ovolval2lem - Mathbox for Glauco Siliprandi

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Theorem ovolval2lem 47926

Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^}. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypothesis**
Ref	Expression
ovolval2lem.1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))

**Assertion**
Ref	Expression
ovolval2lem	⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐹)) = ran (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))

Distinct variable groups: 𝑘,𝐹,𝑛 𝜑,𝑘 Allowed substitution hint: 𝜑(𝑛)

**Proof of Theorem ovolval2lem**
Step	Hyp	Ref	Expression
1		reex 11330	. . . . . . 7 ⊢ ℝ ∈ V
2	1, 1	xpex 7799	. . . . . 6 ⊢ (ℝ × ℝ) ∈ V
3		inss2 4137	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4		mapss 8955	. . . . . 6 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
5	2, 3, 4	mp2an 705	. . . . 5 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
6		ovolval2lem.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7	2	inex2 5285	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
8	7	a1i 11	. . . . . . 7 ⊢ (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
9		nnex 12381	. . . . . . . 8 ⊢ ℕ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
11	8, 10	elmapd 8898	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
12	6, 11	mpbird 260	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
13	5, 12	sselid 3911	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ))
14		1zzd 12767	. . . . 5 ⊢ (𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) → 1 ∈ ℤ)
15		nnuz 13044	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
16		elmapi 8907	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
17	16	adantr 486	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
18		simpr 490	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
19	17, 18	fvovco 46481	. . . . . . . 8 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑘) = ((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘))))
20	19	fveq2d 6920	. . . . . . 7 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))))
21	16	ffvelcdmda 7116	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (ℝ × ℝ))
22		xp1st 8067	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ∈ (ℝ × ℝ) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ)
23	21, 22	syl 18	. . . . . . . 8 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ)
24		xp2nd 8068	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ∈ (ℝ × ℝ) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
25	21, 24	syl 18	. . . . . . . 8 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
26		volicore 47864	. . . . . . . 8 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ) → (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))) ∈ ℝ)
27	23, 25, 26	syl2anc 596	. . . . . . 7 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))) ∈ ℝ)
28	20, 27	eqeltrd 2864	. . . . . 6 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℝ)
29	28	recnd 11376	. . . . 5 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℂ)
30		eqid 2763	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘)))
31		eqid 2763	. . . . 5 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))))
32	14, 15, 29, 30, 31	fsumsermpt 46864	. . . 4 ⊢ (𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))))
33	13, 32	syl 18	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))))
34		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)))
35	34	iftrued 4493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → if((1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)), ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))), 0) = ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))))
36	13, 23	sylan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ)
37	36	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ)
38	13, 25	sylan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
39	38	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
40		ressxr 11392	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
41	40, 37	sselid 3911	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ^*)
42	40, 39	sselid 3911	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ^*)
43		xpss 5674	. . . . . . . . . . . . . . . . . 18 ⊢ (ℝ × ℝ) ⊆ (V × V)
44	43, 21	sselid 3911	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (V × V))
45		1st2ndb 8075	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑘) ∈ (V × V) ↔ (𝐹‘𝑘) = ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
46	44, 45	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
47	13, 46	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
48	47	eqcomd 2769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩ = (𝐹‘𝑘))
49		inss1 4136	. . . . . . . . . . . . . . . . 17 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤ )
51	6, 50	fssd 6757	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℕ⟶ ≤ )
52	51	ffvelcdmda 7116	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ≤ )
53	48, 52	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩ ∈ ≤ )
54		df-br 5109	. . . . . . . . . . . . 13 ⊢ ((1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)) ↔ ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩ ∈ ≤ )
55	53, 54	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)))
56	55	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)))
57		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)))
58	39, 37	lenltd 11495	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → ((2^nd ‘(𝐹‘𝑘)) ≤ (1^st ‘(𝐹‘𝑘)) ↔ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))))
59	57, 58	mpbird 260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) ≤ (1^st ‘(𝐹‘𝑘)))
60	41, 42, 56, 59	xrletrid 13324	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘)))
61		simp3 1156	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘)))
62		simp1 1154	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ)
63		simp2 1155	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
64	62, 63	eqleltd 11493	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → ((1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘)) ↔ ((1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)))))
65	61, 64	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → ((1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))))
66	65	simprd 501	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)))
67	66	iffalsed 4496	. . . . . . . . . . 11 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → if((1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)), ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))), 0) = 0)
68	63	recnd 11376	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) ∈ ℂ)
69	61	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) = (1^st ‘(𝐹‘𝑘)))
70	68, 69	subeq0bd 11781	. . . . . . . . . . 11 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))) = 0)
71	67, 70	eqtr4d 2801	. . . . . . . . . 10 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘))) → if((1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)), ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))), 0) = ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))))
72	37, 39, 60, 71	syl3anc 1399	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → if((1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)), ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))), 0) = ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))))
73	35, 72	pm2.61dan 825	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → if((1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)), ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))), 0) = ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))))
74		volico 47266	. . . . . . . . 9 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ) → (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))) = if((1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)), ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))), 0))
75	36, 38, 74	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))) = if((1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)), ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))), 0))
76	36, 38, 55	abssuble0d 15646	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘((1^st ‘(𝐹‘𝑘)) − (2^nd ‘(𝐹‘𝑘)))) = ((2^nd ‘(𝐹‘𝑘)) − (1^st ‘(𝐹‘𝑘))))
77	73, 75, 76	3eqtr4d 2808	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))) = (abs‘((1^st ‘(𝐹‘𝑘)) − (2^nd ‘(𝐹‘𝑘)))))
78	13	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ))
79		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
80	78, 79, 20	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))))
81	46	fveq2d 6920	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑘)) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩))
82		df-ov 7454	. . . . . . . . . . 11 ⊢ ((1^st ‘(𝐹‘𝑘))(abs ∘ − )(2^nd ‘(𝐹‘𝑘))) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
83	82	eqcomi 2772	. . . . . . . . . 10 ⊢ ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩) = ((1^st ‘(𝐹‘𝑘))(abs ∘ − )(2^nd ‘(𝐹‘𝑘)))
84	83	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩) = ((1^st ‘(𝐹‘𝑘))(abs ∘ − )(2^nd ‘(𝐹‘𝑘))))
85	23	recnd 11376	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ∈ ℂ)
86	25	recnd 11376	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐹‘𝑘)) ∈ ℂ)
87		eqid 2763	. . . . . . . . . . 11 ⊢ (abs ∘ − ) = (abs ∘ − )
88	87	cnmetdval 25144	. . . . . . . . . 10 ⊢ (((1^st ‘(𝐹‘𝑘)) ∈ ℂ ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℂ) → ((1^st ‘(𝐹‘𝑘))(abs ∘ − )(2^nd ‘(𝐹‘𝑘))) = (abs‘((1^st ‘(𝐹‘𝑘)) − (2^nd ‘(𝐹‘𝑘)))))
89	85, 86, 88	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → ((1^st ‘(𝐹‘𝑘))(abs ∘ − )(2^nd ‘(𝐹‘𝑘))) = (abs‘((1^st ‘(𝐹‘𝑘)) − (2^nd ‘(𝐹‘𝑘)))))
90	81, 84, 89	3eqtrd 2802	. . . . . . . 8 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑘)) = (abs‘((1^st ‘(𝐹‘𝑘)) − (2^nd ‘(𝐹‘𝑘)))))
91	78, 79, 90	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑘)) = (abs‘((1^st ‘(𝐹‘𝑘)) − (2^nd ‘(𝐹‘𝑘)))))
92	77, 80, 91	3eqtr4d 2808	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = ((abs ∘ − )‘(𝐹‘𝑘)))
93	92	mpteq2dva 5204	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹‘𝑘))))
94	13, 16	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
95		rr2sscn2 46650	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
96	95	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℂ × ℂ))
97		absf 15548	. . . . . . . 8 ⊢ abs:ℂ⟶ℝ
98		subf 11598	. . . . . . . 8 ⊢ − :(ℂ × ℂ)⟶ℂ
99		fco 6764	. . . . . . . 8 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
100	97, 98, 99	mp2an 705	. . . . . . 7 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
101	100	a1i 11	. . . . . 6 ⊢ (𝜑 → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
102	94, 96, 101	fcomptss 46490	. . . . 5 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐹) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹‘𝑘))))
103	93, 102	eqtr4d 2801	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = ((abs ∘ − ) ∘ 𝐹))
104	103	seqeq3d 14192	. . 3 ⊢ (𝜑 → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
105	33, 104	eqtr2d 2799	. 2 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
106	105	rneqd 5932	1 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐹)) = ran (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⊆ wss 3900 ∩ cin 4092 ifcif 4485 ⟨cop 4840 class class class wbr 5108 ↦ cmpt 5192 × cxp 5656 ran crn 5659 ∘ ccom 5662 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 2^nd c2nd 8034 ↑_m cmap 8885 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 − cmin 11580 ℕcn 12375 [,)cico 13518 ...cfz 13679 seqcseq 14184 abscabs 15444 Σcsu 15897 volcvol 25839

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-z 12734 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699 df-rlim 15700 df-sum 15898 df-rest 17643 df-topgen 17664 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-top 23267 df-topon 23284 df-bases 23319 df-cmp 23760 df-ovol 25840 df-vol 25841

This theorem is used by: (None)

W3C validator