ovolval3 - Mathbox for Glauco Siliprandi

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Theorem ovolval3 45363

Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^} and vol ∘ (,). See ovolval 24990 and ovolval2 45360 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 2733	. . 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 45360	. 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 11201	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ ∈ V
6	5, 5	xpex 7740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℝ × ℝ) ∈ V
7		inss2 4230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8		mapss 8883	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
9	6, 7, 8	mp2an 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
10	9	sseli 3979	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ))
11		elmapi 8843	. . . . . . . . . . . . . . . . . . . 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 7087	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ × ℝ))
14		1st2nd2 8014	. . . . . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩))
17		df-ov 7412	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))) = ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
18	17	eqcomi 2742	. . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))))
21	20	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = (vol‘((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛)))))
22		xp1st 8007	. . . . . . . . . . . . . . . 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 8008	. . . . . . . . . . . . . . . 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 8843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27	26	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28		simpr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
29		ovolfcl 24983	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 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 1145	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)))
32		volioo 25086	. . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
35		ioof 13424	. . . . . . . . . . . . . . . 16 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
36		ffun 6721	. . . . . . . . . . . . . . . 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 11259	. . . . . . . . . . . . . . . 16 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
40	39, 13	sselid 3981	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ^* × ℝ^*))
41	35	fdmi 6730	. . . . . . . . . . . . . . . . 17 ⊢ dom (,) = (ℝ^* × ℝ^*)
42	41	eqcomi 2742	. . . . . . . . . . . . . . . 16 ⊢ (ℝ^* × ℝ^*) = dom (,)
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (ℝ^* × ℝ^*) = dom (,))
44	40, 43	eleqtrd 2836	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ dom (,))
45		fvco 6990	. . . . . . . . . . . . . 14 ⊢ ((Fun (,) ∧ (𝑓‘𝑛) ∈ dom (,)) → ((vol ∘ (,))‘(𝑓‘𝑛)) = (vol‘((,)‘(𝑓‘𝑛))))
46	38, 44, 45	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓‘𝑛)) = (vol‘((,)‘(𝑓‘𝑛))))
47	15	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩))
48		df-ov 7412	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘(𝑓‘𝑛))(abs ∘ − )(2^nd ‘(𝑓‘𝑛))) = ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
49	48	eqcomi 2742	. . . . . . . . . . . . . . 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 11242	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ∈ ℂ)
52	25	recnd 11242	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℂ)
53		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) = (abs ∘ − )
54	53	cnmetdval 24287	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 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 15273	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 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 15378	. . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
61	34, 46, 60	3eqtr4d 2783	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓‘𝑛)) = ((abs ∘ − )‘(𝑓‘𝑛)))
62	61	mpteq2dva 5249	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
63		volioof 44703	. . . . . . . . . . . . 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 6736	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
67		fcompt 7131	. . . . . . . . . . . 12 ⊢ (((vol ∘ (,)):(ℝ^* × ℝ^)⟶(0[,]+∞) ∧ 𝑓:ℕ⟶(ℝ^ × ℝ^*)) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))))
68	64, 66, 67	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))))
69		absf 15284	. . . . . . . . . . . . . 14 ⊢ abs:ℂ⟶ℝ
70		subf 11462	. . . . . . . . . . . . . 14 ⊢ − :(ℂ × ℂ)⟶ℂ
71		fco 6742	. . . . . . . . . . . . . 14 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
72	69, 70, 71	mp2an 691	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
73	72	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
74		rr2sscn2 44076	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
76	12, 75	fssd 6736	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77		fcompt 7131	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
78	73, 76, 77	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
79	62, 68, 78	3eqtr4d 2783	. . . . . . . . . 10 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
80	79	fveq2d 6896	. . . . . . . . 9 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))
81	80	eqeq2d 2744	. . . . . . . 8 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓))))
82	81	anbi2d 630	. . . . . . 7 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))))
83	82	rexbiia 3093	. . . . . 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 3439	. . . . 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 2762	. . . 4 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
86	85	infeq1i 9473	. . 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 2773	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 {crab 3433 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 𝒫 cpw 4603 ⟨cop 4635 ∪ cuni 4909 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 dom cdm 5677 ran crn 5678 ∘ ccom 5681 Fun wfun 6538 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 ↑_m cmap 8820 infcinf 9436 ℂcc 11108 ℝcr 11109 0cc0 11110 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 ℕcn 12212 (,)cioo 13324 [,]cicc 13327 abscabs 15181 vol*covol 24979 volcvol 24980 Σ^{^}csumge0 45078

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-rest 17368 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-top 22396 df-topon 22413 df-bases 22449 df-cmp 22891 df-ovol 24981 df-vol 24982 df-sumge0 45079

This theorem is referenced by: ovolval4lem2 45366

W3C validator