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

Theorem ovolval2lem 46599
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 11244 . . . . . . 7 ℝ ∈ V
21, 1xpex 7772 . . . . . 6 (ℝ × ℝ) ∈ V
3 inss2 4246 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4 mapss 8928 . . . . . 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 5324 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ∈ V
87a1i 11 . . . . . . 7 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
9 nnex 12270 . . . . . . . 8 ℕ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
118, 10elmapd 8879 . . . . . 6 (𝜑 → (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
126, 11mpbird 257 . . . . 5 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
135, 12sselid 3993 . . . 4 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
14 1zzd 12646 . . . . 5 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → 1 ∈ ℤ)
15 nnuz 12919 . . . . 5 ℕ = (ℤ‘1)
16 elmapi 8888 . . . . . . . . . 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 45136 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑘) = ((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘))))
2019fveq2d 6911 . . . . . . 7 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))))
2116ffvelcdmda 7104 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (ℝ × ℝ))
22 xp1st 8045 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
2321, 22syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
24 xp2nd 8046 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
2521, 24syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
26 volicore 46537 . . . . . . . 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 2839 . . . . . 6 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℝ)
2928recnd 11287 . . . . 5 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℂ)
30 eqid 2735 . . . . 5 (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘)))
31 eqid 2735 . . . . 5 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))))
3214, 15, 29, 30, 31fsumsermpt 45535 . . . 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 4539 . . . . . . . . 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 11303 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
4140, 37sselid 3993 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ*)
4240, 39sselid 3993 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ*)
43 xpss 5705 . . . . . . . . . . . . . . . . . 18 (ℝ × ℝ) ⊆ (V × V)
4443, 21sselid 3993 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (V × V))
45 1st2ndb 8053 . . . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ = (𝐹𝑘))
49 inss1 4245 . . . . . . . . . . . . . . . . 17 ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤
5049a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤ )
516, 50fssd 6754 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ ≤ )
5251ffvelcdmda 7104 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ≤ )
5348, 52eqeltrd 2839 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ ∈ ≤ )
54 df-br 5149 . . . . . . . . . . . . 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 11405 . . . . . . . . . . . 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 13194 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
61 simp3 1137 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
62 simp1 1135 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ)
63 simp2 1136 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
6462, 63eqleltd 11403 . . . . . . . . . . . . . 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 4542 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = 0)
6863recnd 11287 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
6961eqcomd 2741 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) = (1st ‘(𝐹𝑘)))
7068, 69subeq0bd 11687 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))) = 0)
7167, 70eqtr4d 2778 . . . . . . . . . 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 1370 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7335, 72pm2.61dan 813 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
74 volico 45939 . . . . . . . . 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 15468 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7773, 75, 763eqtr4d 2785 . . . . . . 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 6911 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
82 df-ov 7434 . . . . . . . . . . 11 ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
8382eqcomi 2744 . . . . . . . . . 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 11287 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℂ)
8625recnd 11287 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
87 eqid 2735 . . . . . . . . . . 11 (abs ∘ − ) = (abs ∘ − )
8887cnmetdval 24807 . . . . . . . . . 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 2779 . . . . . . . 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 2785 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = ((abs ∘ − )‘(𝐹𝑘)))
9392mpteq2dva 5248 . . . . 5 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
9413, 16syl 17 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
95 rr2sscn2 45316 . . . . . . 7 (ℝ × ℝ) ⊆ (ℂ × ℂ)
9695a1i 11 . . . . . 6 (𝜑 → (ℝ × ℝ) ⊆ (ℂ × ℂ))
97 absf 15373 . . . . . . . 8 abs:ℂ⟶ℝ
98 subf 11508 . . . . . . . 8 − :(ℂ × ℂ)⟶ℂ
99 fco 6761 . . . . . . . 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 45146 . . . . 5 (𝜑 → ((abs ∘ − ) ∘ 𝐹) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
10393, 102eqtr4d 2778 . . . 4 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = ((abs ∘ − ) ∘ 𝐹))
104103seqeq3d 14047 . . 3 (𝜑 → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
10533, 104eqtr2d 2776 . 2 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
106105rneqd 5952 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 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963  ifcif 4531  cop 4637   class class class wbr 5148  cmpt 5231   × cxp 5687  ran crn 5690  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156  *cxr 11292   < clt 11293  cle 11294  cmin 11490  cn 12264  [,)cico 13386  ...cfz 13544  seqcseq 14039  abscabs 15270  Σcsu 15719  volcvol 25512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-rest 17469  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-top 22916  df-topon 22933  df-bases 22969  df-cmp 23411  df-ovol 25513  df-vol 25514
This theorem is referenced by: (None)
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