ovolval2lem - Mathbox for Glauco Siliprandi

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Theorem ovolval2lem 45444

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 11203	. . . . . . 7 ⊢ ℝ ∈ V
2	1, 1	xpex 7742	. . . . . 6 ⊢ (ℝ × ℝ) ∈ V
3		inss2 4229	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
4		mapss 8885	. . . . . 6 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
5	2, 3, 4	mp2an 690	. . . . 5 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
6		ovolval2lem.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7	2	inex2 5318	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
8	7	a1i 11	. . . . . . 7 ⊢ (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
9		nnex 12220	. . . . . . . 8 ⊢ ℕ ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
11	8, 10	elmapd 8836	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
12	6, 11	mpbird 256	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
13	5, 12	sselid 3980	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ))
14		1zzd 12595	. . . . 5 ⊢ (𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) → 1 ∈ ℤ)
15		nnuz 12867	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
16		elmapi 8845	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
17	16	adantr 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
18		simpr 485	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
19	17, 18	fvovco 43977	. . . . . . . 8 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑘) = ((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘))))
20	19	fveq2d 6895	. . . . . . 7 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = (vol‘((1^st ‘(𝐹‘𝑘))[,)(2^nd ‘(𝐹‘𝑘)))))
21	16	ffvelcdmda 7086	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (ℝ × ℝ))
22		xp1st 8009	. . . . . . . . 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 8010	. . . . . . . . 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 45382	. . . . . . . 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 11244	. . . . 5 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) ∈ ℂ)
30		eqid 2732	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘)))
31		eqid 2732	. . . . 5 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))))
32	14, 15, 29, 30, 31	fsumsermpt 44380	. . . 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 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)))
35	34	iftrued 4536	. . . . . . . . 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 481	. . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ)
40		ressxr 11260	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
41	40, 37	sselid 3980	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) ∈ ℝ^*)
42	40, 39	sselid 3980	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ^*)
43		xpss 5692	. . . . . . . . . . . . . . . . . 18 ⊢ (ℝ × ℝ) ⊆ (V × V)
44	43, 21	sselid 3980	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ (V × V))
45		1st2ndb 8017	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑘) ∈ (V × V) ↔ (𝐹‘𝑘) = ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
46	44, 45	sylib 217	. . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩ = (𝐹‘𝑘))
49		inss1 4228	. . . . . . . . . . . . . . . . 17 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( ≤ ∩ (ℝ × ℝ)) ⊆ ≤ )
51	6, 50	fssd 6735	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℕ⟶ ≤ )
52	51	ffvelcdmda 7086	. . . . . . . . . . . . . 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 5149	. . . . . . . . . . . . 13 ⊢ ((1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)) ↔ ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩ ∈ ≤ )
55	53, 54	sylibr 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)))
56	55	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → (1^st ‘(𝐹‘𝑘)) ≤ (2^nd ‘(𝐹‘𝑘)))
57		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘))) → ¬ (1^st ‘(𝐹‘𝑘)) < (2^nd ‘(𝐹‘𝑘)))
58	39, 37	lenltd 11362	. . . . . . . . . . . 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 256	. . . . . . . . . . 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 13136	. . . . . . . . . 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 11360	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 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 496	. . . . . . . . . . . 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 4539	. . . . . . . . . . 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 11244	. . . . . . . . . . . 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 2738	. . . . . . . . . . . 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 11642	. . . . . . . . . . 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 2775	. . . . . . . . . 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 1371	. . . . . . . . 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 811	. . . . . . . 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 44784	. . . . . . . . 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 15381	. . . . . . . 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 2782	. . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ))
79		simpr 485	. . . . . . . 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 6895	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑘)) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩))
82		df-ov 7414	. . . . . . . . . . 11 ⊢ ((1^st ‘(𝐹‘𝑘))(abs ∘ − )(2^nd ‘(𝐹‘𝑘))) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
83	82	eqcomi 2741	. . . . . . . . . 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 11244	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐹‘𝑘)) ∈ ℂ)
86	25	recnd 11244	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ) ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐹‘𝑘)) ∈ ℂ)
87		eqid 2732	. . . . . . . . . . 11 ⊢ (abs ∘ − ) = (abs ∘ − )
88	87	cnmetdval 24294	. . . . . . . . . 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 2776	. . . . . . . 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 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(([,) ∘ 𝐹)‘𝑘)) = ((abs ∘ − )‘(𝐹‘𝑘)))
93	92	mpteq2dva 5248	. . . . 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 44161	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
96	95	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℂ × ℂ))
97		absf 15286	. . . . . . . 8 ⊢ abs:ℂ⟶ℝ
98		subf 11464	. . . . . . . 8 ⊢ − :(ℂ × ℂ)⟶ℂ
99		fco 6741	. . . . . . . 8 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
100	97, 98, 99	mp2an 690	. . . . . . 7 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
101	100	a1i 11	. . . . . 6 ⊢ (𝜑 → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
102	94, 96, 101	fcomptss 43987	. . . . 5 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐹) = (𝑘 ∈ ℕ ↦ ((abs ∘ − )‘(𝐹‘𝑘))))
103	93, 102	eqtr4d 2775	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘))) = ((abs ∘ − ) ∘ 𝐹))
104	103	seqeq3d 13976	. . 3 ⊢ (𝜑 → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘(([,) ∘ 𝐹)‘𝑘)))) = seq1( + , ((abs ∘ − ) ∘ 𝐹)))
105	33, 104	eqtr2d 2773	. 2 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
106	105	rneqd 5937	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 ifcif 4528 ⟨cop 4634 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ran crn 5677 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 1^st c1st 7975 2^nd c2nd 7976 ↑_m cmap 8822 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 − cmin 11446 ℕcn 12214 [,)cico 13328 ...cfz 13486 seqcseq 13968 abscabs 15183 Σcsu 15634 volcvol 24987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-rest 17370 df-topgen 17391 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-top 22403 df-topon 22420 df-bases 22456 df-cmp 22898 df-ovol 24988 df-vol 24989

This theorem is referenced by: (None)

W3C validator