Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovolval2lem Structured version   Visualization version   GIF version

Theorem ovolval2lem 43450
 Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypothesis
Ref Expression
ovolval2lem.1 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
Assertion
Ref Expression
ovolval2lem (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐹)) = ran (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
Distinct variable groups:   𝑘,𝐹,𝑛   𝜑,𝑘
Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem ovolval2lem
StepHypRef Expression
1 reex 10635 . . . . . . 7 ℝ ∈ V
21, 1xpex 7469 . . . . . 6 (ℝ × ℝ) ∈ V
3 inss2 4159 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4 mapss 8454 . . . . . 6 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ))
52, 3, 4mp2an 691 . . . . 5 (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ)
6 ovolval2lem.1 . . . . . 6 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
72inex2 5190 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ∈ V
87a1i 11 . . . . . . 7 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
9 nnex 11649 . . . . . . . 8 ℕ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
118, 10elmapd 8421 . . . . . 6 (𝜑 → (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
126, 11mpbird 260 . . . . 5 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
135, 12sseldi 3915 . . . 4 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
14 1zzd 12021 . . . . 5 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → 1 ∈ ℤ)
15 nnuz 12289 . . . . 5 ℕ = (ℤ‘1)
16 elmapi 8429 . . . . . . . . . 10 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
1716adantr 484 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
18 simpr 488 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
1917, 18fvovco 41991 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑘) = ((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘))))
2019fveq2d 6659 . . . . . . 7 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))))
2116ffvelrnda 6838 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (ℝ × ℝ))
22 xp1st 7716 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
2321, 22syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
24 xp2nd 7717 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
2521, 24syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
26 volicore 43388 . . . . . . . 8 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) ∈ ℝ)
2723, 25, 26syl2anc 587 . . . . . . 7 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) ∈ ℝ)
2820, 27eqeltrd 2890 . . . . . 6 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℝ)
2928recnd 10676 . . . . 5 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℂ)
30 eqid 2798 . . . . 5 (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘)))
31 eqid 2798 . . . . 5 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))))
3214, 15, 29, 30, 31fsumsermpt 42389 . . . 4 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))))
3313, 32syl 17 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))))
34 simpr 488 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
3534iftrued 4436 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
3613, 23sylan 583 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
3736adantr 484 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ)
3813, 25sylan 583 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
3938adantr 484 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
40 ressxr 10692 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
4140, 37sseldi 3915 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ*)
4240, 39sseldi 3915 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ*)
43 xpss 5539 . . . . . . . . . . . . . . . . . 18 (ℝ × ℝ) ⊆ (V × V)
4443, 21sseldi 3915 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (V × V))
45 1st2ndb 7724 . . . . . . . . . . . . . . . . 17 ((𝐹𝑘) ∈ (V × V) ↔ (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4644, 45sylib 221 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4713, 46sylan 583 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4847eqcomd 2804 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ = (𝐹𝑘))
49 inss1 4158 . . . . . . . . . . . . . . . . 17 ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤
5049a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤ )
516, 50fssd 6510 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ ≤ )
5251ffvelrnda 6838 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ≤ )
5348, 52eqeltrd 2890 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ ∈ ≤ )
54 df-br 5035 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)) ↔ ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ ∈ ≤ )
5553, 54sylibr 237 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)))
5655adantr 484 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)))
57 simpr 488 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
5839, 37lenltd 10793 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → ((2nd ‘(𝐹𝑘)) ≤ (1st ‘(𝐹𝑘)) ↔ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))))
5957, 58mpbird 260 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ≤ (1st ‘(𝐹𝑘)))
6041, 42, 56, 59xrletrid 12556 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
61 simp3 1135 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
62 simp1 1133 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ)
63 simp2 1134 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
6462, 63eqleltd 10791 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)) ↔ ((1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))))
6561, 64mpbid 235 . . . . . . . . . . . . 13 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))))
6665simprd 499 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
6766iffalsed 4439 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = 0)
6863recnd 10676 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
6961eqcomd 2804 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) = (1st ‘(𝐹𝑘)))
7068, 69subeq0bd 11073 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))) = 0)
7167, 70eqtr4d 2836 . . . . . . . . . 10 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7237, 39, 60, 71syl3anc 1368 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7335, 72pm2.61dan 812 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
74 volico 42793 . . . . . . . . 9 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) = if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0))
7536, 38, 74syl2anc 587 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) = if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0))
7636, 38, 55abssuble0d 14804 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7773, 75, 763eqtr4d 2843 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
7813adantr 484 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
79 simpr 488 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
8078, 79, 20syl2anc 587 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))))
8146fveq2d 6659 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
82 df-ov 7148 . . . . . . . . . . 11 ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
8382eqcomi 2807 . . . . . . . . . 10 ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩) = ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘)))
8483a1i 11 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩) = ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))))
8523recnd 10676 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℂ)
8625recnd 10676 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
87 eqid 2798 . . . . . . . . . . 11 (abs ∘ − ) = (abs ∘ − )
8887cnmetdval 23417 . . . . . . . . . 10 (((1st ‘(𝐹𝑘)) ∈ ℂ ∧ (2nd ‘(𝐹𝑘)) ∈ ℂ) → ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
8985, 86, 88syl2anc 587 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
9081, 84, 893eqtrd 2837 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
9178, 79, 90syl2anc 587 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
9277, 80, 913eqtr4d 2843 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = ((abs ∘ − )‘(𝐹𝑘)))
9392mpteq2dva 5129 . . . . 5 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
9413, 16syl 17 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
95 rr2sscn2 42166 . . . . . . 7 (ℝ × ℝ) ⊆ (ℂ × ℂ)
9695a1i 11 . . . . . 6 (𝜑 → (ℝ × ℝ) ⊆ (ℂ × ℂ))
97 absf 14709 . . . . . . . 8 abs:ℂ⟶ℝ
98 subf 10895 . . . . . . . 8 − :(ℂ × ℂ)⟶ℂ
99 fco 6513 . . . . . . . 8 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
10097, 98, 99mp2an 691 . . . . . . 7 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
101100a1i 11 . . . . . 6 (𝜑 → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
10294, 96, 101fcomptss 42002 . . . . 5 (𝜑 → ((abs ∘ − ) ∘ 𝐹) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
10393, 102eqtr4d 2836 . . . 4 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = ((abs ∘ − ) ∘ 𝐹))
104103seqeq3d 13392 . . 3 (𝜑 → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
10533, 104eqtr2d 2834 . 2 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
106105rneqd 5778 1 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐹)) = ran (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3442   ∩ cin 3882   ⊆ wss 3883  ifcif 4428  ⟨cop 4534   class class class wbr 5034   ↦ cmpt 5114   × cxp 5521  ran crn 5524   ∘ ccom 5527  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  1st c1st 7682  2nd c2nd 7683   ↑m cmap 8407  ℂcc 10542  ℝcr 10543  0cc0 10544  1c1 10545   + caddc 10547  ℝ*cxr 10681   < clt 10682   ≤ cle 10683   − cmin 10877  ℕcn 11643  [,)cico 12748  ...cfz 12905  seqcseq 13384  abscabs 14605  Σcsu 15054  volcvol 24108 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-inf2 9106  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7400  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-map 8409  df-pm 8410  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-fi 8877  df-sup 8908  df-inf 8909  df-oi 8976  df-dju 9332  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-2 11706  df-3 11707  df-n0 11904  df-z 11990  df-uz 12252  df-q 12357  df-rp 12398  df-xneg 12515  df-xadd 12516  df-xmul 12517  df-ioo 12750  df-ico 12752  df-icc 12753  df-fz 12906  df-fzo 13049  df-fl 13177  df-seq 13385  df-exp 13446  df-hash 13707  df-cj 14470  df-re 14471  df-im 14472  df-sqrt 14606  df-abs 14607  df-clim 14857  df-rlim 14858  df-sum 15055  df-rest 16708  df-topgen 16729  df-psmet 20104  df-xmet 20105  df-met 20106  df-bl 20107  df-mopn 20108  df-top 21540  df-topon 21557  df-bases 21592  df-cmp 22033  df-ovol 24109  df-vol 24110 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator