ovolval3 - Mathbox for Glauco Siliprandi

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Theorem ovolval3 46643

Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^} and vol ∘ (,). See ovolval 25431 and ovolval2 46640 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 2736	. . 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 46640	. 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 11225	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ ∈ V
6	5, 5	xpex 7752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℝ × ℝ) ∈ V
7		inss2 4218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8		mapss 8908	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
9	6, 7, 8	mp2an 692	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
10	9	sseli 3959	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ))
11		elmapi 8868	. . . . . . . . . . . . . . . . . . . 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 7079	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ × ℝ))
14		1st2nd2 8032	. . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩))
17		df-ov 7413	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))) = ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
18	17	eqcomi 2745	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))))
21	20	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = (vol‘((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛)))))
22		xp1st 8025	. . . . . . . . . . . . . . . 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 8026	. . . . . . . . . . . . . . . 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 8868	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27	26	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
29		ovolfcl 25424	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)))
32		volioo 25527	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
35		ioof 13469	. . . . . . . . . . . . . . . 16 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
36		ffun 6714	. . . . . . . . . . . . . . . 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 11285	. . . . . . . . . . . . . . . 16 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
40	39, 13	sselid 3961	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ^* × ℝ^*))
41	35	fdmi 6722	. . . . . . . . . . . . . . . . 17 ⊢ dom (,) = (ℝ^* × ℝ^*)
42	41	eqcomi 2745	. . . . . . . . . . . . . . . 16 ⊢ (ℝ^* × ℝ^*) = dom (,)
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (ℝ^* × ℝ^*) = dom (,))
44	40, 43	eleqtrd 2837	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ dom (,))
45		fvco 6982	. . . . . . . . . . . . . 14 ⊢ ((Fun (,) ∧ (𝑓‘𝑛) ∈ dom (,)) → ((vol ∘ (,))‘(𝑓‘𝑛)) = (vol‘((,)‘(𝑓‘𝑛))))
46	38, 44, 45	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓‘𝑛)) = (vol‘((,)‘(𝑓‘𝑛))))
47	15	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩))
48		df-ov 7413	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘(𝑓‘𝑛))(abs ∘ − )(2^nd ‘(𝑓‘𝑛))) = ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
49	48	eqcomi 2745	. . . . . . . . . . . . . . 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 11268	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ∈ ℂ)
52	25	recnd 11268	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℂ)
53		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) = (abs ∘ − )
54	53	cnmetdval 24714	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 15350	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 15455	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
61	34, 46, 60	3eqtr4d 2781	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓‘𝑛)) = ((abs ∘ − )‘(𝑓‘𝑛)))
62	61	mpteq2dva 5219	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
63		volioof 45983	. . . . . . . . . . . . 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 6728	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
67		fcompt 7128	. . . . . . . . . . . 12 ⊢ (((vol ∘ (,)):(ℝ^* × ℝ^)⟶(0[,]+∞) ∧ 𝑓:ℕ⟶(ℝ^ × ℝ^*)) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))))
68	64, 66, 67	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))))
69		absf 15361	. . . . . . . . . . . . . 14 ⊢ abs:ℂ⟶ℝ
70		subf 11489	. . . . . . . . . . . . . 14 ⊢ − :(ℂ × ℂ)⟶ℂ
71		fco 6735	. . . . . . . . . . . . . 14 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
72	69, 70, 71	mp2an 692	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
73	72	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
74		rr2sscn2 45360	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
76	12, 75	fssd 6728	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77		fcompt 7128	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
78	73, 76, 77	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((abs ∘ − ) ∘ 𝑓) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
79	62, 68, 78	3eqtr4d 2781	. . . . . . . . . 10 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
80	79	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))
81	80	eqeq2d 2747	. . . . . . . 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 3082	. . . . . 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 3426	. . . . 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 2760	. . . 4 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
86	85	infeq1i 9496	. . 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 2771	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3061 {crab 3420 Vcvv 3464 ∩ cin 3930 ⊆ wss 3931 𝒫 cpw 4580 ⟨cop 4612 ∪ cuni 4888 class class class wbr 5124 ↦ cmpt 5206 × cxp 5657 dom cdm 5659 ran crn 5660 ∘ ccom 5663 Fun wfun 6530 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 ↑_m cmap 8845 infcinf 9458 ℂcc 11132 ℝcr 11133 0cc0 11134 +∞cpnf 11271 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 − cmin 11471 ℕcn 12245 (,)cioo 13367 [,]cicc 13370 abscabs 15258 vol*covol 25420 volcvol 25421 Σ^{^}csumge0 46358

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-rlim 15510 df-sum 15708 df-rest 17441 df-topgen 17462 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-top 22837 df-topon 22854 df-bases 22889 df-cmp 23330 df-ovol 25422 df-vol 25423 df-sumge0 46359

This theorem is referenced by: ovolval4lem2 46646

W3C validator