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

Theorem ovolval2lem 41178
 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 10065 . . . . . . 7 ℝ ∈ V
21, 1xpex 7004 . . . . . 6 (ℝ × ℝ) ∈ V
3 inss2 3867 . . . . . 6 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4 mapss 7942 . . . . . 6 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ))
52, 3, 4mp2an 708 . . . . 5 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ)
6 ovolval2lem.1 . . . . . 6 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
72inex2 4833 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ∈ V
87a1i 11 . . . . . . 7 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
9 nnex 11064 . . . . . . . 8 ℕ ∈ V
109a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
118, 10elmapd 7913 . . . . . 6 (𝜑 → (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
126, 11mpbird 247 . . . . 5 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
135, 12sseldi 3634 . . . 4 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
14 1zzd 11446 . . . . 5 (𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 1 ∈ ℤ)
15 nnuz 11761 . . . . 5 ℕ = (ℤ‘1)
16 elmapi 7921 . . . . . . . . . 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 476 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
1917, 18fvovco 39695 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑘) = ((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘))))
2019fveq2d 6233 . . . . . . 7 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))))
2116ffvelrnda 6399 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (ℝ × ℝ))
22 xp1st 7242 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
2321, 22syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
24 xp2nd 7243 . . . . . . . . 9 ((𝐹𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
2521, 24syl 17 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
26 volicore 41116 . . . . . . . 8 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) ∈ ℝ)
2723, 25, 26syl2anc 694 . . . . . . 7 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) ∈ ℝ)
2820, 27eqeltrd 2730 . . . . . 6 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℝ)
2928recnd 10106 . . . . 5 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℂ)
30 eqid 2651 . . . . 5 (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘)))
31 eqid 2651 . . . . 5 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))))
3214, 15, 29, 30, 31fsumsermpt 40129 . . . 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 476 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
3534iftrued 4127 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
3613, 23sylan 487 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℝ)
3736adantr 480 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ)
3813, 25sylan 487 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
3938adantr 480 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
40 ressxr 10121 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
4140, 37sseldi 3634 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ*)
4240, 39sseldi 3634 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ*)
43 xpss 5159 . . . . . . . . . . . . . . . . . 18 (ℝ × ℝ) ⊆ (V × V)
4443, 21sseldi 3634 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ (V × V))
45 1st2ndb 7250 . . . . . . . . . . . . . . . . 17 ((𝐹𝑘) ∈ (V × V) ↔ (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4644, 45sylib 208 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4713, 46sylan 487 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
4847eqcomd 2657 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ = (𝐹𝑘))
49 inss1 3866 . . . . . . . . . . . . . . . . 17 ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤
5049a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤ )
516, 50fssd 6095 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ ≤ )
5251ffvelrnda 6399 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ≤ )
5348, 52eqeltrd 2730 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ ∈ ≤ )
54 df-br 4686 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)) ↔ ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩ ∈ ≤ )
5553, 54sylibr 224 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)))
5655adantr 480 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)))
57 simpr 476 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
5839, 37lenltd 10221 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → ((2nd ‘(𝐹𝑘)) ≤ (1st ‘(𝐹𝑘)) ↔ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))))
5957, 58mpbird 247 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ≤ (1st ‘(𝐹𝑘)))
6041, 42, 56, 59xrletrid 12024 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
61 simp3 1083 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘)))
62 simp1 1081 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (1st ‘(𝐹𝑘)) ∈ ℝ)
63 simp2 1082 . . . . . . . . . . . . . . 15 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℝ)
6462, 63eqleltd 10219 . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . 13 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((1st ‘(𝐹𝑘)) ≤ (2nd ‘(𝐹𝑘)) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))))
6665simprd 478 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)))
6766iffalsed 4130 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = 0)
6863recnd 10106 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
6961eqcomd 2657 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → (2nd ‘(𝐹𝑘)) = (1st ‘(𝐹𝑘)))
7068, 69subeq0bd 10494 . . . . . . . . . . 11 (((1st ‘(𝐹𝑘)) ∈ ℝ ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ ∧ (1st ‘(𝐹𝑘)) = (2nd ‘(𝐹𝑘))) → ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))) = 0)
7167, 70eqtr4d 2688 . . . . . . . . . 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 1366 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘))) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7335, 72pm2.61dan 849 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
74 volico 40518 . . . . . . . . 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 694 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))) = if((1st ‘(𝐹𝑘)) < (2nd ‘(𝐹𝑘)), ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))), 0))
7636, 38, 55abssuble0d 14215 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))) = ((2nd ‘(𝐹𝑘)) − (1st ‘(𝐹𝑘))))
7773, 75, 763eqtr4d 2695 . . . . . . 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 476 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
8078, 79, 20syl2anc 694 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1st ‘(𝐹𝑘))[,)(2nd ‘(𝐹𝑘)))))
8146fveq2d 6233 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
82 df-ov 6693 . . . . . . . . . . 11 ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
8382eqcomi 2660 . . . . . . . . . 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 10106 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐹𝑘)) ∈ ℂ)
8625recnd 10106 . . . . . . . . . 10 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐹𝑘)) ∈ ℂ)
87 eqid 2651 . . . . . . . . . . 11 (abs ∘ − ) = (abs ∘ − )
8887cnmetdval 22621 . . . . . . . . . 10 (((1st ‘(𝐹𝑘)) ∈ ℂ ∧ (2nd ‘(𝐹𝑘)) ∈ ℂ) → ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
8985, 86, 88syl2anc 694 . . . . . . . . 9 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → ((1st ‘(𝐹𝑘))(abs ∘ − )(2nd ‘(𝐹𝑘))) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
9081, 84, 893eqtrd 2689 . . . . . . . 8 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
9178, 79, 90syl2anc 694 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑘)) = (abs‘((1st ‘(𝐹𝑘)) − (2nd ‘(𝐹𝑘)))))
9277, 80, 913eqtr4d 2695 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = ((abs ∘ − )‘(𝐹𝑘)))
9392mpteq2dva 4777 . . . . 5 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
9413, 16syl 17 . . . . . 6 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
95 rr2sscn2 39895 . . . . . . 7 (ℝ × ℝ) ⊆ (ℂ × ℂ)
9695a1i 11 . . . . . 6 (𝜑 → (ℝ × ℝ) ⊆ (ℂ × ℂ))
97 absf 14121 . . . . . . . 8 abs:ℂ⟶ℝ
98 subf 10321 . . . . . . . 8 − :(ℂ × ℂ)⟶ℂ
99 fco 6096 . . . . . . . 8 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
10097, 98, 99mp2an 708 . . . . . . 7 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
101100a1i 11 . . . . . 6 (𝜑 → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
10294, 96, 101fcomptss 39709 . . . . 5 (𝜑 → ((abs ∘ − ) ∘ 𝐹) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹𝑘))))
10393, 102eqtr4d 2688 . . . 4 (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = ((abs ∘ − ) ∘ 𝐹))
104103seqeq3d 12849 . . 3 (𝜑 → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
10533, 104eqtr2d 2686 . 2 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
106105rneqd 5385 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 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  ifcif 4119  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  ran crn 5144   ∘ ccom 5147  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209   ↑𝑚 cmap 7899  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977  ℝ*cxr 10111   < clt 10112   ≤ cle 10113   − cmin 10304  ℕcn 11058  [,)cico 12215  ...cfz 12364  seqcseq 12841  abscabs 14018  Σcsu 14460  volcvol 23278 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280 This theorem is referenced by: (None)
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