ovolval2lem - Mathbox for Glauco Siliprandi

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Theorem ovolval2lem 46803

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 11108	. . . . . . 7 ⊢ ℝ ∈ V
2	1, 1	xpex 7695	. . . . . 6 ⊢ (ℝ × ℝ) ∈ V
3		inss2 4187	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4		mapss 8823	. . . . . 6 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
5	2, 3, 4	mp2an 692	. . . . 5 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
6		ovolval2lem.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7	2	inex2 5260	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
8	7	a1i 11	. . . . . . 7 ⊢ (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
9		nnex 12142	. . . . . . . 8 ⊢ ℕ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
11	8, 10	elmapd 8773	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
12	6, 11	mpbird 257	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
13	5, 12	sselid 3928	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ))
14		1zzd 12513	. . . . 5 ⊢ (𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) → 1 ∈ ℤ)
15		nnuz 12781	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
16		elmapi 8782	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
18		simpr 484	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
19	17, 18	fvovco 45353	. . . . . . . 8 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑘) = ((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘))))
20	19	fveq2d 6835	. . . . . . 7 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))))
21	16	ffvelcdmda 7026	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (ℝ × ℝ))
22		xp1st 7962	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ∈ (ℝ × ℝ) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ)
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ)
24		xp2nd 7963	. . . . . . . . 9 ⊢ ((𝐹‘𝑘) ∈ (ℝ × ℝ) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
25	21, 24	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
26		volicore 46741	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))) ∈ ℝ)
28	20, 27	eqeltrd 2833	. . . . . 6 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℝ)
29	28	recnd 11151	. . . . 5 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℂ)
30		eqid 2733	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘)))
31		eqid 2733	. . . . 5 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))))
32	14, 15, 29, 30, 31	fsumsermpt 45741	. . . 4 ⊢ (𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))))
33	13, 32	syl 17	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))))
34		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)))
35	34	iftrued 4484	. . . . . . . . 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 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ)
37	36	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ)
38	13, 25	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
39	38	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
40		ressxr 11167	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
41	40, 37	sselid 3928	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ^*)
42	40, 39	sselid 3928	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ^*)
43		xpss 5637	. . . . . . . . . . . . . . . . . 18 ⊢ (ℝ × ℝ) ⊆ (V × V)
44	43, 21	sselid 3928	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (V × V))
45		1st2ndb 7970	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑘) ∈ (V × V) ↔ (𝐹‘𝑘) = ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
46	44, 45	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
47	13, 46	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
48	47	eqcomd 2739	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩ = (𝐹‘𝑘))
49		inss1 4186	. . . . . . . . . . . . . . . . 17 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤ )
51	6, 50	fssd 6676	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℕ⟶ ≤ )
52	51	ffvelcdmda 7026	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ≤ )
53	48, 52	eqeltrd 2833	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩ ∈ ≤ )
54		df-br 5096	. . . . . . . . . . . . 13 ⊢ ((1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)) ↔ ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩ ∈ ≤ )
55	53, 54	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)))
56	55	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)))
57		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)))
58	39, 37	lenltd 11270	. . . . . . . . . . . 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 257	. . . . . . . . . . 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 13060	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) = (2^nd ‘(𝐹‘𝑘)))
61		simp3 1138	. . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . 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 11268	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 495	. . . . . . . . . . . 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 4487	. . . . . . . . . . 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 11151	. . . . . . . . . . . 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 2739	. . . . . . . . . . . 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 11554	. . . . . . . . . . 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 2771	. . . . . . . . . 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 1373	. . . . . . . . 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 812	. . . . . . . 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 46143	. . . . . . . . 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 584	. . . . . . . 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 15349	. . . . . . . 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 2778	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ))
79		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
80	78, 79, 20	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))))
81	46	fveq2d 6835	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑘)) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩))
82		df-ov 7358	. . . . . . . . . . 11 ⊢ ((1^st ‘(𝐹‘𝑘))(abs ∘ − )(2^nd ‘(𝐹‘𝑘))) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
83	82	eqcomi 2742	. . . . . . . . . 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 11151	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ∈ ℂ)
86	25	recnd 11151	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐹‘𝑘)) ∈ ℂ)
87		eqid 2733	. . . . . . . . . . 11 ⊢ (abs ∘ − ) = (abs ∘ − )
88	87	cnmetdval 24705	. . . . . . . . . 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 584	. . . . . . . . 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 2772	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑘)) = (abs‘((1^st ‘(𝐹‘𝑘)) − (2^nd ‘(𝐹‘𝑘)))))
92	77, 80, 91	3eqtr4d 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = ((abs ∘ − )‘(𝐹‘𝑘)))
93	92	mpteq2dva 5188	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹‘𝑘))))
94	13, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
95		rr2sscn2 45526	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
96	95	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℂ × ℂ))
97		absf 15252	. . . . . . . 8 ⊢ abs:ℂ⟶ℝ
98		subf 11373	. . . . . . . 8 ⊢ − :(ℂ × ℂ)⟶ℂ
99		fco 6683	. . . . . . . 8 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
100	97, 98, 99	mp2an 692	. . . . . . 7 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
101	100	a1i 11	. . . . . 6 ⊢ (𝜑 → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
102	94, 96, 101	fcomptss 45363	. . . . 5 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐹) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹‘𝑘))))
103	93, 102	eqtr4d 2771	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = ((abs ∘ − ) ∘ 𝐹))
104	103	seqeq3d 13923	. . 3 ⊢ (𝜑 → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
105	33, 104	eqtr2d 2769	. 2 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
106	105	rneqd 5884	1 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐹)) = ran (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 Vcvv 3437 ∩ cin 3897 ⊆ wss 3898 ifcif 4476 ⟨cop 4583 class class class wbr 5095 ↦ cmpt 5176 × cxp 5619 ran crn 5622 ∘ ccom 5625 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 1^st c1st 7928 2^nd c2nd 7929 ↑_m cmap 8759 ℂcc 11015 ℝcr 11016 0cc0 11017 1c1 11018 + caddc 11020 ℝ^*cxr 11156 < clt 11157 ≤ cle 11158 − cmin 11355 ℕcn 12136 [,)cico 13254 ...cfz 13414 seqcseq 13915 abscabs 15148 Σcsu 15600 volcvol 25411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-dju 9805 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-n0 12393 df-z 12480 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-fl 13703 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-clim 15402 df-rlim 15403 df-sum 15601 df-rest 17333 df-topgen 17354 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-top 22829 df-topon 22846 df-bases 22881 df-cmp 23322 df-ovol 25412 df-vol 25413

This theorem is referenced by: (None)

W3C validator