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

Theorem ovolval2lem 46663
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 11247 . . . . . . 7 ℝ ∈ V
21, 1xpex 7774 . . . . . 6 (ℝ × ℝ) ∈ V
3 inss2 4237 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4 mapss 8930 . . . . . 6 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ))
52, 3, 4mp2an 692 . . . . 5 (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ)
6 ovolval2lem.1 . . . . . 6 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
72inex2 5317 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ∈ V
87a1i 11 . . . . . . 7 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
9 nnex 12273 . . . . . . . 8 ℕ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
118, 10elmapd 8881 . . . . . 6 (𝜑 → (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
126, 11mpbird 257 . . . . 5 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
135, 12sselid 3980 . . . 4 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
14 1zzd 12650 . . . . 5 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → 1 ∈ ℤ)
15 nnuz 12922 . . . . 5 ℕ = (ℤ‘1)
16 elmapi 8890 . . . . . . . . . 10 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
1716adantr 480 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
18 simpr 484 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
1917, 18fvovco 45203 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑘) = ((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘))))
2019fveq2d 6909 . . . . . . 7 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))))
2116ffvelcdmda 7103 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (ℝ × ℝ))
22 xp1st 8047 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
2321, 22syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
24 xp2nd 8048 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
2521, 24syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
26 volicore 46601 . . . . . . . 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 2840 . . . . . 6 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℝ)
2928recnd 11290 . . . . 5 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℂ)
30 eqid 2736 . . . . 5 (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘)))
31 eqid 2736 . . . . 5 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))))
3214, 15, 29, 30, 31fsumsermpt 45599 . . . 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 484 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
3534iftrued 4532 . . . . . . . . 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 480 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ)
3813, 25sylan 580 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
3938adantr 480 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
40 ressxr 11306 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
4140, 37sselid 3980 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ*)
4240, 39sselid 3980 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ*)
43 xpss 5700 . . . . . . . . . . . . . . . . . 18 (ℝ × ℝ) ⊆ (V × V)
4443, 21sselid 3980 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (V × V))
45 1st2ndb 8055 . . . . . . . . . . . . . . . . 17 ((𝐹𝑘) ∈ (V × V) ↔ (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4644, 45sylib 218 . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ = (𝐹𝑘))
49 inss1 4236 . . . . . . . . . . . . . . . . 17 ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤
5049a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤ )
516, 50fssd 6752 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ ≤ )
5251ffvelcdmda 7103 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ≤ )
5348, 52eqeltrd 2840 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ ∈ ≤ )
54 df-br 5143 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)) ↔ ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ ∈ ≤ )
5553, 54sylibr 234 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)))
5655adantr 480 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)))
57 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
5839, 37lenltd 11408 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → ((2nd ‘(𝐹𝑘)) ≤ (1st ‘(𝐹𝑘)) ↔ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))))
5957, 58mpbird 257 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ≤ (1st ‘(𝐹𝑘)))
6041, 42, 56, 59xrletrid 13198 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
61 simp3 1138 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
62 simp1 1136 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ)
63 simp2 1137 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
6462, 63eqleltd 11406 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 13 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))))
6665simprd 495 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
6766iffalsed 4535 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = 0)
6863recnd 11290 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
6961eqcomd 2742 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) = (1st ‘(𝐹𝑘)))
7068, 69subeq0bd 11690 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))) = 0)
7167, 70eqtr4d 2779 . . . . . . . . . 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 1372 . . . . . . . . 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 46003 . . . . . . . . 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 15472 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7773, 75, 763eqtr4d 2786 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
7813adantr 480 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
79 simpr 484 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
8078, 79, 20syl2anc 584 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))))
8146fveq2d 6909 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
82 df-ov 7435 . . . . . . . . . . 11 ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
8382eqcomi 2745 . . . . . . . . . 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 11290 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℂ)
8625recnd 11290 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
87 eqid 2736 . . . . . . . . . . 11 (abs ∘ − ) = (abs ∘ − )
8887cnmetdval 24792 . . . . . . . . . 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 2780 . . . . . . . 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 2786 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = ((abs ∘ − )‘(𝐹𝑘)))
9392mpteq2dva 5241 . . . . 5 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
9413, 16syl 17 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
95 rr2sscn2 45382 . . . . . . 7 (ℝ × ℝ) ⊆ (ℂ × ℂ)
9695a1i 11 . . . . . 6 (𝜑 → (ℝ × ℝ) ⊆ (ℂ × ℂ))
97 absf 15377 . . . . . . . 8 abs:ℂ⟶ℝ
98 subf 11511 . . . . . . . 8 − :(ℂ × ℂ)⟶ℂ
99 fco 6759 . . . . . . . 8 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
10097, 98, 99mp2an 692 . . . . . . 7 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
101100a1i 11 . . . . . 6 (𝜑 → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
10294, 96, 101fcomptss 45213 . . . . 5 (𝜑 → ((abs ∘ − ) ∘ 𝐹) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
10393, 102eqtr4d 2779 . . . 4 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = ((abs ∘ − ) ∘ 𝐹))
104103seqeq3d 14051 . . 3 (𝜑 → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
10533, 104eqtr2d 2777 . 2 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
106105rneqd 5948 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 395  w3a 1086   = wceq 1539  wcel 2107  Vcvv 3479  cin 3949  wss 3950  ifcif 4524  cop 4631   class class class wbr 5142  cmpt 5224   × cxp 5682  ran crn 5685  ccom 5688  wf 6556  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  m cmap 8867  cc 11154  cr 11155  0cc0 11156  1c1 11157   + caddc 11159  *cxr 11295   < clt 11296  cle 11297  cmin 11493  cn 12267  [,)cico 13390  ...cfz 13548  seqcseq 14043  abscabs 15274  Σcsu 15723  volcvol 25499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fi 9452  df-sup 9483  df-inf 9484  df-oi 9551  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ioo 13392  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-fl 13833  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-rlim 15526  df-sum 15724  df-rest 17468  df-topgen 17489  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-top 22901  df-topon 22918  df-bases 22954  df-cmp 23396  df-ovol 25500  df-vol 25501
This theorem is referenced by: (None)
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