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Theorem ovolval3 39537
 Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 23049 and ovolval2 39534 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 2610 . . 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 39534 . 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 9906 . . . . . . . . . . . . . . . . . . . . . . . 24 ℝ ∈ V
65, 5xpex 6860 . . . . . . . . . . . . . . . . . . . . . . 23 (ℝ × ℝ) ∈ V
7 inss2 3796 . . . . . . . . . . . . . . . . . . . . . . 23 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8 mapss 7786 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ))
96, 7, 8mp2an 704 . . . . . . . . . . . . . . . . . . . . . 22 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ)
109sseli 3564 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
11 elmapi 7765 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1210, 11syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1312ffvelrnda 6267 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
14 1st2nd2 7096 . . . . . . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
17 df-ov 6552 . . . . . . . . . . . . . . . . . . 19 ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
1817eqcomi 2619 . . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
2120fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))))
22 xp1st 7089 . . . . . . . . . . . . . . . . 17 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
2313, 22syl 17 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
24 xp2nd 7090 . . . . . . . . . . . . . . . . 17 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2513, 24syl 17 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
26 elmapi 7765 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2726adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
29 ovolfcl 23042 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3027, 28, 29syl2anc 691 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3130simp3d 1068 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)))
32 volioo 38840 . . . . . . . . . . . . . . . 16 (((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3323, 25, 31, 32syl3anc 1318 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3421, 33eqtrd 2644 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
35 ioof 12142 . . . . . . . . . . . . . . . . 17 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
36 ffun 5961 . . . . . . . . . . . . . . . . 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 9963 . . . . . . . . . . . . . . . . 17 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4039, 13sseldi 3566 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ* × ℝ*))
4135fdmi 5965 . . . . . . . . . . . . . . . . . 18 dom (,) = (ℝ* × ℝ*)
4241eqcomi 2619 . . . . . . . . . . . . . . . . 17 (ℝ* × ℝ*) = dom (,)
4342a1i 11 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (ℝ* × ℝ*) = dom (,))
4440, 43eleqtrd 2690 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom (,))
45 fvco 6184 . . . . . . . . . . . . . . 15 ((Fun (,) ∧ (𝑓𝑛) ∈ dom (,)) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4638, 44, 45syl2anc 691 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4715fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
48 df-ov 6552 . . . . . . . . . . . . . . . . 17 ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
4948eqcomi 2619 . . . . . . . . . . . . . . . 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 9947 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℂ)
5225recnd 9947 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℂ)
53 eqid 2610 . . . . . . . . . . . . . . . . . 18 (abs ∘ − ) = (abs ∘ − )
5453cnmetdval 22384 . . . . . . . . . . . . . . . . 17 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
5551, 52, 54syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
56 abssub 13914 . . . . . . . . . . . . . . . . 17 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5751, 52, 56syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5823, 25, 31abssubge0d 14018 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
5955, 57, 583eqtrd 2648 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6047, 50, 593eqtrd 2648 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6134, 46, 603eqtr4d 2654 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = ((abs ∘ − )‘(𝑓𝑛)))
6261mpteq2dva 4672 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
63 volioof 38880 . . . . . . . . . . . . . 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 5970 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
67 fcompt 6306 . . . . . . . . . . . . 13 (((vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
6864, 66, 67syl2anc 691 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
69 absf 13925 . . . . . . . . . . . . . . 15 abs:ℂ⟶ℝ
70 subf 10162 . . . . . . . . . . . . . . 15 − :(ℂ × ℂ)⟶ℂ
71 fco 5971 . . . . . . . . . . . . . . 15 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7269, 70, 71mp2an 704 . . . . . . . . . . . . . 14 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7372a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
74 rr2sscn2 38523 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℂ × ℂ)
7574a1i 11 . . . . . . . . . . . . . 14 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
7612, 75fssd 5970 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77 fcompt 6306 . . . . . . . . . . . . 13 (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7873, 76, 77syl2anc 691 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7962, 68, 783eqtr4d 2654 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
8079fveq2d 6107 . . . . . . . . . 10 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((abs ∘ − ) ∘ 𝑓)))
8180eqeq2d 2620 . . . . . . . . 9 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓))))
8281anbi2d 736 . . . . . . . 8 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))))
8382rexbiia 3022 . . . . . . 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 3161 . . . . 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 2633 . . . 4 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
8786infeq1i 8267 . . 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 2644 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  infcinf 8230  ℂcc 9813  ℝcr 9814  0cc0 9815  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  (,)cioo 12046  [,]cicc 12049  abscabs 13822  vol*covol 23038  volcvol 23039  Σ^csumge0 39255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256 This theorem is referenced by:  ovolval4lem2  39540
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