Theorem ovolval3 46645

Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^} and vol ∘ (,). See ovolval 25374 and ovolval2 46642 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 2729	. . 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 46642	. 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 11159	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ ∈ V
6	5, 5	xpex 7729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℝ × ℝ) ∈ V
7		inss2 4201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8		mapss 8862	. . . . . . . . . . . . . . . . . . . . . 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 3942	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ))
11		elmapi 8822	. . . . . . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ × ℝ))
14		1st2nd2 8007	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (𝑓‘𝑛) = ⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩)
15	13, 14	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) = ⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩)
16	15	fveq2d 6862	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((,)‘⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩))
17		df-ov 7390	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))) = ((,)‘⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩)
18	17	eqcomi 2738	. . . . . . . . . . . . . . . . 17 ⊢ ((,)‘⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩) = ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛)))
19	18	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩) = ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))))
20	16, 19	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛))))
21	20	fveq2d 6862	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = (vol‘((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛)))))
22		xp1st 8000	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓‘𝑛)) ∈ ℝ)
23	13, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓‘𝑛)) ∈ ℝ)
24		xp2nd 8001	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓‘𝑛)) ∈ ℝ)
25	13, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓‘𝑛)) ∈ ℝ)
26		elmapi 8822	. . . . . . . . . . . . . . . . . 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 25367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓‘𝑛)) ∈ ℝ ∧ (1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛))))
30	27, 28, 29	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓‘𝑛)) ∈ ℝ ∧ (1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛))))
31	30	simp3d 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛)))
32		volioo 25470	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘(𝑓‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝑓‘𝑛)) ∈ ℝ ∧ (1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛))) → (vol‘((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛)))) = ((2nd ‘(𝑓‘𝑛)) − (1st ‘(𝑓‘𝑛))))
33	23, 25, 31, 32	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((1st ‘(𝑓‘𝑛))(,)(2nd ‘(𝑓‘𝑛)))) = ((2nd ‘(𝑓‘𝑛)) − (1st ‘(𝑓‘𝑛))))
34	21, 33	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = ((2nd ‘(𝑓‘𝑛)) − (1st ‘(𝑓‘𝑛))))
35		ioof 13408	. . . . . . . . . . . . . . . 16 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
36		ffun 6691	. . . . . . . . . . . . . . . 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 11219	. . . . . . . . . . . . . . . 16 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
40	39, 13	sselid 3944	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ* × ℝ*))
41	35	fdmi 6699	. . . . . . . . . . . . . . . . 17 ⊢ dom (,) = (ℝ* × ℝ*)
42	41	eqcomi 2738	. . . . . . . . . . . . . . . 16 ⊢ (ℝ* × ℝ*) = dom (,)
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (ℝ* × ℝ*) = dom (,))
44	40, 43	eleqtrd 2830	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ dom (,))
45		fvco 6959	. . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((abs ∘ − )‘⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩))
48		df-ov 7390	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘(𝑓‘𝑛))(abs ∘ − )(2nd ‘(𝑓‘𝑛))) = ((abs ∘ − )‘⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩)
49	48	eqcomi 2738	. . . . . . . . . . . . . . 15 ⊢ ((abs ∘ − )‘⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩) = ((1st ‘(𝑓‘𝑛))(abs ∘ − )(2nd ‘(𝑓‘𝑛)))
50	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘⟨(1st ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛))⟩) = ((1st ‘(𝑓‘𝑛))(abs ∘ − )(2nd ‘(𝑓‘𝑛))))
51	23	recnd 11202	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓‘𝑛)) ∈ ℂ)
52	25	recnd 11202	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓‘𝑛)) ∈ ℂ)
53		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) = (abs ∘ − )
54	53	cnmetdval 24658	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘(𝑓‘𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓‘𝑛)) ∈ ℂ) → ((1st ‘(𝑓‘𝑛))(abs ∘ − )(2nd ‘(𝑓‘𝑛))) = (abs‘((1st ‘(𝑓‘𝑛)) − (2nd ‘(𝑓‘𝑛)))))
55	51, 52, 54	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓‘𝑛))(abs ∘ − )(2nd ‘(𝑓‘𝑛))) = (abs‘((1st ‘(𝑓‘𝑛)) − (2nd ‘(𝑓‘𝑛)))))
56		abssub 15293	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘(𝑓‘𝑛)) ∈ ℂ ∧ (2nd ‘(𝑓‘𝑛)) ∈ ℂ) → (abs‘((1st ‘(𝑓‘𝑛)) − (2nd ‘(𝑓‘𝑛)))) = (abs‘((2nd ‘(𝑓‘𝑛)) − (1st ‘(𝑓‘𝑛)))))
57	51, 52, 56	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((1st ‘(𝑓‘𝑛)) − (2nd ‘(𝑓‘𝑛)))) = (abs‘((2nd ‘(𝑓‘𝑛)) − (1st ‘(𝑓‘𝑛)))))
58	23, 25, 31	abssubge0d 15400	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (abs‘((2nd ‘(𝑓‘𝑛)) − (1st ‘(𝑓‘𝑛)))) = ((2nd ‘(𝑓‘𝑛)) − (1st ‘(𝑓‘𝑛))))
59	55, 57, 58	3eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝑓‘𝑛))(abs ∘ − )(2nd ‘(𝑓‘𝑛))) = ((2nd ‘(𝑓‘𝑛)) − (1st ‘(𝑓‘𝑛))))
60	47, 50, 59	3eqtrd 2768	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((2nd ‘(𝑓‘𝑛)) − (1st ‘(𝑓‘𝑛))))
61	34, 46, 60	3eqtr4d 2774	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓‘𝑛)) = ((abs ∘ − )‘(𝑓‘𝑛)))
62	61	mpteq2dva 5200	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
63		volioof 45985	. . . . . . . . . . . . 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 6705	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
67		fcompt 7105	. . . . . . . . . . . 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 15304	. . . . . . . . . . . . . 14 ⊢ abs:ℂ⟶ℝ
70		subf 11423	. . . . . . . . . . . . . 14 ⊢ − :(ℂ × ℂ)⟶ℂ
71		fco 6712	. . . . . . . . . . . . . 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 45362	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
76	12, 75	fssd 6705	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77		fcompt 7105	. . . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
80	79	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((abs ∘ − ) ∘ 𝑓)))
81	80	eqeq2d 2740	. . . . . . . 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 3074	. . . . . 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 3411	. . . . 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 2753	. . . 4 ⊢ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
86	85	infeq1i 9430	. . 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 2764	1 ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3405  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  ⟨cop 4595  ∪ cuni 4871   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636  dom cdm 5638  ran crn 5639   ∘ ccom 5642  Fun wfun 6505  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967   ↑_m cmap 8799  infcinf 9392  ℂcc 11066  ℝcr 11067  0cc0 11068  +∞cpnf 11205  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209   − cmin 11405  ℕcn 12186  (,)cioo 13306  [,]cicc 13309  abscabs 15200  vol*covol 25363  volcvol 25364  Σ^{^}csumge0 46360

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fi 9362  df-sup 9393  df-inf 9394  df-oi 9463  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-q 12908  df-rp 12952  df-xneg 13072  df-xadd 13073  df-xmul 13074  df-ioo 13310  df-ico 13312  df-icc 13313  df-fz 13469  df-fzo 13616  df-fl 13754  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-rlim 15455  df-sum 15653  df-rest 17385  df-topgen 17406  df-psmet 21256  df-xmet 21257  df-met 21258  df-bl 21259  df-mopn 21260  df-top 22781  df-topon 22798  df-bases 22833  df-cmp 23274  df-ovol 25365  df-vol 25366  df-sumge0 46361

This theorem is referenced by:  ovolval4lem2  46648