ovolval3 - Mathbox for Glauco Siliprandi

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Theorem ovolval3 46603

Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^} and vol ∘ (,). See ovolval 25522 and ovolval2 46600 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 2735	. . 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 46600	. 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 11244	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ ∈ V
6	5, 5	xpex 7772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℝ × ℝ) ∈ V
7		inss2 4246	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
8		mapss 8928	. . . . . . . . . . . . . . . . . . . . . 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 3991	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ))
11		elmapi 8888	. . . . . . . . . . . . . . . . . . . 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 7104	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ × ℝ))
14		1st2nd2 8052	. . . . . . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩))
17		df-ov 7434	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))) = ((,)‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
18	17	eqcomi 2744	. . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝑓‘𝑛)) = ((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛))))
21	20	fveq2d 6911	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = (vol‘((1^st ‘(𝑓‘𝑛))(,)(2^nd ‘(𝑓‘𝑛)))))
22		xp1st 8045	. . . . . . . . . . . . . . . 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 8046	. . . . . . . . . . . . . . . 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 8888	. . . . . . . . . . . . . . . . . 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 25515	. . . . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)))
32		volioo 25618	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝑓‘𝑛))) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
35		ioof 13484	. . . . . . . . . . . . . . . 16 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
36		ffun 6740	. . . . . . . . . . . . . . . 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 11304	. . . . . . . . . . . . . . . 16 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
40	39, 13	sselid 3993	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓‘𝑛) ∈ (ℝ^* × ℝ^*))
41	35	fdmi 6748	. . . . . . . . . . . . . . . . 17 ⊢ dom (,) = (ℝ^* × ℝ^*)
42	41	eqcomi 2744	. . . . . . . . . . . . . . . 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 7007	. . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩))
48		df-ov 7434	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘(𝑓‘𝑛))(abs ∘ − )(2^nd ‘(𝑓‘𝑛))) = ((abs ∘ − )‘⟨(1^st ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛))⟩)
49	48	eqcomi 2744	. . . . . . . . . . . . . . 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 11287	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝑓‘𝑛)) ∈ ℂ)
52	25	recnd 11287	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝑓‘𝑛)) ∈ ℂ)
53		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) = (abs ∘ − )
54	53	cnmetdval 24807	. . . . . . . . . . . . . . . 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 15362	. . . . . . . . . . . . . . . 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 15467	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑛)) = ((2^nd ‘(𝑓‘𝑛)) − (1^st ‘(𝑓‘𝑛))))
61	34, 46, 60	3eqtr4d 2785	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((vol ∘ (,))‘(𝑓‘𝑛)) = ((abs ∘ − )‘(𝑓‘𝑛)))
62	61	mpteq2dva 5248	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (𝑛 ∈ ℕ ↦ ((vol ∘ (,))‘(𝑓‘𝑛))) = (𝑛 ∈ ℕ ↦ ((abs ∘ − )‘(𝑓‘𝑛))))
63		volioof 45943	. . . . . . . . . . . . 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 6754	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
67		fcompt 7153	. . . . . . . . . . . 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 15373	. . . . . . . . . . . . . 14 ⊢ abs:ℂ⟶ℝ
70		subf 11508	. . . . . . . . . . . . . 14 ⊢ − :(ℂ × ℂ)⟶ℂ
71		fco 6761	. . . . . . . . . . . . . 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 45316	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
76	12, 75	fssd 6754	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℂ × ℂ))
77		fcompt 7153	. . . . . . . . . . . 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 2785	. . . . . . . . . 10 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓))
80	79	fveq2d 6911	. . . . . . . . 9 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))
81	80	eqeq2d 2746	. . . . . . . 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 3090	. . . . . 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 2764	. . . 4 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
86	85	infeq1i 9516	. . 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 2775	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 {crab 3433 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 ⟨cop 4637 ∪ cuni 4912 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 dom cdm 5689 ran crn 5690 ∘ ccom 5693 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 ↑_m cmap 8865 infcinf 9479 ℂcc 11151 ℝcr 11152 0cc0 11153 +∞cpnf 11290 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 (,)cioo 13384 [,]cicc 13387 abscabs 15270 vol*covol 25511 volcvol 25512 Σ^{^}csumge0 46318

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-rest 17469 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cmp 23411 df-ovol 25513 df-vol 25514 df-sumge0 46319

This theorem is referenced by: ovolval4lem2 46606

W3C validator