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Theorem ovolval3 44878
Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 24837 and ovolval2 44875 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 2736 . . 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 44875 . 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 11142 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ∈ V
65, 5xpex 7687 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ × ℝ) ∈ V
7 inss2 4189 . . . . . . . . . . . . . . . . . . . . . 22 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8 mapss 8827 . . . . . . . . . . . . . . . . . . . . . 22 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ))
96, 7, 8mp2an 690 . . . . . . . . . . . . . . . . . . . . 21 (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ)
109sseli 3940 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ))
11 elmapi 8787 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1210, 11syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1312ffvelcdmda 7035 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
14 1st2nd2 7960 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑛) ∈ (ℝ × ℝ) → (𝑓𝑛) = ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
1513, 14syl 17 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) = ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
1615fveq2d 6846 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
17 df-ov 7360 . . . . . . . . . . . . . . . . . 18 ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
1817eqcomi 2745 . . . . . . . . . . . . . . . . 17 ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))
1918a1i 11 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
2016, 19eqtrd 2776 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
2120fveq2d 6846 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))))
22 xp1st 7953 . . . . . . . . . . . . . . . 16 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
2313, 22syl 17 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
24 xp2nd 7954 . . . . . . . . . . . . . . . 16 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2513, 24syl 17 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
26 elmapi 8787 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2726adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
29 ovolfcl 24830 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3027, 28, 29syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3130simp3d 1144 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)))
32 volioo 24933 . . . . . . . . . . . . . . 15 (((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3323, 25, 31, 32syl3anc 1371 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3421, 33eqtrd 2776 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
35 ioof 13364 . . . . . . . . . . . . . . . 16 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
36 ffun 6671 . . . . . . . . . . . . . . . 16 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
3735, 36ax-mp 5 . . . . . . . . . . . . . . 15 Fun (,)
3837a1i 11 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → Fun (,))
39 rexpssxrxp 11200 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4039, 13sselid 3942 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ* × ℝ*))
4135fdmi 6680 . . . . . . . . . . . . . . . . 17 dom (,) = (ℝ* × ℝ*)
4241eqcomi 2745 . . . . . . . . . . . . . . . 16 (ℝ* × ℝ*) = dom (,)
4342a1i 11 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (ℝ* × ℝ*) = dom (,))
4440, 43eleqtrd 2840 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom (,))
45 fvco 6939 . . . . . . . . . . . . . 14 ((Fun (,) ∧ (𝑓𝑛) ∈ dom (,)) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4638, 44, 45syl2anc 584 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4715fveq2d 6846 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
48 df-ov 7360 . . . . . . . . . . . . . . . 16 ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
4948eqcomi 2745 . . . . . . . . . . . . . . 15 ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩) = ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛)))
5049a1i 11 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩) = ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))))
5123recnd 11183 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℂ)
5225recnd 11183 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℂ)
53 eqid 2736 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) = (abs ∘ − )
5453cnmetdval 24134 . . . . . . . . . . . . . . . 16 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
5551, 52, 54syl2anc 584 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
56 abssub 15211 . . . . . . . . . . . . . . . 16 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5751, 52, 56syl2anc 584 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5823, 25, 31abssubge0d 15316 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
5955, 57, 583eqtrd 2780 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6047, 50, 593eqtrd 2780 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6134, 46, 603eqtr4d 2786 . . . . . . . . . . . 12 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = ((abs ∘ − )‘(𝑓𝑛)))
6261mpteq2dva 5205 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
63 volioof 44218 . . . . . . . . . . . . 13 (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)
6463a1i 11 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞))
6539a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
6612, 65fssd 6686 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
67 fcompt 7079 . . . . . . . . . . . 12 (((vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
6864, 66, 67syl2anc 584 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
69 absf 15222 . . . . . . . . . . . . . 14 abs:ℂ⟶ℝ
70 subf 11403 . . . . . . . . . . . . . 14 − :(ℂ × ℂ)⟶ℂ
71 fco 6692 . . . . . . . . . . . . . 14 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7269, 70, 71mp2an 690 . . . . . . . . . . . . 13 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7372a1i 11 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
74 rr2sscn2 43590 . . . . . . . . . . . . . 14 (ℝ × ℝ) ⊆ (ℂ × ℂ)
7574a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
7612, 75fssd 6686 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77 fcompt 7079 . . . . . . . . . . . 12 (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7873, 76, 77syl2anc 584 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7962, 68, 783eqtr4d 2786 . . . . . . . . . 10 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
8079fveq2d 6846 . . . . . . . . 9 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((abs ∘ − ) ∘ 𝑓)))
8180eqeq2d 2747 . . . . . . . 8 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓))))
8281anbi2d 629 . . . . . . 7 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))))
8382rexbiia 3095 . . . . . 6 (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓))))
8483rabbii 3413 . . . . 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)))}
854, 84eqtr2i 2765 . . . 4 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
8685infeq1i 9414 . . 3 inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}, ℝ*, < ) = inf(𝑀, ℝ*, < )
8786a1i 11 . 2 (𝜑 → inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}, ℝ*, < ) = inf(𝑀, ℝ*, < ))
883, 87eqtrd 2776 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  {crab 3407  Vcvv 3445  cin 3909  wss 3910  𝒫 cpw 4560  cop 4592   cuni 4865   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  ccom 5637  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  infcinf 9377  cc 11049  cr 11050  0cc0 11051  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cmin 11385  cn 12153  (,)cioo 13264  [,]cicc 13267  abscabs 15119  vol*covol 24826  volcvol 24827  Σ^csumge0 44593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-sumge0 44594
This theorem is referenced by:  ovolval4lem2  44881
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