ovolval3 - Mathbox for Glauco Siliprandi

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Theorem ovolval3 47090

Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^} and vol ∘ (,). See ovolval 25458 and ovolval2 47087 for alternative expressions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
ovolval3.a	⊢ (𝜑 → 𝐴 ⊆ 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.

Step	Hyp	Ref	Expression
1		ovolval3.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		eqid 2739	. . 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)))}
3	1, 2	ovolval2 47087	. 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 11120	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ ∈ V
6	5, 5	xpex 7696	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℝ × ℝ) ∈ V
7		inss2 4166	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8		mapss 8827	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
9	6, 7, 8	mp2an 698	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
10	9	sseli 3911	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ))
11		elmapi 8786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
12	10, 11	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
13	12	ffvelcdmda 7025	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ × ℝ))
14		1st2nd2 7970	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (𝑓‘𝑛) = ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
15	13, 14	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) = ⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
16	15	fveq2d 6831	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩))
17		df-ov 7359	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))) = ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
18	17	eqcomi 2748	. . . . . . . . . . . . . . . . 17 ⊢ ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩) = ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛)))
19	18	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩) = ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))))
20	16, 19	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))))
21	20	fveq2d 6831	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = (vol‘((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛)))))
22		xp1st 7963	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝑓‘𝑛)) ∈ ℝ)
23	13, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ∈ ℝ)
24		xp2nd 7964	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℝ)
25	13, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℝ)
26		elmapi 8786	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27	26	adantr 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 25451	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝑓‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝑓‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))))
30	27, 28, 29	syl2anc 590	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝑓‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝑓‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))))
31	30	simp3d 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)))
32		volioo 25554	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(𝑓‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝑓‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))) → (vol‘((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛)))) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
33	23, 25, 31, 32	syl3anc 1379	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛)))) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
34	21, 33	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
35		ioof 13391	. . . . . . . . . . . . . . . 16 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
36		ffun 6658	. . . . . . . . . . . . . . . 16 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → Fun (,))
37	35, 36	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ Fun (,)
38	37	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → Fun (,))
39		rexpssxrxp 11181	. . . . . . . . . . . . . . . 16 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
40	39, 13	sselid 3913	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ^* × ℝ^*))
41	35	fdmi 6666	. . . . . . . . . . . . . . . . 17 ⊢ dom (,) = (ℝ^* × ℝ^*)
42	41	eqcomi 2748	. . . . . . . . . . . . . . . 16 ⊢ (ℝ^* × ℝ^*) = dom (,)
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (ℝ^* × ℝ^*) = dom (,))
44	40, 43	eleqtrd 2841	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ dom (,))
45		fvco 6925	. . . . . . . . . . . . . 14 ⊢ ((Fun (,) ∧ (𝑓‘𝑛) ∈ dom (,)) → ((vol ∘ (,))‘(𝑓‘𝑛)) = (vol‘((,)‘(𝑓‘𝑛))))
46	38, 44, 45	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓‘𝑛)) = (vol‘((,)‘(𝑓‘𝑛))))
47	15	fveq2d 6831	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩))
48		df-ov 7359	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘(𝑓‘𝑛))(abs ∘ − )(2^nd ‘(𝑓‘𝑛))) = ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
49	48	eqcomi 2748	. . . . . . . . . . . . . . 15 ⊢ ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩) = ((1^st ‘(𝑓‘𝑛))(abs ∘ − )(2^nd ‘(𝑓‘𝑛)))
50	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩) = ((1^st ‘(𝑓‘𝑛))(abs ∘ − )(2^nd ‘(𝑓‘𝑛))))
51	23	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ∈ ℂ)
52	25	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℂ)
53		eqid 2739	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) = (abs ∘ − )
54	53	cnmetdval 24753	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ‘(𝑓‘𝑛)) ∈ ℂ ∧ (2^nd ‘(𝑓‘𝑛)) ∈ ℂ) → ((1^st ‘(𝑓‘𝑛))(abs ∘ − )(2^nd ‘(𝑓‘𝑛))) = (abs‘((1^st ‘(𝑓‘𝑛)) − (2^nd ‘(𝑓‘𝑛)))))
55	51, 52, 54	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝑓‘𝑛))(abs ∘ − )(2^nd ‘(𝑓‘𝑛))) = (abs‘((1^st ‘(𝑓‘𝑛)) − (2^nd ‘(𝑓‘𝑛)))))
56		abssub 15280	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ‘(𝑓‘𝑛)) ∈ ℂ ∧ (2^nd ‘(𝑓‘𝑛)) ∈ ℂ) → (abs‘((1^st ‘(𝑓‘𝑛)) − (2^nd ‘(𝑓‘𝑛)))) = (abs‘((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛)))))
57	51, 52, 56	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((1^st ‘(𝑓‘𝑛)) − (2^nd ‘(𝑓‘𝑛)))) = (abs‘((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛)))))
58	23, 25, 31	abssubge0d 15387	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛)))) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
59	55, 57, 58	3eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝑓‘𝑛))(abs ∘ − )(2^nd ‘(𝑓‘𝑛))) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
60	47, 50, 59	3eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
61	34, 46, 60	3eqtr4d 2784	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓‘𝑛)) = ((abs ∘ − )‘(𝑓‘𝑛)))
62	61	mpteq2dva 5165	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
63		volioof 46430	. . . . . . . . . . . . 13 ⊢ (vol ∘ (,)):(ℝ^* × ℝ^*)⟶(0[,]+∞)
64	63	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (vol ∘ (,)):(ℝ^* × ℝ^*)⟶(0[,]+∞))
65	39	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*))
66	12, 65	fssd 6672	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
67		fcompt 7075	. . . . . . . . . . . 12 ⊢ (((vol ∘ (,)):(ℝ^* × ℝ^)⟶(0[,]+∞) ∧ 𝑓:ℕ⟶(ℝ^ × ℝ^*)) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))))
68	64, 66, 67	syl2anc 590	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))))
69		absf 15291	. . . . . . . . . . . . . 14 ⊢ abs:ℂ⟶ℝ
70		subf 11386	. . . . . . . . . . . . . 14 ⊢ − :(ℂ × ℂ)⟶ℂ
71		fco 6679	. . . . . . . . . . . . . 14 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
72	69, 70, 71	mp2an 698	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
73	72	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
74		rr2sscn2 45810	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
76	12, 75	fssd 6672	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77		fcompt 7075	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
78	73, 76, 77	syl2anc 590	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
79	62, 68, 78	3eqtr4d 2784	. . . . . . . . . 10 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
80	79	fveq2d 6831	. . . . . . . . 9 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))
81	80	eqeq2d 2750	. . . . . . . 8 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓))))
82	81	anbi2d 636	. . . . . . 7 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))))
83	82	rexbiia 3084	. . . . . 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))))
84	83	rabbii 3396	. . . . 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)))}
85	4, 84	eqtr2i 2763	. . . 4 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
86	85	infeq1i 9382	. . 3 ⊢ inf({𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))}, ℝ^, < ) = inf(𝑀, ℝ^, < )
87	86	a1i 11	. 2 ⊢ (𝜑 → inf({𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))}, ℝ^, < ) = inf(𝑀, ℝ^, < ))
88	3, 87	eqtrd 2774	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∃wrex 3063 {crab 3391 Vcvv 3431 ∩ cin 3882 ⊆ wss 3883 𝒫 cpw 4529 ⟨cop 4561 ∪ cuni 4838 class class class wbr 5072 ↦ cmpt 5153 × cxp 5616 dom cdm 5618 ran crn 5619 ∘ ccom 5622 Fun wfun 6479 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 ↑_m cmap 8763 infcinf 9344 ℂcc 11027 ℝcr 11028 0cc0 11029 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 − cmin 11368 ℕcn 12165 (,)cioo 13289 [,]cicc 13292 abscabs 15187 vol*covol 25447 volcvol 25448 Σ^{^}csumge0 46805

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-rest 17376 df-topgen 17397 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-top 22877 df-topon 22894 df-bases 22929 df-cmp 23370 df-ovol 25449 df-vol 25450 df-sumge0 46806

This theorem is referenced by: ovolval4lem2 47093

W3C validator