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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovolval2lem Structured version   Visualization version   GIF version

Theorem ovolval2lem 45359
Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Ξ£^. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypothesis
Ref Expression
ovolval2lem.1 (πœ‘ β†’ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
Assertion
Ref Expression
ovolval2lem (πœ‘ β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝐹)) = ran (𝑛 ∈ β„• ↦ Ξ£π‘˜ ∈ (1...𝑛)(volβ€˜(([,) ∘ 𝐹)β€˜π‘˜))))
Distinct variable groups:   π‘˜,𝐹,𝑛   πœ‘,π‘˜
Allowed substitution hint:   πœ‘(𝑛)

Proof of Theorem ovolval2lem
StepHypRef Expression
1 reex 11201 . . . . . . 7 ℝ ∈ V
21, 1xpex 7740 . . . . . 6 (ℝ Γ— ℝ) ∈ V
3 inss2 4230 . . . . . 6 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
4 mapss 8883 . . . . . 6 (((ℝ Γ— ℝ) ∈ V ∧ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)) β†’ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) βŠ† ((ℝ Γ— ℝ) ↑m β„•))
52, 3, 4mp2an 691 . . . . 5 (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) βŠ† ((ℝ Γ— ℝ) ↑m β„•)
6 ovolval2lem.1 . . . . . 6 (πœ‘ β†’ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
72inex2 5319 . . . . . . . 8 ( ≀ ∩ (ℝ Γ— ℝ)) ∈ V
87a1i 11 . . . . . . 7 (πœ‘ β†’ ( ≀ ∩ (ℝ Γ— ℝ)) ∈ V)
9 nnex 12218 . . . . . . . 8 β„• ∈ V
109a1i 11 . . . . . . 7 (πœ‘ β†’ β„• ∈ V)
118, 10elmapd 8834 . . . . . 6 (πœ‘ β†’ (𝐹 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ↔ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))))
126, 11mpbird 257 . . . . 5 (πœ‘ β†’ 𝐹 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•))
135, 12sselid 3981 . . . 4 (πœ‘ β†’ 𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•))
14 1zzd 12593 . . . . 5 (𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) β†’ 1 ∈ β„€)
15 nnuz 12865 . . . . 5 β„• = (β„€β‰₯β€˜1)
16 elmapi 8843 . . . . . . . . . 10 (𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))
1716adantr 482 . . . . . . . . 9 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))
18 simpr 486 . . . . . . . . 9 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ π‘˜ ∈ β„•)
1917, 18fvovco 43892 . . . . . . . 8 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (([,) ∘ 𝐹)β€˜π‘˜) = ((1st β€˜(πΉβ€˜π‘˜))[,)(2nd β€˜(πΉβ€˜π‘˜))))
2019fveq2d 6896 . . . . . . 7 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)) = (volβ€˜((1st β€˜(πΉβ€˜π‘˜))[,)(2nd β€˜(πΉβ€˜π‘˜)))))
2116ffvelcdmda 7087 . . . . . . . . 9 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ (ℝ Γ— ℝ))
22 xp1st 8007 . . . . . . . . 9 ((πΉβ€˜π‘˜) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
2321, 22syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
24 xp2nd 8008 . . . . . . . . 9 ((πΉβ€˜π‘˜) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
2521, 24syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
26 volicore 45297 . . . . . . . 8 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ) β†’ (volβ€˜((1st β€˜(πΉβ€˜π‘˜))[,)(2nd β€˜(πΉβ€˜π‘˜)))) ∈ ℝ)
2723, 25, 26syl2anc 585 . . . . . . 7 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (volβ€˜((1st β€˜(πΉβ€˜π‘˜))[,)(2nd β€˜(πΉβ€˜π‘˜)))) ∈ ℝ)
2820, 27eqeltrd 2834 . . . . . 6 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)) ∈ ℝ)
2928recnd 11242 . . . . 5 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)) ∈ β„‚)
30 eqid 2733 . . . . 5 (𝑛 ∈ β„• ↦ Ξ£π‘˜ ∈ (1...𝑛)(volβ€˜(([,) ∘ 𝐹)β€˜π‘˜))) = (𝑛 ∈ β„• ↦ Ξ£π‘˜ ∈ (1...𝑛)(volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)))
31 eqid 2733 . . . . 5 seq1( + , (π‘˜ ∈ β„• ↦ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)))) = seq1( + , (π‘˜ ∈ β„• ↦ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜))))
3214, 15, 29, 30, 31fsumsermpt 44295 . . . 4 (𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) β†’ (𝑛 ∈ β„• ↦ Ξ£π‘˜ ∈ (1...𝑛)(volβ€˜(([,) ∘ 𝐹)β€˜π‘˜))) = seq1( + , (π‘˜ ∈ β„• ↦ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)))))
3313, 32syl 17 . . 3 (πœ‘ β†’ (𝑛 ∈ β„• ↦ Ξ£π‘˜ ∈ (1...𝑛)(volβ€˜(([,) ∘ 𝐹)β€˜π‘˜))) = seq1( + , (π‘˜ ∈ β„• ↦ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)))))
34 simpr 486 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)))
3534iftrued 4537 . . . . . . . . 9 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ if((1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)), ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))), 0) = ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))))
3613, 23sylan 581 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
3736adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
3813, 25sylan 581 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
3938adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
40 ressxr 11258 . . . . . . . . . . . 12 ℝ βŠ† ℝ*
4140, 37sselid 3981 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ*)
4240, 39sselid 3981 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ*)
43 xpss 5693 . . . . . . . . . . . . . . . . . 18 (ℝ Γ— ℝ) βŠ† (V Γ— V)
4443, 21sselid 3981 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ (V Γ— V))
45 1st2ndb 8015 . . . . . . . . . . . . . . . . 17 ((πΉβ€˜π‘˜) ∈ (V Γ— V) ↔ (πΉβ€˜π‘˜) = ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩)
4644, 45sylib 217 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) = ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩)
4713, 46sylan 581 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) = ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩)
4847eqcomd 2739 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩ = (πΉβ€˜π‘˜))
49 inss1 4229 . . . . . . . . . . . . . . . . 17 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† ≀
5049a1i 11 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† ≀ )
516, 50fssd 6736 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐹:β„•βŸΆ ≀ )
5251ffvelcdmda 7087 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΉβ€˜π‘˜) ∈ ≀ )
5348, 52eqeltrd 2834 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩ ∈ ≀ )
54 df-br 5150 . . . . . . . . . . . . 13 ((1st β€˜(πΉβ€˜π‘˜)) ≀ (2nd β€˜(πΉβ€˜π‘˜)) ↔ ⟨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩ ∈ ≀ )
5553, 54sylibr 233 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (1st β€˜(πΉβ€˜π‘˜)) ≀ (2nd β€˜(πΉβ€˜π‘˜)))
5655adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ (1st β€˜(πΉβ€˜π‘˜)) ≀ (2nd β€˜(πΉβ€˜π‘˜)))
57 simpr 486 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)))
5839, 37lenltd 11360 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ ((2nd β€˜(πΉβ€˜π‘˜)) ≀ (1st β€˜(πΉβ€˜π‘˜)) ↔ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))))
5957, 58mpbird 257 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ≀ (1st β€˜(πΉβ€˜π‘˜)))
6041, 42, 56, 59xrletrid 13134 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜)))
61 simp3 1139 . . . . . . . . . . . . . 14 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜)))
62 simp1 1137 . . . . . . . . . . . . . . 15 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
63 simp2 1138 . . . . . . . . . . . . . . 15 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ)
6462, 63eqleltd 11358 . . . . . . . . . . . . . 14 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ ((1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜)) ↔ ((1st β€˜(πΉβ€˜π‘˜)) ≀ (2nd β€˜(πΉβ€˜π‘˜)) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)))))
6561, 64mpbid 231 . . . . . . . . . . . . 13 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ ((1st β€˜(πΉβ€˜π‘˜)) ≀ (2nd β€˜(πΉβ€˜π‘˜)) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))))
6665simprd 497 . . . . . . . . . . . 12 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)))
6766iffalsed 4540 . . . . . . . . . . 11 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ if((1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)), ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))), 0) = 0)
6863recnd 11242 . . . . . . . . . . . 12 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ β„‚)
6961eqcomd 2739 . . . . . . . . . . . 12 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ (2nd β€˜(πΉβ€˜π‘˜)) = (1st β€˜(πΉβ€˜π‘˜)))
7068, 69subeq0bd 11640 . . . . . . . . . . 11 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))) = 0)
7167, 70eqtr4d 2776 . . . . . . . . . 10 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘˜)) = (2nd β€˜(πΉβ€˜π‘˜))) β†’ if((1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)), ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))), 0) = ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))))
7237, 39, 60, 71syl3anc 1372 . . . . . . . . 9 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ Β¬ (1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜))) β†’ if((1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)), ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))), 0) = ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))))
7335, 72pm2.61dan 812 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ if((1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)), ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))), 0) = ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))))
74 volico 44699 . . . . . . . . 9 (((1st β€˜(πΉβ€˜π‘˜)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ ℝ) β†’ (volβ€˜((1st β€˜(πΉβ€˜π‘˜))[,)(2nd β€˜(πΉβ€˜π‘˜)))) = if((1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)), ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))), 0))
7536, 38, 74syl2anc 585 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (volβ€˜((1st β€˜(πΉβ€˜π‘˜))[,)(2nd β€˜(πΉβ€˜π‘˜)))) = if((1st β€˜(πΉβ€˜π‘˜)) < (2nd β€˜(πΉβ€˜π‘˜)), ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))), 0))
7636, 38, 55abssuble0d 15379 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (absβ€˜((1st β€˜(πΉβ€˜π‘˜)) βˆ’ (2nd β€˜(πΉβ€˜π‘˜)))) = ((2nd β€˜(πΉβ€˜π‘˜)) βˆ’ (1st β€˜(πΉβ€˜π‘˜))))
7773, 75, 763eqtr4d 2783 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (volβ€˜((1st β€˜(πΉβ€˜π‘˜))[,)(2nd β€˜(πΉβ€˜π‘˜)))) = (absβ€˜((1st β€˜(πΉβ€˜π‘˜)) βˆ’ (2nd β€˜(πΉβ€˜π‘˜)))))
7813adantr 482 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ 𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•))
79 simpr 486 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ π‘˜ ∈ β„•)
8078, 79, 20syl2anc 585 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)) = (volβ€˜((1st β€˜(πΉβ€˜π‘˜))[,)(2nd β€˜(πΉβ€˜π‘˜)))))
8146fveq2d 6896 . . . . . . . . 9 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ ((abs ∘ βˆ’ )β€˜(πΉβ€˜π‘˜)) = ((abs ∘ βˆ’ )β€˜βŸ¨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩))
82 df-ov 7412 . . . . . . . . . . 11 ((1st β€˜(πΉβ€˜π‘˜))(abs ∘ βˆ’ )(2nd β€˜(πΉβ€˜π‘˜))) = ((abs ∘ βˆ’ )β€˜βŸ¨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩)
8382eqcomi 2742 . . . . . . . . . 10 ((abs ∘ βˆ’ )β€˜βŸ¨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩) = ((1st β€˜(πΉβ€˜π‘˜))(abs ∘ βˆ’ )(2nd β€˜(πΉβ€˜π‘˜)))
8483a1i 11 . . . . . . . . 9 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ ((abs ∘ βˆ’ )β€˜βŸ¨(1st β€˜(πΉβ€˜π‘˜)), (2nd β€˜(πΉβ€˜π‘˜))⟩) = ((1st β€˜(πΉβ€˜π‘˜))(abs ∘ βˆ’ )(2nd β€˜(πΉβ€˜π‘˜))))
8523recnd 11242 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (1st β€˜(πΉβ€˜π‘˜)) ∈ β„‚)
8625recnd 11242 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ (2nd β€˜(πΉβ€˜π‘˜)) ∈ β„‚)
87 eqid 2733 . . . . . . . . . . 11 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
8887cnmetdval 24287 . . . . . . . . . 10 (((1st β€˜(πΉβ€˜π‘˜)) ∈ β„‚ ∧ (2nd β€˜(πΉβ€˜π‘˜)) ∈ β„‚) β†’ ((1st β€˜(πΉβ€˜π‘˜))(abs ∘ βˆ’ )(2nd β€˜(πΉβ€˜π‘˜))) = (absβ€˜((1st β€˜(πΉβ€˜π‘˜)) βˆ’ (2nd β€˜(πΉβ€˜π‘˜)))))
8985, 86, 88syl2anc 585 . . . . . . . . 9 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ ((1st β€˜(πΉβ€˜π‘˜))(abs ∘ βˆ’ )(2nd β€˜(πΉβ€˜π‘˜))) = (absβ€˜((1st β€˜(πΉβ€˜π‘˜)) βˆ’ (2nd β€˜(πΉβ€˜π‘˜)))))
9081, 84, 893eqtrd 2777 . . . . . . . 8 ((𝐹 ∈ ((ℝ Γ— ℝ) ↑m β„•) ∧ π‘˜ ∈ β„•) β†’ ((abs ∘ βˆ’ )β€˜(πΉβ€˜π‘˜)) = (absβ€˜((1st β€˜(πΉβ€˜π‘˜)) βˆ’ (2nd β€˜(πΉβ€˜π‘˜)))))
9178, 79, 90syl2anc 585 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ ((abs ∘ βˆ’ )β€˜(πΉβ€˜π‘˜)) = (absβ€˜((1st β€˜(πΉβ€˜π‘˜)) βˆ’ (2nd β€˜(πΉβ€˜π‘˜)))))
9277, 80, 913eqtr4d 2783 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)) = ((abs ∘ βˆ’ )β€˜(πΉβ€˜π‘˜)))
9392mpteq2dva 5249 . . . . 5 (πœ‘ β†’ (π‘˜ ∈ β„• ↦ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜))) = (π‘˜ ∈ β„• ↦ ((abs ∘ βˆ’ )β€˜(πΉβ€˜π‘˜))))
9413, 16syl 17 . . . . . 6 (πœ‘ β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))
95 rr2sscn2 44076 . . . . . . 7 (ℝ Γ— ℝ) βŠ† (β„‚ Γ— β„‚)
9695a1i 11 . . . . . 6 (πœ‘ β†’ (ℝ Γ— ℝ) βŠ† (β„‚ Γ— β„‚))
97 absf 15284 . . . . . . . 8 abs:β„‚βŸΆβ„
98 subf 11462 . . . . . . . 8 βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚
99 fco 6742 . . . . . . . 8 ((abs:β„‚βŸΆβ„ ∧ βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚) β†’ (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„)
10097, 98, 99mp2an 691 . . . . . . 7 (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„
101100a1i 11 . . . . . 6 (πœ‘ β†’ (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„)
10294, 96, 101fcomptss 43902 . . . . 5 (πœ‘ β†’ ((abs ∘ βˆ’ ) ∘ 𝐹) = (π‘˜ ∈ β„• ↦ ((abs ∘ βˆ’ )β€˜(πΉβ€˜π‘˜))))
10393, 102eqtr4d 2776 . . . 4 (πœ‘ β†’ (π‘˜ ∈ β„• ↦ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜))) = ((abs ∘ βˆ’ ) ∘ 𝐹))
104103seqeq3d 13974 . . 3 (πœ‘ β†’ seq1( + , (π‘˜ ∈ β„• ↦ (volβ€˜(([,) ∘ 𝐹)β€˜π‘˜)))) = seq1( + , ((abs ∘ βˆ’ ) ∘ 𝐹)))
10533, 104eqtr2d 2774 . 2 (πœ‘ β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ 𝐹)) = (𝑛 ∈ β„• ↦ Ξ£π‘˜ ∈ (1...𝑛)(volβ€˜(([,) ∘ 𝐹)β€˜π‘˜))))
106105rneqd 5938 1 (πœ‘ β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝐹)) = ran (𝑛 ∈ β„• ↦ Ξ£π‘˜ ∈ (1...𝑛)(volβ€˜(([,) ∘ 𝐹)β€˜π‘˜))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949  ifcif 4529  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232   Γ— cxp 5675  ran crn 5678   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974   ↑m cmap 8820  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113  β„*cxr 11247   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  β„•cn 12212  [,)cico 13326  ...cfz 13484  seqcseq 13966  abscabs 15181  Ξ£csu 15632  volcvol 24980
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-rlim 15433  df-sum 15633  df-rest 17368  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449  df-cmp 22891  df-ovol 24981  df-vol 24982
This theorem is referenced by: (None)
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