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Theorem ovoliunlem1 25403

Description: Lemma for ovoliun 25406. (Contributed by Mario Carneiro, 12-Jun-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

**Hypotheses**
Ref	Expression
ovoliun.t	⊢ 𝑇 = seq1( + , 𝐺)
ovoliun.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
ovoliun.r	⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
ovoliun.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
ovoliun.s	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛)))
ovoliun.u	⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ovoliun.h	⊢ 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1^st ‘(𝐽‘𝑘)))‘(2^nd ‘(𝐽‘𝑘))))
ovoliun.j	⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
ovoliun.f	⊢ (𝜑 → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
ovoliun.x1	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
ovoliun.x2	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
ovoliun.k	⊢ (𝜑 → 𝐾 ∈ ℕ)
ovoliun.l1	⊢ (𝜑 → 𝐿 ∈ ℤ)
ovoliun.l2	⊢ (𝜑 → ∀𝑤 ∈ (1...𝐾)(1^st ‘(𝐽‘𝑤)) ≤ 𝐿)

**Assertion**
Ref	Expression
ovoliunlem1	⊢ (𝜑 → (𝑈‘𝐾) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵))

Distinct variable groups: 𝐴,𝑘 𝑘,𝑛,𝐵 𝑘,𝐹,𝑛 𝑤,𝑘,𝐽,𝑛 𝑛,𝐾,𝑤 𝑘,𝐿,𝑛,𝑤 𝑛,𝐻 𝜑,𝑘,𝑛 𝑆,𝑘 𝑘,𝐺 𝑇,𝑘 𝑛,𝐺 𝑇,𝑛 Allowed substitution hints: 𝜑(𝑤) 𝐴(𝑤,𝑛) 𝐵(𝑤) 𝑆(𝑤,𝑛) 𝑇(𝑤) 𝑈(𝑤,𝑘,𝑛) 𝐹(𝑤) 𝐺(𝑤) 𝐻(𝑤,𝑘) 𝐾(𝑘)

Proof of Theorem ovoliunlem1 Dummy variables 𝑗 𝑚 𝑥 𝑦 𝑧 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6863	. . . . . . . 8 ⊢ (𝑗 = (𝐽‘𝑚) → (𝐹‘(1^st ‘𝑗)) = (𝐹‘(1^st ‘(𝐽‘𝑚))))
2		fveq2 6858	. . . . . . . 8 ⊢ (𝑗 = (𝐽‘𝑚) → (2^nd ‘𝑗) = (2^nd ‘(𝐽‘𝑚)))
3	1, 2	fveq12d 6865	. . . . . . 7 ⊢ (𝑗 = (𝐽‘𝑚) → ((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) = ((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))
4	3	fveq2d 6862	. . . . . 6 ⊢ (𝑗 = (𝐽‘𝑚) → (2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) = (2^nd ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))))
5	3	fveq2d 6862	. . . . . 6 ⊢ (𝑗 = (𝐽‘𝑚) → (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) = (1^st ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))))
6	4, 5	oveq12d 7405	. . . . 5 ⊢ (𝑗 = (𝐽‘𝑚) → ((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) = ((2^nd ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))) − (1^st ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))))
7		fzfid 13938	. . . . 5 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
8		ovoliun.j	. . . . . . 7 ⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
9		f1of1 6799	. . . . . . 7 ⊢ (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ–1-1→(ℕ × ℕ))
10	8, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽:ℕ–1-1→(ℕ × ℕ))
11		fz1ssnn 13516	. . . . . 6 ⊢ (1...𝐾) ⊆ ℕ
12		f1ores 6814	. . . . . 6 ⊢ ((𝐽:ℕ–1-1→(ℕ × ℕ) ∧ (1...𝐾) ⊆ ℕ) → (𝐽 ↾ (1...𝐾)):(1...𝐾)–1-1-onto→(𝐽 “ (1...𝐾)))
13	10, 11, 12	sylancl 586	. . . . 5 ⊢ (𝜑 → (𝐽 ↾ (1...𝐾)):(1...𝐾)–1-1-onto→(𝐽 “ (1...𝐾)))
14		fvres 6877	. . . . . 6 ⊢ (𝑚 ∈ (1...𝐾) → ((𝐽 ↾ (1...𝐾))‘𝑚) = (𝐽‘𝑚))
15	14	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → ((𝐽 ↾ (1...𝐾))‘𝑚) = (𝐽‘𝑚))
16		ovoliun.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
17	16	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
18		imassrn 6042	. . . . . . . . . . . . . . 15 ⊢ (𝐽 “ (1...𝐾)) ⊆ ran 𝐽
19		f1of 6800	. . . . . . . . . . . . . . . . 17 ⊢ (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ⟶(ℕ × ℕ))
20	8, 19	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽:ℕ⟶(ℕ × ℕ))
21	20	frnd 6696	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐽 ⊆ (ℕ × ℕ))
22	18, 21	sstrid 3958	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ))
23	22	sselda 3946	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → 𝑗 ∈ (ℕ × ℕ))
24		xp1st 8000	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℕ × ℕ) → (1^st ‘𝑗) ∈ ℕ)
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (1^st ‘𝑗) ∈ ℕ)
26	17, 25	ffvelcdmd 7057	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (𝐹‘(1^st ‘𝑗)) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
27		elovolmlem 25375	. . . . . . . . . . 11 ⊢ ((𝐹‘(1^st ‘𝑗)) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ (𝐹‘(1^st ‘𝑗)):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28	26, 27	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (𝐹‘(1^st ‘𝑗)):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
29		xp2nd 8001	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℕ × ℕ) → (2^nd ‘𝑗) ∈ ℕ)
30	23, 29	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (2^nd ‘𝑗) ∈ ℕ)
31	28, 30	ffvelcdmd 7057	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → ((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) ∈ ( ≤ ∩ (ℝ × ℝ)))
32	31	elin2d 4168	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → ((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) ∈ (ℝ × ℝ))
33		xp2nd 8001	. . . . . . . 8 ⊢ (((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) ∈ (ℝ × ℝ) → (2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) ∈ ℝ)
34	32, 33	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) ∈ ℝ)
35		xp1st 8000	. . . . . . . 8 ⊢ (((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) ∈ (ℝ × ℝ) → (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) ∈ ℝ)
36	32, 35	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) ∈ ℝ)
37	34, 36	resubcld 11606	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → ((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) ∈ ℝ)
38	37	recnd 11202	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → ((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) ∈ ℂ)
39	6, 7, 13, 15, 38	fsumf1o 15689	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) = Σ𝑚 ∈ (1...𝐾)((2^nd ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))) − (1^st ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))))
40	16	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
41	20	ffvelcdmda 7056	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐽‘𝑘) ∈ (ℕ × ℕ))
42		xp1st 8000	. . . . . . . . . . . 12 ⊢ ((𝐽‘𝑘) ∈ (ℕ × ℕ) → (1^st ‘(𝐽‘𝑘)) ∈ ℕ)
43	41, 42	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐽‘𝑘)) ∈ ℕ)
44	40, 43	ffvelcdmd 7057	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(1^st ‘(𝐽‘𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
45		elovolmlem 25375	. . . . . . . . . 10 ⊢ ((𝐹‘(1^st ‘(𝐽‘𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ (𝐹‘(1^st ‘(𝐽‘𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
46	44, 45	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(1^st ‘(𝐽‘𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
47		xp2nd 8001	. . . . . . . . . 10 ⊢ ((𝐽‘𝑘) ∈ (ℕ × ℕ) → (2^nd ‘(𝐽‘𝑘)) ∈ ℕ)
48	41, 47	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐽‘𝑘)) ∈ ℕ)
49	46, 48	ffvelcdmd 7057	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(1^st ‘(𝐽‘𝑘)))‘(2^nd ‘(𝐽‘𝑘))) ∈ ( ≤ ∩ (ℝ × ℝ)))
50		ovoliun.h	. . . . . . . 8 ⊢ 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1^st ‘(𝐽‘𝑘)))‘(2^nd ‘(𝐽‘𝑘))))
51	49, 50	fmptd 7086	. . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
52		elfznn 13514	. . . . . . 7 ⊢ (𝑚 ∈ (1...𝐾) → 𝑚 ∈ ℕ)
53		eqid 2729	. . . . . . . 8 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
54	53	ovolfsval 25371	. . . . . . 7 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑚) = ((2^nd ‘(𝐻‘𝑚)) − (1^st ‘(𝐻‘𝑚))))
55	51, 52, 54	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (((abs ∘ − ) ∘ 𝐻)‘𝑚) = ((2^nd ‘(𝐻‘𝑚)) − (1^st ‘(𝐻‘𝑚))))
56	52	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → 𝑚 ∈ ℕ)
57		2fveq3 6863	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (1^st ‘(𝐽‘𝑘)) = (1^st ‘(𝐽‘𝑚)))
58	57	fveq2d 6862	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝐹‘(1^st ‘(𝐽‘𝑘))) = (𝐹‘(1^st ‘(𝐽‘𝑚))))
59		2fveq3 6863	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (2^nd ‘(𝐽‘𝑘)) = (2^nd ‘(𝐽‘𝑚)))
60	58, 59	fveq12d 6865	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((𝐹‘(1^st ‘(𝐽‘𝑘)))‘(2^nd ‘(𝐽‘𝑘))) = ((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))
61		fvex 6871	. . . . . . . . . 10 ⊢ ((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))) ∈ V
62	60, 50, 61	fvmpt 6968	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝐻‘𝑚) = ((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))
63	56, 62	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (𝐻‘𝑚) = ((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))
64	63	fveq2d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (2^nd ‘(𝐻‘𝑚)) = (2^nd ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))))
65	63	fveq2d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (1^st ‘(𝐻‘𝑚)) = (1^st ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))))
66	64, 65	oveq12d 7405	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → ((2^nd ‘(𝐻‘𝑚)) − (1^st ‘(𝐻‘𝑚))) = ((2^nd ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))) − (1^st ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))))
67	55, 66	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (((abs ∘ − ) ∘ 𝐻)‘𝑚) = ((2^nd ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))) − (1^st ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))))
68		ovoliun.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ)
69		nnuz 12836	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
70	68, 69	eleqtrdi 2838	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘1))
71		ffvelcdm 7053	. . . . . . . . . . 11 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ( ≤ ∩ (ℝ × ℝ)))
72	51, 52, 71	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (𝐻‘𝑚) ∈ ( ≤ ∩ (ℝ × ℝ)))
73	72	elin2d 4168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (𝐻‘𝑚) ∈ (ℝ × ℝ))
74		xp2nd 8001	. . . . . . . . 9 ⊢ ((𝐻‘𝑚) ∈ (ℝ × ℝ) → (2^nd ‘(𝐻‘𝑚)) ∈ ℝ)
75	73, 74	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (2^nd ‘(𝐻‘𝑚)) ∈ ℝ)
76		xp1st 8000	. . . . . . . . 9 ⊢ ((𝐻‘𝑚) ∈ (ℝ × ℝ) → (1^st ‘(𝐻‘𝑚)) ∈ ℝ)
77	73, 76	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (1^st ‘(𝐻‘𝑚)) ∈ ℝ)
78	75, 77	resubcld 11606	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → ((2^nd ‘(𝐻‘𝑚)) − (1^st ‘(𝐻‘𝑚))) ∈ ℝ)
79	78	recnd 11202	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → ((2^nd ‘(𝐻‘𝑚)) − (1^st ‘(𝐻‘𝑚))) ∈ ℂ)
80	66, 79	eqeltrrd 2829	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → ((2^nd ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))) − (1^st ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))) ∈ ℂ)
81	67, 70, 80	fsumser 15696	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ (1...𝐾)((2^nd ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))) − (1^st ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝐾))
82	39, 81	eqtrd 2764	. . 3 ⊢ (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝐾))
83		ovoliun.u	. . . 4 ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
84	83	fveq1i 6859	. . 3 ⊢ (𝑈‘𝐾) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝐾)
85	82, 84	eqtr4di 2782	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) = (𝑈‘𝐾))
86		f1oeng 8942	. . . . . . 7 ⊢ (((1...𝐾) ∈ Fin ∧ (𝐽 ↾ (1...𝐾)):(1...𝐾)–1-1-onto→(𝐽 “ (1...𝐾))) → (1...𝐾) ≈ (𝐽 “ (1...𝐾)))
87	7, 13, 86	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (1...𝐾) ≈ (𝐽 “ (1...𝐾)))
88	87	ensymd 8976	. . . . 5 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ≈ (1...𝐾))
89		enfii 9150	. . . . 5 ⊢ (((1...𝐾) ∈ Fin ∧ (𝐽 “ (1...𝐾)) ≈ (1...𝐾)) → (𝐽 “ (1...𝐾)) ∈ Fin)
90	7, 88, 89	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ∈ Fin)
91	90, 37	fsumrecl 15700	. . 3 ⊢ (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) ∈ ℝ)
92		fzfid 13938	. . . . 5 ⊢ (𝜑 → (1...𝐿) ∈ Fin)
93		elfznn 13514	. . . . . 6 ⊢ (𝑛 ∈ (1...𝐿) → 𝑛 ∈ ℕ)
94		ovoliun.v	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
95	93, 94	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (vol*‘𝐴) ∈ ℝ)
96	92, 95	fsumrecl 15700	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) ∈ ℝ)
97		ovoliun.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
98	97	rpred 12995	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
99		2nn 12259	. . . . . . . 8 ⊢ 2 ∈ ℕ
100		nnnn0 12449	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
101		nnexpcl 14039	. . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
102	99, 100, 101	sylancr 587	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
103	93, 102	syl 17	. . . . . 6 ⊢ (𝑛 ∈ (1...𝐿) → (2↑𝑛) ∈ ℕ)
104		nndivre 12227	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ (2↑𝑛) ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ)
105	98, 103, 104	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐵 / (2↑𝑛)) ∈ ℝ)
106	92, 105	fsumrecl 15700	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) ∈ ℝ)
107	96, 106	readdcld 11203	. . 3 ⊢ (𝜑 → (Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) + Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛))) ∈ ℝ)
108		ovoliun.r	. . . 4 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
109	108, 98	readdcld 11203	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^*, < ) + 𝐵) ∈ ℝ)
110		relxp 5656	. . . . . . . . . . . . . . 15 ⊢ Rel ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))
111		relres 5976	. . . . . . . . . . . . . . 15 ⊢ Rel ((𝐽 “ (1...𝐾)) ↾ {𝑛})
112		elsni 4606	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ {𝑛} → 𝑥 = 𝑛)
113	112	opeq1d 4843	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑛} → ⟨𝑥, 𝑦⟩ = ⟨𝑛, 𝑦⟩)
114	113	eleq1d 2813	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ {𝑛} → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) ↔ ⟨𝑛, 𝑦⟩ ∈ (𝐽 “ (1...𝐾))))
115		vex 3451	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑛 ∈ V
116		vex 3451	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑦 ∈ V
117	115, 116	elimasn 6061	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}) ↔ ⟨𝑛, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)))
118	114, 117	bitr4di 289	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑛} → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) ↔ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})))
119	118	pm5.32i 574	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ {𝑛} ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾))) ↔ (𝑥 ∈ {𝑛} ∧ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})))
120	116	opelresi 5958	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐽 “ (1...𝐾)) ↾ {𝑛}) ↔ (𝑥 ∈ {𝑛} ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾))))
121		opelxp 5674	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ↔ (𝑥 ∈ {𝑛} ∧ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})))
122	119, 120, 121	3bitr4ri 304	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐽 “ (1...𝐾)) ↾ {𝑛}))
123	110, 111, 122	eqrelriiv 5753	. . . . . . . . . . . . . 14 ⊢ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ↾ {𝑛})
124		df-res 5650	. . . . . . . . . . . . . 14 ⊢ ((𝐽 “ (1...𝐾)) ↾ {𝑛}) = ((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V))
125	123, 124	eqtri 2752	. . . . . . . . . . . . 13 ⊢ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V))
126	125	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V)))
127	126	iuneq2dv 4980	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ∪ 𝑛 ∈ (1...𝐿)((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V)))
128		iunin2 5035	. . . . . . . . . . 11 ⊢ ∪ 𝑛 ∈ (1...𝐿)((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V)) = ((𝐽 “ (1...𝐾)) ∩ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V))
129	127, 128	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ∩ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V)))
130		relxp 5656	. . . . . . . . . . . . . 14 ⊢ Rel (ℕ × ℕ)
131		relss 5744	. . . . . . . . . . . . . 14 ⊢ ((𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ) → (Rel (ℕ × ℕ) → Rel (𝐽 “ (1...𝐾))))
132	22, 130, 131	mpisyl 21	. . . . . . . . . . . . 13 ⊢ (𝜑 → Rel (𝐽 “ (1...𝐾)))
133		ovoliun.l2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑤 ∈ (1...𝐾)(1^st ‘(𝐽‘𝑤)) ≤ 𝐿)
134	20	ffnd 6689	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 Fn ℕ)
135		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = (𝐽‘𝑤) → (1^st ‘𝑗) = (1^st ‘(𝐽‘𝑤)))
136	135	breq1d 5117	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (𝐽‘𝑤) → ((1^st ‘𝑗) ≤ 𝐿 ↔ (1^st ‘(𝐽‘𝑤)) ≤ 𝐿))
137	136	ralima 7211	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 Fn ℕ ∧ (1...𝐾) ⊆ ℕ) → (∀𝑗 ∈ (𝐽 “ (1...𝐾))(1^st ‘𝑗) ≤ 𝐿 ↔ ∀𝑤 ∈ (1...𝐾)(1^st ‘(𝐽‘𝑤)) ≤ 𝐿))
138	134, 11, 137	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∀𝑗 ∈ (𝐽 “ (1...𝐾))(1^st ‘𝑗) ≤ 𝐿 ↔ ∀𝑤 ∈ (1...𝐾)(1^st ‘(𝐽‘𝑤)) ≤ 𝐿))
139	133, 138	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑗 ∈ (𝐽 “ (1...𝐾))(1^st ‘𝑗) ≤ 𝐿)
140	139	r19.21bi 3229	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (1^st ‘𝑗) ≤ 𝐿)
141	25, 69	eleqtrdi 2838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (1^st ‘𝑗) ∈ (ℤ_≥‘1))
142		ovoliun.l1	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿 ∈ ℤ)
143	142	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → 𝐿 ∈ ℤ)
144		elfz5 13477	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑗) ∈ (ℤ_≥‘1) ∧ 𝐿 ∈ ℤ) → ((1^st ‘𝑗) ∈ (1...𝐿) ↔ (1^st ‘𝑗) ≤ 𝐿))
145	141, 143, 144	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → ((1^st ‘𝑗) ∈ (1...𝐿) ↔ (1^st ‘𝑗) ≤ 𝐿))
146	140, 145	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (1^st ‘𝑗) ∈ (1...𝐿))
147	146	ralrimiva 3125	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑗 ∈ (𝐽 “ (1...𝐾))(1^st ‘𝑗) ∈ (1...𝐿))
148		vex 3451	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
149	148, 116	op1std 7978	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑗) = 𝑥)
150	149	eleq1d 2813	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ⟨𝑥, 𝑦⟩ → ((1^st ‘𝑗) ∈ (1...𝐿) ↔ 𝑥 ∈ (1...𝐿)))
151	150	rspccv 3585	. . . . . . . . . . . . . . 15 ⊢ (∀𝑗 ∈ (𝐽 “ (1...𝐾))(1^st ‘𝑗) ∈ (1...𝐿) → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) → 𝑥 ∈ (1...𝐿)))
152	147, 151	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) → 𝑥 ∈ (1...𝐿)))
153		opelxp 5674	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V) ↔ (𝑥 ∈ ∪ 𝑛 ∈ (1...𝐿){𝑛} ∧ 𝑦 ∈ V))
154	116	biantru 529	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (1...𝐿){𝑛} ↔ (𝑥 ∈ ∪ 𝑛 ∈ (1...𝐿){𝑛} ∧ 𝑦 ∈ V))
155		iunid 5024	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ (1...𝐿){𝑛} = (1...𝐿)
156	155	eleq2i 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (1...𝐿){𝑛} ↔ 𝑥 ∈ (1...𝐿))
157	153, 154, 156	3bitr2i 299	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑦⟩ ∈ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V) ↔ 𝑥 ∈ (1...𝐿))
158	152, 157	imbitrrdi 252	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) → ⟨𝑥, 𝑦⟩ ∈ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V)))
159	132, 158	relssdv 5751	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ⊆ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V))
160		xpiundir 5710	. . . . . . . . . . . 12 ⊢ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V) = ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V)
161	159, 160	sseqtrdi 3987	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ⊆ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V))
162		dfss2 3932	. . . . . . . . . . 11 ⊢ ((𝐽 “ (1...𝐾)) ⊆ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V) ↔ ((𝐽 “ (1...𝐾)) ∩ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V)) = (𝐽 “ (1...𝐾)))
163	161, 162	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐽 “ (1...𝐾)) ∩ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V)) = (𝐽 “ (1...𝐾)))
164	129, 163	eqtrd 2764	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = (𝐽 “ (1...𝐾)))
165	164, 90	eqeltrd 2828	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin)
166		ssiun2 5011	. . . . . . . 8 ⊢ (𝑛 ∈ (1...𝐿) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ⊆ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})))
167		ssfi 9137	. . . . . . . 8 ⊢ ((∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin ∧ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ⊆ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin)
168	165, 166, 167	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin)
169		2ndconst 8080	. . . . . . . . . 10 ⊢ (𝑛 ∈ V → (2^nd ↾ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))):({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))–1-1-onto→((𝐽 “ (1...𝐾)) “ {𝑛}))
170	169	elv 3452	. . . . . . . . 9 ⊢ (2^nd ↾ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))):({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))–1-1-onto→((𝐽 “ (1...𝐾)) “ {𝑛})
171		f1oeng 8942	. . . . . . . . 9 ⊢ ((({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin ∧ (2^nd ↾ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))):({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))–1-1-onto→((𝐽 “ (1...𝐾)) “ {𝑛})) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ≈ ((𝐽 “ (1...𝐾)) “ {𝑛}))
172	168, 170, 171	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ≈ ((𝐽 “ (1...𝐾)) “ {𝑛}))
173	172	ensymd 8976	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ≈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})))
174		enfii 9150	. . . . . . 7 ⊢ ((({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin ∧ ((𝐽 “ (1...𝐾)) “ {𝑛}) ≈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ∈ Fin)
175	168, 173, 174	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ∈ Fin)
176		ffvelcdm 7053	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
177	16, 93, 176	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
178		elovolmlem 25375	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
179	177, 178	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
180	179	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
181		imassrn 6042	. . . . . . . . . . . . . 14 ⊢ ((𝐽 “ (1...𝐾)) “ {𝑛}) ⊆ ran (𝐽 “ (1...𝐾))
182	22	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ))
183		rnss 5903	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ) → ran (𝐽 “ (1...𝐾)) ⊆ ran (ℕ × ℕ))
184	182, 183	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran (𝐽 “ (1...𝐾)) ⊆ ran (ℕ × ℕ))
185		rnxpid 6146	. . . . . . . . . . . . . . 15 ⊢ ran (ℕ × ℕ) = ℕ
186	184, 185	sseqtrdi 3987	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran (𝐽 “ (1...𝐾)) ⊆ ℕ)
187	181, 186	sstrid 3958	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ⊆ ℕ)
188	187	sseld 3945	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}) → 𝑖 ∈ ℕ))
189	188	impr 454	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → 𝑖 ∈ ℕ)
190	180, 189	ffvelcdmd 7057	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((𝐹‘𝑛)‘𝑖) ∈ ( ≤ ∩ (ℝ × ℝ)))
191	190	elin2d 4168	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((𝐹‘𝑛)‘𝑖) ∈ (ℝ × ℝ))
192		xp2nd 8001	. . . . . . . . 9 ⊢ (((𝐹‘𝑛)‘𝑖) ∈ (ℝ × ℝ) → (2^nd ‘((𝐹‘𝑛)‘𝑖)) ∈ ℝ)
193	191, 192	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → (2^nd ‘((𝐹‘𝑛)‘𝑖)) ∈ ℝ)
194		xp1st 8000	. . . . . . . . 9 ⊢ (((𝐹‘𝑛)‘𝑖) ∈ (ℝ × ℝ) → (1^st ‘((𝐹‘𝑛)‘𝑖)) ∈ ℝ)
195	191, 194	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → (1^st ‘((𝐹‘𝑛)‘𝑖)) ∈ ℝ)
196	193, 195	resubcld 11606	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ ℝ)
197	196	anassrs 467	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})) → ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ ℝ)
198	175, 197	fsumrecl 15700	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ ℝ)
199	98, 102, 104	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ)
200	94, 199	readdcld 11203	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) ∈ ℝ)
201	93, 200	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) ∈ ℝ)
202		eqid 2729	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ∘ (𝐹‘𝑛)) = ((abs ∘ − ) ∘ (𝐹‘𝑛))
203		ovoliun.s	. . . . . . . . . . . 12 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛)))
204	202, 203	ovolsf 25373	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
205	179, 204	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 𝑆:ℕ⟶(0[,)+∞))
206	205	frnd 6696	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran 𝑆 ⊆ (0[,)+∞))
207		icossxr 13393	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ^*
208	206, 207	sstrdi 3959	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran 𝑆 ⊆ ℝ^*)
209		supxrcl 13275	. . . . . . . 8 ⊢ (ran 𝑆 ⊆ ℝ^* → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
210	208, 209	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
211		mnfxr 11231	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
212	211	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → -∞ ∈ ℝ^*)
213	95	rexrd 11224	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (vol‘𝐴) ∈ ℝ^)
214	95	mnfltd 13084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → -∞ < (vol*‘𝐴))
215		ovoliun.x1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
216	93, 215	sylan2 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
217	203	ovollb 25380	. . . . . . . . 9 ⊢ (((𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛))) → (vol‘𝐴) ≤ sup(ran 𝑆, ℝ^, < ))
218	179, 216, 217	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (vol‘𝐴) ≤ sup(ran 𝑆, ℝ^, < ))
219	212, 213, 210, 214, 218	xrltletrd 13121	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → -∞ < sup(ran 𝑆, ℝ^*, < ))
220		ovoliun.x2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
221	93, 220	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
222		xrre 13129	. . . . . . 7 ⊢ (((sup(ran 𝑆, ℝ^, < ) ∈ ℝ^ ∧ ((vol‘𝐴) + (𝐵 / (2↑𝑛))) ∈ ℝ) ∧ (-∞ < sup(ran 𝑆, ℝ^, < ) ∧ sup(ran 𝑆, ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))) → sup(ran 𝑆, ℝ^*, < ) ∈ ℝ)
223	210, 201, 219, 221, 222	syl22anc 838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ^*, < ) ∈ ℝ)
224		1zzd 12564	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 1 ∈ ℤ)
225	202	ovolfsval 25371	. . . . . . . . 9 ⊢ (((𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑖 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐹‘𝑛))‘𝑖) = ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))))
226	179, 225	sylan 580	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐹‘𝑛))‘𝑖) = ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))))
227	202	ovolfsf 25372	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ (𝐹‘𝑛)):ℕ⟶(0[,)+∞))
228	179, 227	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ((abs ∘ − ) ∘ (𝐹‘𝑛)):ℕ⟶(0[,)+∞))
229	228	ffvelcdmda 7056	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐹‘𝑛))‘𝑖) ∈ (0[,)+∞))
230	226, 229	eqeltrrd 2829	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ (0[,)+∞))
231		elrege0 13415	. . . . . . . . . 10 ⊢ (((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ (0[,)+∞) ↔ (((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ ℝ ∧ 0 ≤ ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖)))))
232	230, 231	sylib 218	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → (((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ ℝ ∧ 0 ≤ ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖)))))
233	232	simpld 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ ℝ)
234	232	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → 0 ≤ ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))))
235		supxrub 13284	. . . . . . . . . . . . . . 15 ⊢ ((ran 𝑆 ⊆ ℝ^* ∧ 𝑧 ∈ ran 𝑆) → 𝑧 ≤ sup(ran 𝑆, ℝ^*, < ))
236	208, 235	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑧 ∈ ran 𝑆) → 𝑧 ≤ sup(ran 𝑆, ℝ^*, < ))
237	236	ralrimiva 3125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ∀𝑧 ∈ ran 𝑆 𝑧 ≤ sup(ran 𝑆, ℝ^*, < ))
238		brralrspcev 5167	. . . . . . . . . . . . 13 ⊢ ((sup(ran 𝑆, ℝ^, < ) ∈ ℝ ∧ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ sup(ran 𝑆, ℝ^, < )) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥)
239	223, 237, 238	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥)
240	205	ffnd 6689	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 𝑆 Fn ℕ)
241		breq1 5110	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑆‘𝑘) ≤ 𝑥))
242	241	ralrn 7060	. . . . . . . . . . . . . 14 ⊢ (𝑆 Fn ℕ → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥))
243	240, 242	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥))
244	243	rexbidv 3157	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥))
245	239, 244	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)
246	69, 203, 224, 226, 233, 234, 245	isumsup2 15812	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 𝑆 ⇝ sup(ran 𝑆, ℝ, < ))
247	203, 246	eqbrtrrid 5143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛))) ⇝ sup(ran 𝑆, ℝ, < ))
248		climrel 15458	. . . . . . . . . 10 ⊢ Rel ⇝
249	248	releldmi 5912	. . . . . . . . 9 ⊢ (seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛))) ⇝ sup(ran 𝑆, ℝ, < ) → seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛))) ∈ dom ⇝ )
250	247, 249	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛))) ∈ dom ⇝ )
251	69, 224, 175, 187, 226, 233, 234, 250	isumless 15811	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ≤ Σ𝑖 ∈ ℕ ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))))
252	69, 203, 224, 226, 233, 234, 245	isumsup 15813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ℕ ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) = sup(ran 𝑆, ℝ, < ))
253		rge0ssre 13417	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
254	206, 253	sstrdi 3959	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran 𝑆 ⊆ ℝ)
255		1nn 12197	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
256	205	fdmd 6698	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → dom 𝑆 = ℕ)
257	255, 256	eleqtrrid 2835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 1 ∈ dom 𝑆)
258	257	ne0d 4305	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → dom 𝑆 ≠ ∅)
259		dm0rn0 5888	. . . . . . . . . . 11 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
260	259	necon3bii 2977	. . . . . . . . . 10 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
261	258, 260	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran 𝑆 ≠ ∅)
262		supxrre 13287	. . . . . . . . 9 ⊢ ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥) → sup(ran 𝑆, ℝ^*, < ) = sup(ran 𝑆, ℝ, < ))
263	254, 261, 239, 262	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ^*, < ) = sup(ran 𝑆, ℝ, < ))
264	252, 263	eqtr4d 2767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ℕ ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) = sup(ran 𝑆, ℝ^*, < ))
265	251, 264	breqtrd 5133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ≤ sup(ran 𝑆, ℝ^*, < ))
266	198, 223, 201, 265, 221	letrd 11331	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
267	92, 198, 201, 266	fsumle 15765	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ≤ Σ𝑛 ∈ (1...𝐿)((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
268		vex 3451	. . . . . . . . . . 11 ⊢ 𝑖 ∈ V
269	115, 268	op1std 7978	. . . . . . . . . 10 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (1^st ‘𝑗) = 𝑛)
270	269	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (𝐹‘(1^st ‘𝑗)) = (𝐹‘𝑛))
271	115, 268	op2ndd 7979	. . . . . . . . 9 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (2^nd ‘𝑗) = 𝑖)
272	270, 271	fveq12d 6865	. . . . . . . 8 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → ((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) = ((𝐹‘𝑛)‘𝑖))
273	272	fveq2d 6862	. . . . . . 7 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) = (2^nd ‘((𝐹‘𝑛)‘𝑖)))
274	272	fveq2d 6862	. . . . . . 7 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) = (1^st ‘((𝐹‘𝑛)‘𝑖)))
275	273, 274	oveq12d 7405	. . . . . 6 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → ((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) = ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))))
276	196	recnd 11202	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ ℂ)
277	275, 92, 175, 276	fsum2d 15737	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) = Σ𝑗 ∈ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))))
278	164	sumeq1d 15666	. . . . 5 ⊢ (𝜑 → Σ𝑗 ∈ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) = Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))))
279	277, 278	eqtrd 2764	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)Σ𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) = Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))))
280	95	recnd 11202	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (vol*‘𝐴) ∈ ℂ)
281	105	recnd 11202	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐵 / (2↑𝑛)) ∈ ℂ)
282	92, 280, 281	fsumadd 15706	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)((vol‘𝐴) + (𝐵 / (2↑𝑛))) = (Σ𝑛 ∈ (1...𝐿)(vol‘𝐴) + Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛))))
283	267, 279, 282	3brtr3d 5138	. . 3 ⊢ (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) ≤ (Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) + Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛))))
284		1zzd 12564	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
285		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
286		ovoliun.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
287	286	fvmpt2 6979	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (vol‘𝐴) ∈ ℝ) → (𝐺‘𝑛) = (vol‘𝐴))
288	285, 94, 287	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (vol*‘𝐴))
289	288, 94	eqeltrd 2828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
290	69, 284, 289	serfre 13996	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
291		ovoliun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , 𝐺)
292	291	feq1i 6679	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
293	290, 292	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
294	293	frnd 6696	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
295		ressxr 11218	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
296	294, 295	sstrdi 3959	. . . . 5 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
297	93, 288	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐺‘𝑛) = (vol*‘𝐴))
298		1red 11175	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
299		ffvelcdm 7053	. . . . . . . . . . . . 13 ⊢ ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 1 ∈ ℕ) → (𝐽‘1) ∈ (ℕ × ℕ))
300	20, 255, 299	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐽‘1) ∈ (ℕ × ℕ))
301		xp1st 8000	. . . . . . . . . . . 12 ⊢ ((𝐽‘1) ∈ (ℕ × ℕ) → (1^st ‘(𝐽‘1)) ∈ ℕ)
302	300, 301	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘(𝐽‘1)) ∈ ℕ)
303	302	nnred 12201	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐽‘1)) ∈ ℝ)
304	142	zred 12638	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℝ)
305	302	nnge1d 12234	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (1^st ‘(𝐽‘1)))
306		2fveq3 6863	. . . . . . . . . . . 12 ⊢ (𝑤 = 1 → (1^st ‘(𝐽‘𝑤)) = (1^st ‘(𝐽‘1)))
307	306	breq1d 5117	. . . . . . . . . . 11 ⊢ (𝑤 = 1 → ((1^st ‘(𝐽‘𝑤)) ≤ 𝐿 ↔ (1^st ‘(𝐽‘1)) ≤ 𝐿))
308		eluzfz1 13492	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝐾))
309	70, 308	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ (1...𝐾))
310	307, 133, 309	rspcdva 3589	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐽‘1)) ≤ 𝐿)
311	298, 303, 304, 305, 310	letrd 11331	. . . . . . . . 9 ⊢ (𝜑 → 1 ≤ 𝐿)
312		elnnz1 12559	. . . . . . . . 9 ⊢ (𝐿 ∈ ℕ ↔ (𝐿 ∈ ℤ ∧ 1 ≤ 𝐿))
313	142, 311, 312	sylanbrc 583	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℕ)
314	313, 69	eleqtrdi 2838	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (ℤ_≥‘1))
315	297, 314, 280	fsumser 15696	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) = (seq1( + , 𝐺)‘𝐿))
316		seqfn 13978	. . . . . . . . 9 ⊢ (1 ∈ ℤ → seq1( + , 𝐺) Fn (ℤ_≥‘1))
317	284, 316	syl 17	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺) Fn (ℤ_≥‘1))
318		fnfvelrn 7052	. . . . . . . 8 ⊢ ((seq1( + , 𝐺) Fn (ℤ_≥‘1) ∧ 𝐿 ∈ (ℤ_≥‘1)) → (seq1( + , 𝐺)‘𝐿) ∈ ran seq1( + , 𝐺))
319	317, 314, 318	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝐿) ∈ ran seq1( + , 𝐺))
320	291	rneqi 5901	. . . . . . 7 ⊢ ran 𝑇 = ran seq1( + , 𝐺)
321	319, 320	eleqtrrdi 2839	. . . . . 6 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝐿) ∈ ran 𝑇)
322	315, 321	eqeltrd 2828	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) ∈ ran 𝑇)
323		supxrub 13284	. . . . 5 ⊢ ((ran 𝑇 ⊆ ℝ^* ∧ Σ𝑛 ∈ (1...𝐿)(vol‘𝐴) ∈ ran 𝑇) → Σ𝑛 ∈ (1...𝐿)(vol‘𝐴) ≤ sup(ran 𝑇, ℝ^*, < ))
324	296, 322, 323	syl2anc 584	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol‘𝐴) ≤ sup(ran 𝑇, ℝ^, < ))
325	98	recnd 11202	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
326		geo2sum 15839	. . . . . 6 ⊢ ((𝐿 ∈ ℕ ∧ 𝐵 ∈ ℂ) → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) = (𝐵 − (𝐵 / (2↑𝐿))))
327	313, 325, 326	syl2anc 584	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) = (𝐵 − (𝐵 / (2↑𝐿))))
328	313	nnnn0d 12503	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℕ₀)
329		nnexpcl 14039	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝐿 ∈ ℕ₀) → (2↑𝐿) ∈ ℕ)
330	99, 328, 329	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐿) ∈ ℕ)
331	330	nnrpd 12993	. . . . . . . 8 ⊢ (𝜑 → (2↑𝐿) ∈ ℝ⁺)
332	97, 331	rpdivcld 13012	. . . . . . 7 ⊢ (𝜑 → (𝐵 / (2↑𝐿)) ∈ ℝ⁺)
333	332	rpge0d 12999	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐵 / (2↑𝐿)))
334	98, 330	nndivred 12240	. . . . . . 7 ⊢ (𝜑 → (𝐵 / (2↑𝐿)) ∈ ℝ)
335	98, 334	subge02d 11770	. . . . . 6 ⊢ (𝜑 → (0 ≤ (𝐵 / (2↑𝐿)) ↔ (𝐵 − (𝐵 / (2↑𝐿))) ≤ 𝐵))
336	333, 335	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝐵 − (𝐵 / (2↑𝐿))) ≤ 𝐵)
337	327, 336	eqbrtrd 5129	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) ≤ 𝐵)
338	96, 106, 108, 98, 324, 337	le2addd 11797	. . 3 ⊢ (𝜑 → (Σ𝑛 ∈ (1...𝐿)(vol‘𝐴) + Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛))) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))
339	91, 107, 109, 283, 338	letrd 11331	. 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝐽 “ (1...𝐾))((2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) − (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)))) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵))
340	85, 339	eqbrtrrd 5131	1 ⊢ (𝜑 → (𝑈‘𝐾) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 {csn 4589 ⟨cop 4595 ∪ cuni 4871 ∪ ciun 4955 class class class wbr 5107 ↦ cmpt 5188 × cxp 5636 dom cdm 5638 ran crn 5639 ↾ cres 5640 “ cima 5641 ∘ ccom 5642 Rel wrel 5643 Fn wfn 6506 ⟶wf 6507 –1-1→wf1 6508 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 ↑_m cmap 8799 ≈ cen 8915 Fincfn 8918 supcsup 9391 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 (,)cioo 13306 [,)cico 13308 ...cfz 13468 seqcseq 13966 ↑cexp 14026 abscabs 15200 ⇝ cli 15450 Σcsu 15652 vol*covol 25363

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-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 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-rp 12952 df-ioo 13310 df-ico 13312 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-ovol 25365

This theorem is referenced by: ovoliunlem2 25404

W3C validator