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Theorem ovolval3 41365
Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 23440 and ovolval2 41362 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 2758 . . 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 41362 . 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 10217 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ∈ V
65, 5xpex 7125 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ × ℝ) ∈ V
7 inss2 3975 . . . . . . . . . . . . . . . . . . . . . 22 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8 mapss 8064 . . . . . . . . . . . . . . . . . . . . . 22 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ))
96, 7, 8mp2an 710 . . . . . . . . . . . . . . . . . . . . 21 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ)
109sseli 3738 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
11 elmapi 8043 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1210, 11syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1312ffvelrnda 6520 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
14 1st2nd2 7370 . . . . . . . . . . . . . . . . . 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 6354 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
17 df-ov 6814 . . . . . . . . . . . . . . . . . 18 ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))) = ((,)‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
1817eqcomi 2767 . . . . . . . . . . . . . . . . 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 2792 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓𝑛)) = ((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛))))
2120fveq2d 6354 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))))
22 xp1st 7363 . . . . . . . . . . . . . . . 16 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
2313, 22syl 17 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
24 xp2nd 7364 . . . . . . . . . . . . . . . 16 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2513, 24syl 17 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
26 elmapi 8043 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2726adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28 simpr 479 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
29 ovolfcl 23433 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3027, 28, 29syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
3130simp3d 1139 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)))
32 volioo 23535 . . . . . . . . . . . . . . 15 (((1st ‘(𝑓𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓𝑛)) ∈ ℝ ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3323, 25, 31, 32syl3anc 1477 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((1st ‘(𝑓𝑛))(,)(2nd ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
3421, 33eqtrd 2792 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
35 ioof 12462 . . . . . . . . . . . . . . . 16 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
36 ffun 6207 . . . . . . . . . . . . . . . 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 10274 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4039, 13sseldi 3740 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ* × ℝ*))
4135fdmi 6211 . . . . . . . . . . . . . . . . 17 dom (,) = (ℝ* × ℝ*)
4241eqcomi 2767 . . . . . . . . . . . . . . . 16 (ℝ* × ℝ*) = dom (,)
4342a1i 11 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (ℝ* × ℝ*) = dom (,))
4440, 43eleqtrd 2839 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ dom (,))
45 fvco 6434 . . . . . . . . . . . . . 14 ((Fun (,) ∧ (𝑓𝑛) ∈ dom (,)) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4638, 44, 45syl2anc 696 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = (vol‘((,)‘(𝑓𝑛))))
4715fveq2d 6354 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩))
48 df-ov 6814 . . . . . . . . . . . . . . . 16 ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((abs ∘ − )‘⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
4948eqcomi 2767 . . . . . . . . . . . . . . 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 10258 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℂ)
5225recnd 10258 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℂ)
53 eqid 2758 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) = (abs ∘ − )
5453cnmetdval 22773 . . . . . . . . . . . . . . . 16 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
5551, 52, 54syl2anc 696 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))))
56 abssub 14263 . . . . . . . . . . . . . . . 16 (((1st ‘(𝑓𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓𝑛)) ∈ ℂ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5751, 52, 56syl2anc 696 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((1st ‘(𝑓𝑛)) − (2nd ‘(𝑓𝑛)))) = (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))))
5823, 25, 31abssubge0d 14367 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛)))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
5955, 57, 583eqtrd 2796 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓𝑛))(abs ∘ − )(2nd ‘(𝑓𝑛))) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6047, 50, 593eqtrd 2796 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑛)) = ((2nd ‘(𝑓𝑛)) − (1st ‘(𝑓𝑛))))
6134, 46, 603eqtr4d 2802 . . . . . . . . . . . 12 ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓𝑛)) = ((abs ∘ − )‘(𝑓𝑛)))
6261mpteq2dva 4894 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
63 volioof 40705 . . . . . . . . . . . . 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 6216 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
67 fcompt 6561 . . . . . . . . . . . 12 (((vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞) ∧ 𝑓:ℕ⟶(ℝ* × ℝ*)) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
6864, 66, 67syl2anc 696 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓𝑛))))
69 absf 14274 . . . . . . . . . . . . . 14 abs:ℂ⟶ℝ
70 subf 10473 . . . . . . . . . . . . . 14 − :(ℂ × ℂ)⟶ℂ
71 fco 6217 . . . . . . . . . . . . . 14 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
7269, 70, 71mp2an 710 . . . . . . . . . . . . 13 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
7372a1i 11 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
74 rr2sscn2 40078 . . . . . . . . . . . . . 14 (ℝ × ℝ) ⊆ (ℂ × ℂ)
7574a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
7612, 75fssd 6216 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77 fcompt 6561 . . . . . . . . . . . 12 (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7873, 76, 77syl2anc 696 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓𝑛))))
7962, 68, 783eqtr4d 2802 . . . . . . . . . 10 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
8079fveq2d 6354 . . . . . . . . 9 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((abs ∘ − ) ∘ 𝑓)))
8180eqeq2d 2768 . . . . . . . 8 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓))))
8281anbi2d 742 . . . . . . 7 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))))
8382rexbiia 3176 . . . . . 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 3323 . . . . 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 2781 . . . 4 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
8685infeq1i 8547 . . 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 2792 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1630  wcel 2137  wrex 3049  {crab 3052  Vcvv 3338  cin 3712  wss 3713  𝒫 cpw 4300  cop 4325   cuni 4586   class class class wbr 4802  cmpt 4879   × cxp 5262  dom cdm 5264  ran crn 5265  ccom 5268  Fun wfun 6041  wf 6043  cfv 6047  (class class class)co 6811  1st c1st 7329  2nd c2nd 7330  𝑚 cmap 8021  infcinf 8510  cc 10124  cr 10125  0cc0 10126  +∞cpnf 10261  *cxr 10263   < clt 10264  cle 10265  cmin 10456  cn 11210  (,)cioo 12366  [,]cicc 12369  abscabs 14171  vol*covol 23429  volcvol 23430  Σ^csumge0 41080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203  ax-pre-sup 10204
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-of 7060  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-map 8023  df-pm 8024  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fi 8480  df-sup 8511  df-inf 8512  df-oi 8578  df-card 8953  df-cda 9180  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-nn 11211  df-2 11269  df-3 11270  df-n0 11483  df-z 11568  df-uz 11878  df-q 11980  df-rp 12024  df-xneg 12137  df-xadd 12138  df-xmul 12139  df-ioo 12370  df-ico 12372  df-icc 12373  df-fz 12518  df-fzo 12658  df-fl 12785  df-seq 12994  df-exp 13053  df-hash 13310  df-cj 14036  df-re 14037  df-im 14038  df-sqrt 14172  df-abs 14173  df-clim 14416  df-rlim 14417  df-sum 14614  df-rest 16283  df-topgen 16304  df-psmet 19938  df-xmet 19939  df-met 19940  df-bl 19941  df-mopn 19942  df-top 20899  df-topon 20916  df-bases 20950  df-cmp 21390  df-ovol 23431  df-vol 23432  df-sumge0 41081
This theorem is referenced by:  ovolval4lem2  41368
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