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

Theorem ovolval2lem 42421
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 10463 . . . . . . 7 ℝ ∈ V
21, 1xpex 7324 . . . . . 6 (ℝ × ℝ) ∈ V
3 inss2 4121 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4 mapss 8292 . . . . . 6 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ))
52, 3, 4mp2an 688 . . . . 5 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ)
6 ovolval2lem.1 . . . . . 6 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
72inex2 5107 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ∈ V
87a1i 11 . . . . . . 7 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
9 nnex 11481 . . . . . . . 8 ℕ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
118, 10elmapd 8261 . . . . . 6 (𝜑 → (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
126, 11mpbird 258 . . . . 5 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
135, 12sseldi 3882 . . . 4 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
14 1zzd 11851 . . . . 5 (𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 1 ∈ ℤ)
15 nnuz 12119 . . . . 5 ℕ = (ℤ‘1)
16 elmapi 8269 . . . . . . . . . 10 (𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
1716adantr 481 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
18 simpr 485 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
1917, 18fvovco 40948 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑘) = ((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘))))
2019fveq2d 6534 . . . . . . 7 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))))
2116ffvelrnda 6707 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (ℝ × ℝ))
22 xp1st 7568 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
2321, 22syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
24 xp2nd 7569 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
2521, 24syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
26 volicore 42359 . . . . . . . 8 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) ∈ ℝ)
2723, 25, 26syl2anc 584 . . . . . . 7 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) ∈ ℝ)
2820, 27eqeltrd 2881 . . . . . 6 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℝ)
2928recnd 10504 . . . . 5 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℂ)
30 eqid 2793 . . . . 5 (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘)))
31 eqid 2793 . . . . 5 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))))
3214, 15, 29, 30, 31fsumsermpt 41356 . . . 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 485 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
3534iftrued 4383 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
3613, 23sylan 580 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
3736adantr 481 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ)
3813, 25sylan 580 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
3938adantr 481 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
40 ressxr 10520 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
4140, 37sseldi 3882 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ*)
4240, 39sseldi 3882 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ*)
43 xpss 5451 . . . . . . . . . . . . . . . . . 18 (ℝ × ℝ) ⊆ (V × V)
4443, 21sseldi 3882 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (V × V))
45 1st2ndb 7576 . . . . . . . . . . . . . . . . 17 ((𝐹𝑘) ∈ (V × V) ↔ (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4644, 45sylib 219 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4713, 46sylan 580 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4847eqcomd 2799 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ = (𝐹𝑘))
49 inss1 4120 . . . . . . . . . . . . . . . . 17 ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤
5049a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤ )
516, 50fssd 6388 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ ≤ )
5251ffvelrnda 6707 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ≤ )
5348, 52eqeltrd 2881 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ ∈ ≤ )
54 df-br 4957 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)) ↔ ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ ∈ ≤ )
5553, 54sylibr 235 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)))
5655adantr 481 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)))
57 simpr 485 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
5839, 37lenltd 10622 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → ((2nd ‘(𝐹𝑘)) ≤ (1st ‘(𝐹𝑘)) ↔ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))))
5957, 58mpbird 258 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ≤ (1st ‘(𝐹𝑘)))
6041, 42, 56, 59xrletrid 12387 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
61 simp3 1129 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
62 simp1 1127 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ)
63 simp2 1128 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
6462, 63eqleltd 10620 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 13 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))))
6665simprd 496 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
6766iffalsed 4386 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = 0)
6863recnd 10504 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
6961eqcomd 2799 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) = (1st ‘(𝐹𝑘)))
7068, 69subeq0bd 10903 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))) = 0)
7167, 70eqtr4d 2832 . . . . . . . . . 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 1362 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7335, 72pm2.61dan 809 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
74 volico 41764 . . . . . . . . 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 584 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) = if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0))
7636, 38, 55abssuble0d 14614 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7773, 75, 763eqtr4d 2839 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
7813adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
79 simpr 485 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
8078, 79, 20syl2anc 584 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))))
8146fveq2d 6534 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
82 df-ov 7010 . . . . . . . . . . 11 ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
8382eqcomi 2802 . . . . . . . . . 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 10504 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℂ)
8625recnd 10504 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
87 eqid 2793 . . . . . . . . . . 11 (abs ∘ − ) = (abs ∘ − )
8887cnmetdval 23050 . . . . . . . . . 10 (((1st ‘(𝐹𝑘)) ∈ ℂ ∧ (2nd ‘(𝐹𝑘)) ∈ ℂ) → ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
8985, 86, 88syl2anc 584 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
9081, 84, 893eqtrd 2833 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
9178, 79, 90syl2anc 584 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
9277, 80, 913eqtr4d 2839 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = ((abs ∘ − )‘(𝐹𝑘)))
9392mpteq2dva 5049 . . . . 5 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
9413, 16syl 17 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
95 rr2sscn2 41128 . . . . . . 7 (ℝ × ℝ) ⊆ (ℂ × ℂ)
9695a1i 11 . . . . . 6 (𝜑 → (ℝ × ℝ) ⊆ (ℂ × ℂ))
97 absf 14519 . . . . . . . 8 abs:ℂ⟶ℝ
98 subf 10724 . . . . . . . 8 − :(ℂ × ℂ)⟶ℂ
99 fco 6391 . . . . . . . 8 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
10097, 98, 99mp2an 688 . . . . . . 7 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
101100a1i 11 . . . . . 6 (𝜑 → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
10294, 96, 101fcomptss 40959 . . . . 5 (𝜑 → ((abs ∘ − ) ∘ 𝐹) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
10393, 102eqtr4d 2832 . . . 4 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = ((abs ∘ − ) ∘ 𝐹))
104103seqeq3d 13215 . . 3 (𝜑 → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
10533, 104eqtr2d 2830 . 2 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
106105rneqd 5682 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 396  w3a 1078   = wceq 1520  wcel 2079  Vcvv 3432  cin 3853  wss 3854  ifcif 4375  cop 4472   class class class wbr 4956  cmpt 5035   × cxp 5433  ran crn 5436  ccom 5439  wf 6213  cfv 6217  (class class class)co 7007  1st c1st 7534  2nd c2nd 7535  𝑚 cmap 8247  cc 10370  cr 10371  0cc0 10372  1c1 10373   + caddc 10375  *cxr 10509   < clt 10510  cle 10511  cmin 10706  cn 11475  [,)cico 12579  ...cfz 12731  seqcseq 13207  abscabs 14415  Σcsu 14864  volcvol 23735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-inf2 8939  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449  ax-pre-sup 10450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-fal 1533  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-se 5395  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-isom 6226  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-of 7258  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-2o 7945  df-oadd 7948  df-er 8130  df-map 8249  df-pm 8250  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-fi 8711  df-sup 8742  df-inf 8743  df-oi 8810  df-dju 9165  df-card 9203  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-div 11135  df-nn 11476  df-2 11537  df-3 11538  df-n0 11735  df-z 11819  df-uz 12083  df-q 12187  df-rp 12229  df-xneg 12346  df-xadd 12347  df-xmul 12348  df-ioo 12581  df-ico 12583  df-icc 12584  df-fz 12732  df-fzo 12873  df-fl 13000  df-seq 13208  df-exp 13268  df-hash 13529  df-cj 14280  df-re 14281  df-im 14282  df-sqrt 14416  df-abs 14417  df-clim 14667  df-rlim 14668  df-sum 14865  df-rest 16513  df-topgen 16534  df-psmet 20207  df-xmet 20208  df-met 20209  df-bl 20210  df-mopn 20211  df-top 21174  df-topon 21191  df-bases 21226  df-cmp 21667  df-ovol 23736  df-vol 23737
This theorem is referenced by: (None)
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