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Theorem ovolval3 39320
Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 22993 and ovolval2 39317 for alternative expressions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovolval3.a (𝜑𝐴 ⊆ ℝ)
ovolval3.m 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
Assertion
Ref Expression
ovolval3 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Distinct variable groups:   𝐴,𝑓,𝑦   𝜑,𝑓,𝑦
Allowed substitution hints:   𝑀(𝑦,𝑓)

Proof of Theorem ovolval3
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 ovolval3.a . . 3 (𝜑𝐴 ⊆ ℝ)
2 eqid 2609 . . 3 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))} = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}
31, 2ovolval2 39317 . 2 (𝜑 → (vol*‘𝐴) = inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}, ℝ*, < ))
4 ovolval3.m . . . . 5 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
5 reex 9883 . . . . . . . . . . . . . . . . . . . . . . . 24 ℝ ∈ V
65, 5xpex 6837 . . . . . . . . . . . . . . . . . . . . . . 23 (ℝ × ℝ) ∈ V
7 inss2 3795 . . . . . . . . . . . . . . . . . . . . . . 23 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8 mapss 7763 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ))
96, 7, 8mp2an 703 . . . . . . . . . . . . . . . . . . . . . 22 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ)
109sseli 3563 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
11 elmapi 7742 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1210, 11syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1312ffvelrnda 6251 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
14 1st2nd2 7073 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑛) ∈ (ℝ × ℝ) → (𝑓𝑛) = ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
1513, 14syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) = ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
1615fveq2d 6091 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
17 df-ov 6529 . . . . . . . . . . . . . . . . . . 19 ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
1817eqcomi 2618 . . . . . . . . . . . . . . . . . 18 ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))
1918a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
2016, 19eqtrd 2643 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
2120fveq2d 6091 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))))
22 xp1st 7066 . . . . . . . . . . . . . . . . 17 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
2313, 22syl 17 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
24 xp2nd 7067 . . . . . . . . . . . . . . . . 17 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2513, 24syl 17 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
26 elmapi 7742 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2726adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28 simpr 475 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
29 ovolfcl 22986 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3027, 28, 29syl2anc 690 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3130simp3d 1067 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)))
32 volioo 38623 . . . . . . . . . . . . . . . 16 (((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3323, 25, 31, 32syl3anc 1317 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3421, 33eqtrd 2643 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
35 ioof 12100 . . . . . . . . . . . . . . . . 17 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
36 ffun 5946 . . . . . . . . . . . . . . . . 17 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
3735, 36ax-mp 5 . . . . . . . . . . . . . . . 16 Fun (,)
3837a1i 11 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → Fun (,))
39 rexpssxrxp 9940 . . . . . . . . . . . . . . . . 17 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4039, 13sseldi 3565 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ* × ℝ*))
4135fdmi 5950 . . . . . . . . . . . . . . . . . 18 dom (,) = (ℝ* × ℝ*)
4241eqcomi 2618 . . . . . . . . . . . . . . . . 17 (ℝ* × ℝ*) = dom (,)
4342a1i 11 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (ℝ* × ℝ*) = dom (,))
4440, 43eleqtrd 2689 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom (,))
45 fvco 6168 . . . . . . . . . . . . . . 15 ((Fun (,) ∧ (𝑓𝑛) ∈ dom (,)) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4638, 44, 45syl2anc 690 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4715fveq2d 6091 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
48 df-ov 6529 . . . . . . . . . . . . . . . . 17 ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
4948eqcomi 2618 . . . . . . . . . . . . . . . 16 ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩) = ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛)))
5049a1i 11 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩) = ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))))
5123recnd 9924 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℂ)
5225recnd 9924 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℂ)
53 eqid 2609 . . . . . . . . . . . . . . . . . 18 (abs ∘ − ) = (abs ∘ − )
5453cnmetdval 22331 . . . . . . . . . . . . . . . . 17 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
5551, 52, 54syl2anc 690 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
56 abssub 13862 . . . . . . . . . . . . . . . . 17 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5751, 52, 56syl2anc 690 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5823, 25, 31abssubge0d 13966 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
5955, 57, 583eqtrd 2647 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6047, 50, 593eqtrd 2647 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6134, 46, 603eqtr4d 2653 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = ((abs ∘ − )‘(𝑓𝑛)))
6261mpteq2dva 4666 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
63 volioof 38663 . . . . . . . . . . . . . 14 (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)
6463a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞))
6539a1i 11 . . . . . . . . . . . . . 14 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
6612, 65fssd 5955 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
67 fcompt 6290 . . . . . . . . . . . . 13 (((vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
6864, 66, 67syl2anc 690 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
69 absf 13873 . . . . . . . . . . . . . . 15 abs:ℂ⟶ℝ
70 subf 10134 . . . . . . . . . . . . . . 15 − :(ℂ × ℂ)⟶ℂ
71 fco 5956 . . . . . . . . . . . . . . 15 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7269, 70, 71mp2an 703 . . . . . . . . . . . . . 14 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7372a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
74 rr2sscn2 38306 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℂ × ℂ)
7574a1i 11 . . . . . . . . . . . . . 14 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
7612, 75fssd 5955 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77 fcompt 6290 . . . . . . . . . . . . 13 (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7873, 76, 77syl2anc 690 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7962, 68, 783eqtr4d 2653 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
8079fveq2d 6091 . . . . . . . . . 10 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((abs ∘ − ) ∘ 𝑓)))
8180eqeq2d 2619 . . . . . . . . 9 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓))))
8281anbi2d 735 . . . . . . . 8 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))))
8382rexbiia 3021 . . . . . . 7 (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓))))
8483a1i 11 . . . . . 6 (𝑦 ∈ ℝ* → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))))
8584rabbiia 3160 . . . . 5 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}
864, 85eqtr2i 2632 . . . 4 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
8786infeq1i 8244 . . 3 inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}, ℝ*, < ) = inf(𝑀, ℝ*, < )
8887a1i 11 . 2 (𝜑 → inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}, ℝ*, < ) = inf(𝑀, ℝ*, < ))
893, 88eqtrd 2643 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wrex 2896  {crab 2899  Vcvv 3172  cin 3538  wss 3539  𝒫 cpw 4107  cop 4130   cuni 4366   class class class wbr 4577  cmpt 4637   × cxp 5025  dom cdm 5027  ran crn 5028  ccom 5031  Fun wfun 5783  wf 5785  cfv 5789  (class class class)co 6526  1st c1st 7034  2nd c2nd 7035  𝑚 cmap 7721  infcinf 8207  cc 9790  cr 9791  0cc0 9792  +∞cpnf 9927  *cxr 9929   < clt 9930  cle 9931  cmin 10117  cn 10869  (,)cioo 12004  [,]cicc 12007  abscabs 13770  vol*covol 22982  volcvol 22983  Σ^csumge0 39038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-2 10928  df-3 10929  df-n0 11142  df-z 11213  df-uz 11522  df-q 11623  df-rp 11667  df-xneg 11780  df-xadd 11781  df-xmul 11782  df-ioo 12008  df-ico 12010  df-icc 12011  df-fz 12155  df-fzo 12292  df-fl 12412  df-seq 12621  df-exp 12680  df-hash 12937  df-cj 13635  df-re 13636  df-im 13637  df-sqrt 13771  df-abs 13772  df-clim 14015  df-rlim 14016  df-sum 14213  df-rest 15854  df-topgen 15875  df-psmet 19507  df-xmet 19508  df-met 19509  df-bl 19510  df-mopn 19511  df-top 20468  df-bases 20469  df-topon 20470  df-cmp 20947  df-ovol 22984  df-vol 22985  df-sumge0 39039
This theorem is referenced by:  ovolval4lem2  39323
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