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Theorem ovolval3 41381
 Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 23461 and ovolval2 41378 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 2771 . . 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 41378 . 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 10229 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ∈ V
65, 5xpex 7109 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ × ℝ) ∈ V
7 inss2 3982 . . . . . . . . . . . . . . . . . . . . . 22 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8 mapss 8054 . . . . . . . . . . . . . . . . . . . . . 22 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ))
96, 7, 8mp2an 672 . . . . . . . . . . . . . . . . . . . . 21 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ)
109sseli 3748 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
11 elmapi 8031 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1210, 11syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1312ffvelrnda 6502 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
14 1st2nd2 7354 . . . . . . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
17 df-ov 6796 . . . . . . . . . . . . . . . . . 18 ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
1817eqcomi 2780 . . . . . . . . . . . . . . . . 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 2805 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
2120fveq2d 6336 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))))
22 xp1st 7347 . . . . . . . . . . . . . . . 16 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
2313, 22syl 17 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
24 xp2nd 7348 . . . . . . . . . . . . . . . 16 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2513, 24syl 17 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
26 elmapi 8031 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2726adantr 466 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28 simpr 471 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
29 ovolfcl 23454 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3027, 28, 29syl2anc 573 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3130simp3d 1138 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)))
32 volioo 23557 . . . . . . . . . . . . . . 15 (((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3323, 25, 31, 32syl3anc 1476 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3421, 33eqtrd 2805 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
35 ioof 12477 . . . . . . . . . . . . . . . 16 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
36 ffun 6188 . . . . . . . . . . . . . . . 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 10286 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4039, 13sseldi 3750 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ* × ℝ*))
4135fdmi 6192 . . . . . . . . . . . . . . . . 17 dom (,) = (ℝ* × ℝ*)
4241eqcomi 2780 . . . . . . . . . . . . . . . 16 (ℝ* × ℝ*) = dom (,)
4342a1i 11 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (ℝ* × ℝ*) = dom (,))
4440, 43eleqtrd 2852 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom (,))
45 fvco 6416 . . . . . . . . . . . . . 14 ((Fun (,) ∧ (𝑓𝑛) ∈ dom (,)) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4638, 44, 45syl2anc 573 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4715fveq2d 6336 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
48 df-ov 6796 . . . . . . . . . . . . . . . 16 ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
4948eqcomi 2780 . . . . . . . . . . . . . . 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 10270 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℂ)
5225recnd 10270 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℂ)
53 eqid 2771 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) = (abs ∘ − )
5453cnmetdval 22794 . . . . . . . . . . . . . . . 16 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
5551, 52, 54syl2anc 573 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
56 abssub 14274 . . . . . . . . . . . . . . . 16 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5751, 52, 56syl2anc 573 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5823, 25, 31abssubge0d 14378 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
5955, 57, 583eqtrd 2809 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6047, 50, 593eqtrd 2809 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6134, 46, 603eqtr4d 2815 . . . . . . . . . . . 12 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = ((abs ∘ − )‘(𝑓𝑛)))
6261mpteq2dva 4878 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
63 volioof 40721 . . . . . . . . . . . . 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 6197 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
67 fcompt 6543 . . . . . . . . . . . 12 (((vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
6864, 66, 67syl2anc 573 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
69 absf 14285 . . . . . . . . . . . . . 14 abs:ℂ⟶ℝ
70 subf 10485 . . . . . . . . . . . . . 14 − :(ℂ × ℂ)⟶ℂ
71 fco 6198 . . . . . . . . . . . . . 14 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7269, 70, 71mp2an 672 . . . . . . . . . . . . 13 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7372a1i 11 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
74 rr2sscn2 40098 . . . . . . . . . . . . . 14 (ℝ × ℝ) ⊆ (ℂ × ℂ)
7574a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
7612, 75fssd 6197 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77 fcompt 6543 . . . . . . . . . . . 12 (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7873, 76, 77syl2anc 573 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7962, 68, 783eqtr4d 2815 . . . . . . . . . 10 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
8079fveq2d 6336 . . . . . . . . 9 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((abs ∘ − ) ∘ 𝑓)))
8180eqeq2d 2781 . . . . . . . 8 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓))))
8281anbi2d 614 . . . . . . 7 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))))
8382rexbiia 3188 . . . . . 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 3335 . . . . 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 2794 . . . 4 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
8685infeq1i 8540 . . 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 2805 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  {crab 3065  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723  𝒫 cpw 4297  ⟨cop 4322  ∪ cuni 4574   class class class wbr 4786   ↦ cmpt 4863   × cxp 5247  dom cdm 5249  ran crn 5250   ∘ ccom 5253  Fun wfun 6025  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314   ↑𝑚 cmap 8009  infcinf 8503  ℂcc 10136  ℝcr 10137  0cc0 10138  +∞cpnf 10273  ℝ*cxr 10275   < clt 10276   ≤ cle 10277   − cmin 10468  ℕcn 11222  (,)cioo 12380  [,]cicc 12383  abscabs 14182  vol*covol 23450  volcvol 23451  Σ^csumge0 41096 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-rlim 14428  df-sum 14625  df-rest 16291  df-topgen 16312  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-top 20919  df-topon 20936  df-bases 20971  df-cmp 21411  df-ovol 23452  df-vol 23453  df-sumge0 41097 This theorem is referenced by:  ovolval4lem2  41384
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