑓) = (π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩))
3736fveq1d 6893 . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩) = (π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
3938fvmpt2 7009 . . . . . . . . . . . . . . . . . . 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 689 . . . . . . . . . . . . . . . . . 18 (π‘₯ ∈ β„• β†’ ((π‘₯ ∈ β„• ↦ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)β€˜π‘₯) = ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
4137, 40sylan9eq 2792 . . . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . . . 17 (π‘“β€˜π‘₯) ∈ V
4443, 43op1st 7985 . . . . . . . . . . . . . . . 16 (1st β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩) = (π‘“β€˜π‘₯)
4542, 44eqtrdi 2788 . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . . 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 7986 . . . . . . . . . . . . . . . 16 (2nd β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩) = (π‘“β€˜π‘₯)
4947, 48eqtrdi 2788 . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . 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 631 . . . . . . . . . . . . 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 3168 . . . . . . . . . . 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 3145 . . . . . . . . 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 25209 . . . . . . . . . 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 591 . . . . . . . . 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 256 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 𝐴 βŠ† βˆͺ ran ([,] ∘ (𝑓 ∘f I 𝑓)))
62 eqid 2732 . . . . . . . . 9 seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓)))
6362ovollb2 25230 . . . . . . . 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 584 . . . . . . 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 5715 . . . . . . . . . . . . . . . 16 (((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) ∧ π‘₯ ∈ β„•) β†’ ⟨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩ ∈ (β„‚ Γ— β„‚))
66 absf 15288 . . . . . . . . . . . . . . . . . . 19 abs:β„‚βŸΆβ„
67 subf 11466 . . . . . . . . . . . . . . . . . . 19 βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚
68 fco 6741 . . . . . . . . . . . . . . . . . . 19 ((abs:β„‚βŸΆβ„ ∧ βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚) β†’ (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„)
6966, 67, 68mp2an 690 . . . . . . . . . . . . . . . . . 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 6960 . . . . . . . . . . . . . . . 16 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (abs ∘ βˆ’ ) = (𝑦 ∈ (β„‚ Γ— β„‚) ↦ ((abs ∘ βˆ’ )β€˜π‘¦)))
72 fveq2 6891 . . . . . . . . . . . . . . . . 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 7414 . . . . . . . . . . . . . . . . 17 ((π‘“β€˜π‘₯)(abs ∘ βˆ’ )(π‘“β€˜π‘₯)) = ((abs ∘ βˆ’ )β€˜βŸ¨(π‘“β€˜π‘₯), (π‘“β€˜π‘₯)⟩)
7472, 73eqtr4di 2790 . . . . . . . . . . . . . . . 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 7129 . . . . . . . . . . . . . . 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 24508 . . . . . . . . . . . . . . . . 17 (abs ∘ βˆ’ ) ∈ (Metβ€˜β„‚)
77 met0 24069 . . . . . . . . . . . . . . . . 17 (((abs ∘ βˆ’ ) ∈ (Metβ€˜β„‚) ∧ (π‘“β€˜π‘₯) ∈ β„‚) β†’ ((π‘“β€˜π‘₯)(abs ∘ βˆ’ )(π‘“β€˜π‘₯)) = 0)
7876, 21, 77sylancr 587 . . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓)) = (π‘₯ ∈ β„• ↦ 0))
81 fconstmpt 5738 . . . . . . . . . . . . . 14 (β„• Γ— {0}) = (π‘₯ ∈ β„• ↦ 0)
8280, 81eqtr4di 2790 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓)) = (β„• Γ— {0}))
8382seqeq3d 13978 . . . . . . . . . . . 12 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))) = seq1( + , (β„• Γ— {0})))
84 1z 12596 . . . . . . . . . . . . 13 1 ∈ β„€
85 nnuz 12869 . . . . . . . . . . . . . 14 β„• = (β„€β‰₯β€˜1)
8685ser0f 14025 . . . . . . . . . . . . 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 2788 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))) = (β„• Γ— {0}))
8988rneqd 5937 . . . . . . . . . 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 12227 . . . . . . . . . . 11 1 ∈ β„•
91 ne0i 4334 . . . . . . . . . . 11 (1 ∈ β„• β†’ β„• β‰  βˆ…)
92 rnxp 6169 . . . . . . . . . . 11 (β„• β‰  βˆ… β†’ ran (β„• Γ— {0}) = {0})
9390, 91, 92mp2b 10 . . . . . . . . . 10 ran (β„• Γ— {0}) = {0}
9489, 93eqtrdi 2788 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ (𝑓 ∘f I 𝑓))) = {0})
9594supeq1d 9443 . . . . . . . 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 13124 . . . . . . . . 9 < Or ℝ*
97 0xr 11265 . . . . . . . . 9 0 ∈ ℝ*
98 supsn 9469 . . . . . . . . 9 (( < Or ℝ* ∧ 0 ∈ ℝ*) β†’ sup({0}, ℝ*, < ) = 0)
9996, 97, 98mp2an 690 . . . . . . . 8 sup({0}, ℝ*, < ) = 0
10095, 99eqtrdi 2788 . . . . . . 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 25222 . . . . . . 7 (𝐴 βŠ† ℝ β†’ 0 ≀ (vol*β€˜π΄))
103102adantr 481 . . . . . 6 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ 0 ≀ (vol*β€˜π΄))
104 ovolcl 25219 . . . . . . . 8 (𝐴 βŠ† ℝ β†’ (vol*β€˜π΄) ∈ ℝ*)
105104adantr 481 . . . . . . 7 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (vol*β€˜π΄) ∈ ℝ*)
106 xrletri3 13137 . . . . . . 7 (((vol*β€˜π΄) ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ ((vol*β€˜π΄) = 0 ↔ ((vol*β€˜π΄) ≀ 0 ∧ 0 ≀ (vol*β€˜π΄))))
107105, 97, 106sylancl 586 . . . . . 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 711 . . . . 5 ((𝐴 βŠ† ℝ ∧ 𝑓:ℕ–1-1-onto→𝐴) β†’ (vol*β€˜π΄) = 0)
109108ex 413 . . . 4 (𝐴 βŠ† ℝ β†’ (𝑓:ℕ–1-1-onto→𝐴 β†’ (vol*β€˜π΄) = 0))
110109exlimdv 1936 . . 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 9001 . 2 (𝐴 β‰ˆ β„• β†’ β„• β‰ˆ 𝐴)
113111, 112impel 506 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 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   ∩ 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 6538  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411   ∘f cof 7670  1st c1st 7975  2nd c2nd 7976   β‰ˆ cen 8938  supcsup 9437  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115  β„*cxr 11251   < clt 11252   ≀ cle 11253   βˆ’ cmin 11448  β„•cn 12216  β„€cz 12562  [,]cicc 13331  seqcseq 13970  abscabs 15185  Metcmet 21130  vol*covol 25203
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-xadd 13097  df-ioo 13332  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-xmet 21137  df-met 21138  df-ovol 25205
This theorem is referenced by:  ovolq  25232  ovolctb2  25233  ovoliunnfl  36833
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