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Theorem ovoliunlem1 25522

Description: Lemma for ovoliun 25525. (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 6906	. . . . . . . 8 ⊢ (𝑗 = (𝐽‘𝑚) → (𝐹‘(1^st ‘𝑗)) = (𝐹‘(1^st ‘(𝐽‘𝑚))))
2		fveq2 6901	. . . . . . . 8 ⊢ (𝑗 = (𝐽‘𝑚) → (2^nd ‘𝑗) = (2^nd ‘(𝐽‘𝑚)))
3	1, 2	fveq12d 6908	. . . . . . 7 ⊢ (𝑗 = (𝐽‘𝑚) → ((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) = ((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))
4	3	fveq2d 6905	. . . . . 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 6905	. . . . . 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 7442	. . . . 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 13993	. . . . 5 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
8		ovoliun.j	. . . . . . 7 ⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
9		f1of1 6842	. . . . . . 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 13586	. . . . . 6 ⊢ (1...𝐾) ⊆ ℕ
12		f1ores 6857	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → (𝐽 ↾ (1...𝐾)):(1...𝐾)–1-1-onto→(𝐽 “ (1...𝐾)))
14		fvres 6920	. . . . . 6 ⊢ (𝑚 ∈ (1...𝐾) → ((𝐽 ↾ (1...𝐾))‘𝑚) = (𝐽‘𝑚))
15	14	adantl 480	. . . . 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 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
18		imassrn 6080	. . . . . . . . . . . . . . 15 ⊢ (𝐽 “ (1...𝐾)) ⊆ ran 𝐽
19		f1of 6843	. . . . . . . . . . . . . . . . 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 6736	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐽 ⊆ (ℕ × ℕ))
22	18, 21	sstrid 3991	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ))
23	22	sselda 3979	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → 𝑗 ∈ (ℕ × ℕ))
24		xp1st 8035	. . . . . . . . . . . . 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 7099	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (𝐹‘(1^st ‘𝑗)) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
27		elovolmlem 25494	. . . . . . . . . . 11 ⊢ ((𝐹‘(1^st ‘𝑗)) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ (𝐹‘(1^st ‘𝑗)):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28	26, 27	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (𝐹‘(1^st ‘𝑗)):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
29		xp2nd 8036	. . . . . . . . . . 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 7099	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → ((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) ∈ ( ≤ ∩ (ℝ × ℝ)))
32	31	elin2d 4200	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → ((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) ∈ (ℝ × ℝ))
33		xp2nd 8036	. . . . . . . 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 8035	. . . . . . . 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 11692	. . . . . 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 11292	. . . . 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 15727	. . . 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 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
41	20	ffvelcdmda 7098	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐽‘𝑘) ∈ (ℕ × ℕ))
42		xp1st 8035	. . . . . . . . . . . 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 7099	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(1^st ‘(𝐽‘𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
45		elovolmlem 25494	. . . . . . . . . 10 ⊢ ((𝐹‘(1^st ‘(𝐽‘𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ (𝐹‘(1^st ‘(𝐽‘𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
46	44, 45	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(1^st ‘(𝐽‘𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
47		xp2nd 8036	. . . . . . . . . 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 7099	. . . . . . . 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 7128	. . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
52		elfznn 13584	. . . . . . 7 ⊢ (𝑚 ∈ (1...𝐾) → 𝑚 ∈ ℕ)
53		eqid 2726	. . . . . . . 8 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
54	53	ovolfsval 25490	. . . . . . 7 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑚) = ((2^nd ‘(𝐻‘𝑚)) − (1^st ‘(𝐻‘𝑚))))
55	51, 52, 54	syl2an 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (((abs ∘ − ) ∘ 𝐻)‘𝑚) = ((2^nd ‘(𝐻‘𝑚)) − (1^st ‘(𝐻‘𝑚))))
56	52	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → 𝑚 ∈ ℕ)
57		2fveq3 6906	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (1^st ‘(𝐽‘𝑘)) = (1^st ‘(𝐽‘𝑚)))
58	57	fveq2d 6905	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝐹‘(1^st ‘(𝐽‘𝑘))) = (𝐹‘(1^st ‘(𝐽‘𝑚))))
59		2fveq3 6906	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (2^nd ‘(𝐽‘𝑘)) = (2^nd ‘(𝐽‘𝑚)))
60	58, 59	fveq12d 6908	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((𝐹‘(1^st ‘(𝐽‘𝑘)))‘(2^nd ‘(𝐽‘𝑘))) = ((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))))
61		fvex 6914	. . . . . . . . . 10 ⊢ ((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚))) ∈ V
62	60, 50, 61	fvmpt 7009	. . . . . . . . 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 6905	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (2^nd ‘(𝐻‘𝑚)) = (2^nd ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))))
65	63	fveq2d 6905	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (1^st ‘(𝐻‘𝑚)) = (1^st ‘((𝐹‘(1^st ‘(𝐽‘𝑚)))‘(2^nd ‘(𝐽‘𝑚)))))
66	64, 65	oveq12d 7442	. . . . . 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 2766	. . . . 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 12917	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
70	68, 69	eleqtrdi 2836	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘1))
71		ffvelcdm 7095	. . . . . . . . . . 11 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ( ≤ ∩ (ℝ × ℝ)))
72	51, 52, 71	syl2an 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (𝐻‘𝑚) ∈ ( ≤ ∩ (ℝ × ℝ)))
73	72	elin2d 4200	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → (𝐻‘𝑚) ∈ (ℝ × ℝ))
74		xp2nd 8036	. . . . . . . . 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 8035	. . . . . . . . 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 11692	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → ((2^nd ‘(𝐻‘𝑚)) − (1^st ‘(𝐻‘𝑚))) ∈ ℝ)
79	78	recnd 11292	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝐾)) → ((2^nd ‘(𝐻‘𝑚)) − (1^st ‘(𝐻‘𝑚))) ∈ ℂ)
80	66, 79	eqeltrrd 2827	. . . . 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 15734	. . . 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 2766	. . 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 6902	. . 3 ⊢ (𝑈‘𝐾) = (seq1( + , ((abs ∘ − ) ∘ 𝐻))‘𝐾)
85	82, 84	eqtr4di 2784	. 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 9002	. . . . . . 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 582	. . . . . 6 ⊢ (𝜑 → (1...𝐾) ≈ (𝐽 “ (1...𝐾)))
88	87	ensymd 9036	. . . . 5 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ≈ (1...𝐾))
89		enfii 9223	. . . . 5 ⊢ (((1...𝐾) ∈ Fin ∧ (𝐽 “ (1...𝐾)) ≈ (1...𝐾)) → (𝐽 “ (1...𝐾)) ∈ Fin)
90	7, 88, 89	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ∈ Fin)
91	90, 37	fsumrecl 15738	. . 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 13993	. . . . 5 ⊢ (𝜑 → (1...𝐿) ∈ Fin)
93		elfznn 13584	. . . . . 6 ⊢ (𝑛 ∈ (1...𝐿) → 𝑛 ∈ ℕ)
94		ovoliun.v	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
95	93, 94	sylan2 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (vol*‘𝐴) ∈ ℝ)
96	92, 95	fsumrecl 15738	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) ∈ ℝ)
97		ovoliun.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
98	97	rpred 13070	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
99		2nn 12337	. . . . . . . 8 ⊢ 2 ∈ ℕ
100		nnnn0 12531	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
101		nnexpcl 14094	. . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
102	99, 100, 101	sylancr 585	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
103	93, 102	syl 17	. . . . . 6 ⊢ (𝑛 ∈ (1...𝐿) → (2↑𝑛) ∈ ℕ)
104		nndivre 12305	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ (2↑𝑛) ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ)
105	98, 103, 104	syl2an 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐵 / (2↑𝑛)) ∈ ℝ)
106	92, 105	fsumrecl 15738	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) ∈ ℝ)
107	96, 106	readdcld 11293	. . 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 11293	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^*, < ) + 𝐵) ∈ ℝ)
110		relxp 5700	. . . . . . . . . . . . . . 15 ⊢ Rel ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))
111		relres 6015	. . . . . . . . . . . . . . 15 ⊢ Rel ((𝐽 “ (1...𝐾)) ↾ {𝑛})
112		elsni 4650	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ {𝑛} → 𝑥 = 𝑛)
113	112	opeq1d 4885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑛} → ⟨𝑥, 𝑦⟩ = ⟨𝑛, 𝑦⟩)
114	113	eleq1d 2811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ {𝑛} → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) ↔ ⟨𝑛, 𝑦⟩ ∈ (𝐽 “ (1...𝐾))))
115		vex 3466	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑛 ∈ V
116		vex 3466	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑦 ∈ V
117	115, 116	elimasn 6099	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}) ↔ ⟨𝑛, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)))
118	114, 117	bitr4di 288	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑛} → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) ↔ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})))
119	118	pm5.32i 573	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ {𝑛} ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾))) ↔ (𝑥 ∈ {𝑛} ∧ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})))
120	116	opelresi 5997	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐽 “ (1...𝐾)) ↾ {𝑛}) ↔ (𝑥 ∈ {𝑛} ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾))))
121		opelxp 5718	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ↔ (𝑥 ∈ {𝑛} ∧ 𝑦 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛})))
122	119, 120, 121	3bitr4ri 303	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐽 “ (1...𝐾)) ↾ {𝑛}))
123	110, 111, 122	eqrelriiv 5796	. . . . . . . . . . . . . 14 ⊢ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ↾ {𝑛})
124		df-res 5694	. . . . . . . . . . . . . 14 ⊢ ((𝐽 “ (1...𝐾)) ↾ {𝑛}) = ((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V))
125	123, 124	eqtri 2754	. . . . . . . . . . . . 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 5025	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ∪ 𝑛 ∈ (1...𝐿)((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V)))
128		iunin2 5079	. . . . . . . . . . 11 ⊢ ∪ 𝑛 ∈ (1...𝐿)((𝐽 “ (1...𝐾)) ∩ ({𝑛} × V)) = ((𝐽 “ (1...𝐾)) ∩ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V))
129	127, 128	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = ((𝐽 “ (1...𝐾)) ∩ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V)))
130		relxp 5700	. . . . . . . . . . . . . 14 ⊢ Rel (ℕ × ℕ)
131		relss 5787	. . . . . . . . . . . . . 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 6729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 Fn ℕ)
135		fveq2 6901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = (𝐽‘𝑤) → (1^st ‘𝑗) = (1^st ‘(𝐽‘𝑤)))
136	135	breq1d 5163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (𝐽‘𝑤) → ((1^st ‘𝑗) ≤ 𝐿 ↔ (1^st ‘(𝐽‘𝑤)) ≤ 𝐿))
137	136	ralima 7255	. . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∀𝑗 ∈ (𝐽 “ (1...𝐾))(1^st ‘𝑗) ≤ 𝐿 ↔ ∀𝑤 ∈ (1...𝐾)(1^st ‘(𝐽‘𝑤)) ≤ 𝐿))
139	133, 138	mpbird 256	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑗 ∈ (𝐽 “ (1...𝐾))(1^st ‘𝑗) ≤ 𝐿)
140	139	r19.21bi 3239	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (1^st ‘𝑗) ≤ 𝐿)
141	25, 69	eleqtrdi 2836	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (1^st ‘𝑗) ∈ (ℤ_≥‘1))
142		ovoliun.l1	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿 ∈ ℤ)
143	142	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → 𝐿 ∈ ℤ)
144		elfz5 13547	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑗) ∈ (ℤ_≥‘1) ∧ 𝐿 ∈ ℤ) → ((1^st ‘𝑗) ∈ (1...𝐿) ↔ (1^st ‘𝑗) ≤ 𝐿))
145	141, 143, 144	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → ((1^st ‘𝑗) ∈ (1...𝐿) ↔ (1^st ‘𝑗) ≤ 𝐿))
146	140, 145	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐽 “ (1...𝐾))) → (1^st ‘𝑗) ∈ (1...𝐿))
147	146	ralrimiva 3136	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑗 ∈ (𝐽 “ (1...𝐾))(1^st ‘𝑗) ∈ (1...𝐿))
148		vex 3466	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
149	148, 116	op1std 8013	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑗) = 𝑥)
150	149	eleq1d 2811	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ⟨𝑥, 𝑦⟩ → ((1^st ‘𝑗) ∈ (1...𝐿) ↔ 𝑥 ∈ (1...𝐿)))
151	150	rspccv 3605	. . . . . . . . . . . . . . 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 5718	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V) ↔ (𝑥 ∈ ∪ 𝑛 ∈ (1...𝐿){𝑛} ∧ 𝑦 ∈ V))
154	116	biantru 528	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (1...𝐿){𝑛} ↔ (𝑥 ∈ ∪ 𝑛 ∈ (1...𝐿){𝑛} ∧ 𝑦 ∈ V))
155		iunid 5068	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ (1...𝐿){𝑛} = (1...𝐿)
156	155	eleq2i 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (1...𝐿){𝑛} ↔ 𝑥 ∈ (1...𝐿))
157	153, 154, 156	3bitr2i 298	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑦⟩ ∈ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V) ↔ 𝑥 ∈ (1...𝐿))
158	152, 157	imbitrrdi 251	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐽 “ (1...𝐾)) → ⟨𝑥, 𝑦⟩ ∈ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V)))
159	132, 158	relssdv 5794	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ⊆ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V))
160		xpiundir 5753	. . . . . . . . . . . 12 ⊢ (∪ 𝑛 ∈ (1...𝐿){𝑛} × V) = ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V)
161	159, 160	sseqtrdi 4030	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 “ (1...𝐾)) ⊆ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V))
162		dfss2 3965	. . . . . . . . . . 11 ⊢ ((𝐽 “ (1...𝐾)) ⊆ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V) ↔ ((𝐽 “ (1...𝐾)) ∩ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V)) = (𝐽 “ (1...𝐾)))
163	161, 162	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐽 “ (1...𝐾)) ∩ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × V)) = (𝐽 “ (1...𝐾)))
164	129, 163	eqtrd 2766	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) = (𝐽 “ (1...𝐾)))
165	164, 90	eqeltrd 2826	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin)
166		ssiun2 5055	. . . . . . . 8 ⊢ (𝑛 ∈ (1...𝐿) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ⊆ ∪ 𝑛 ∈ (1...𝐿)({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})))
167		ssfi 9211	. . . . . . . 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 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ∈ Fin)
169		2ndconst 8115	. . . . . . . . . 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 3468	. . . . . . . . 9 ⊢ (2^nd ↾ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))):({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛}))–1-1-onto→((𝐽 “ (1...𝐾)) “ {𝑛})
171		f1oeng 9002	. . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})) ≈ ((𝐽 “ (1...𝐾)) “ {𝑛}))
173	172	ensymd 9036	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ≈ ({𝑛} × ((𝐽 “ (1...𝐾)) “ {𝑛})))
174		enfii 9223	. . . . . . 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 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ∈ Fin)
176		ffvelcdm 7095	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
177	16, 93, 176	syl2an 594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
178		elovolmlem 25494	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
179	177, 178	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
180	179	adantrr 715	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
181		imassrn 6080	. . . . . . . . . . . . . 14 ⊢ ((𝐽 “ (1...𝐾)) “ {𝑛}) ⊆ ran (𝐽 “ (1...𝐾))
182	22	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐽 “ (1...𝐾)) ⊆ (ℕ × ℕ))
183		rnss 5945	. . . . . . . . . . . . . . . 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 6184	. . . . . . . . . . . . . . 15 ⊢ ran (ℕ × ℕ) = ℕ
186	184, 185	sseqtrdi 4030	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran (𝐽 “ (1...𝐾)) ⊆ ℕ)
187	181, 186	sstrid 3991	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ((𝐽 “ (1...𝐾)) “ {𝑛}) ⊆ ℕ)
188	187	sseld 3978	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}) → 𝑖 ∈ ℕ))
189	188	impr 453	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → 𝑖 ∈ ℕ)
190	180, 189	ffvelcdmd 7099	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((𝐹‘𝑛)‘𝑖) ∈ ( ≤ ∩ (ℝ × ℝ)))
191	190	elin2d 4200	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((𝐹‘𝑛)‘𝑖) ∈ (ℝ × ℝ))
192		xp2nd 8036	. . . . . . . . 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 8035	. . . . . . . . 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 11692	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝐿) ∧ 𝑖 ∈ ((𝐽 “ (1...𝐾)) “ {𝑛}))) → ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ ℝ)
197	196	anassrs 466	. . . . . 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 15738	. . . . 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 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ)
200	94, 199	readdcld 11293	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) ∈ ℝ)
201	93, 200	sylan2 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ((vol*‘𝐴) + (𝐵 / (2↑𝑛))) ∈ ℝ)
202		eqid 2726	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ∘ (𝐹‘𝑛)) = ((abs ∘ − ) ∘ (𝐹‘𝑛))
203		ovoliun.s	. . . . . . . . . . . 12 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛)))
204	202, 203	ovolsf 25492	. . . . . . . . . . 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 6736	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran 𝑆 ⊆ (0[,)+∞))
207		icossxr 13463	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ^*
208	206, 207	sstrdi 3992	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran 𝑆 ⊆ ℝ^*)
209		supxrcl 13348	. . . . . . . 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 11321	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
212	211	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → -∞ ∈ ℝ^*)
213	95	rexrd 11314	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (vol‘𝐴) ∈ ℝ^)
214	95	mnfltd 13158	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → -∞ < (vol*‘𝐴))
215		ovoliun.x1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
216	93, 215	sylan2 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
217	203	ovollb 25499	. . . . . . . . 9 ⊢ (((𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛))) → (vol‘𝐴) ≤ sup(ran 𝑆, ℝ^, < ))
218	179, 216, 217	syl2anc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (vol‘𝐴) ≤ sup(ran 𝑆, ℝ^, < ))
219	212, 213, 210, 214, 218	xrltletrd 13194	. . . . . . 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 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
222		xrre 13202	. . . . . . 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 837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ^*, < ) ∈ ℝ)
224		1zzd 12645	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 1 ∈ ℤ)
225	202	ovolfsval 25490	. . . . . . . . 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 578	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐹‘𝑛))‘𝑖) = ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))))
227	202	ovolfsf 25491	. . . . . . . . . . . . 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 7098	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → (((abs ∘ − ) ∘ (𝐹‘𝑛))‘𝑖) ∈ (0[,)+∞))
230	226, 229	eqeltrrd 2827	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ (0[,)+∞))
231		elrege0 13485	. . . . . . . . . 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 217	. . . . . . . . 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 493	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) ∈ ℝ)
234	232	simprd 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑖 ∈ ℕ) → 0 ≤ ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))))
235		supxrub 13357	. . . . . . . . . . . . . . 15 ⊢ ((ran 𝑆 ⊆ ℝ^* ∧ 𝑧 ∈ ran 𝑆) → 𝑧 ≤ sup(ran 𝑆, ℝ^*, < ))
236	208, 235	sylan 578	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝐿)) ∧ 𝑧 ∈ ran 𝑆) → 𝑧 ≤ sup(ran 𝑆, ℝ^*, < ))
237	236	ralrimiva 3136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ∀𝑧 ∈ ran 𝑆 𝑧 ≤ sup(ran 𝑆, ℝ^*, < ))
238		brralrspcev 5213	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥)
240	205	ffnd 6729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 𝑆 Fn ℕ)
241		breq1 5156	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑆‘𝑘) ≤ 𝑥))
242	241	ralrn 7102	. . . . . . . . . . . . . 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 3169	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥))
245	239, 244	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)
246	69, 203, 224, 226, 233, 234, 245	isumsup2 15850	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 𝑆 ⇝ sup(ran 𝑆, ℝ, < ))
247	203, 246	eqbrtrrid 5189	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛))) ⇝ sup(ran 𝑆, ℝ, < ))
248		climrel 15494	. . . . . . . . . 10 ⊢ Rel ⇝
249	248	releldmi 5954	. . . . . . . . 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 15849	. . . . . . 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 15851	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → Σ𝑖 ∈ ℕ ((2^nd ‘((𝐹‘𝑛)‘𝑖)) − (1^st ‘((𝐹‘𝑛)‘𝑖))) = sup(ran 𝑆, ℝ, < ))
253		rge0ssre 13487	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
254	206, 253	sstrdi 3992	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran 𝑆 ⊆ ℝ)
255		1nn 12275	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
256	205	fdmd 6738	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → dom 𝑆 = ℕ)
257	255, 256	eleqtrrid 2833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → 1 ∈ dom 𝑆)
258	257	ne0d 4338	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → dom 𝑆 ≠ ∅)
259		dm0rn0 5931	. . . . . . . . . . 11 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
260	259	necon3bii 2983	. . . . . . . . . 10 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
261	258, 260	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → ran 𝑆 ≠ ∅)
262		supxrre 13360	. . . . . . . . 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 1368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → sup(ran 𝑆, ℝ^*, < ) = sup(ran 𝑆, ℝ, < ))
264	252, 263	eqtr4d 2769	. . . . . . 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 5179	. . . . . 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 11421	. . . . 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 15803	. . . 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 3466	. . . . . . . . . . 11 ⊢ 𝑖 ∈ V
269	115, 268	op1std 8013	. . . . . . . . . 10 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (1^st ‘𝑗) = 𝑛)
270	269	fveq2d 6905	. . . . . . . . 9 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (𝐹‘(1^st ‘𝑗)) = (𝐹‘𝑛))
271	115, 268	op2ndd 8014	. . . . . . . . 9 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (2^nd ‘𝑗) = 𝑖)
272	270, 271	fveq12d 6908	. . . . . . . 8 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → ((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗)) = ((𝐹‘𝑛)‘𝑖))
273	272	fveq2d 6905	. . . . . . 7 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (2^nd ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) = (2^nd ‘((𝐹‘𝑛)‘𝑖)))
274	272	fveq2d 6905	. . . . . . 7 ⊢ (𝑗 = ⟨𝑛, 𝑖⟩ → (1^st ‘((𝐹‘(1^st ‘𝑗))‘(2^nd ‘𝑗))) = (1^st ‘((𝐹‘𝑛)‘𝑖)))
275	273, 274	oveq12d 7442	. . . . . 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 11292	. . . . . 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 15775	. . . . 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 15705	. . . . 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 2766	. . . 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 11292	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (vol*‘𝐴) ∈ ℂ)
281	105	recnd 11292	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐵 / (2↑𝑛)) ∈ ℂ)
282	92, 280, 281	fsumadd 15744	. . . 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 5184	. . 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 12645	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
285		simpr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
286		ovoliun.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
287	286	fvmpt2 7020	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (vol‘𝐴) ∈ ℝ) → (𝐺‘𝑛) = (vol‘𝐴))
288	285, 94, 287	syl2anc 582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (vol*‘𝐴))
289	288, 94	eqeltrd 2826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
290	69, 284, 289	serfre 14051	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
291		ovoliun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , 𝐺)
292	291	feq1i 6719	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
293	290, 292	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
294	293	frnd 6736	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
295		ressxr 11308	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
296	294, 295	sstrdi 3992	. . . . 5 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
297	93, 288	sylan2 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝐿)) → (𝐺‘𝑛) = (vol*‘𝐴))
298		1red 11265	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
299		ffvelcdm 7095	. . . . . . . . . . . . 13 ⊢ ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 1 ∈ ℕ) → (𝐽‘1) ∈ (ℕ × ℕ))
300	20, 255, 299	sylancl 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐽‘1) ∈ (ℕ × ℕ))
301		xp1st 8035	. . . . . . . . . . . 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 12279	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐽‘1)) ∈ ℝ)
304	142	zred 12718	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℝ)
305	302	nnge1d 12312	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (1^st ‘(𝐽‘1)))
306		2fveq3 6906	. . . . . . . . . . . 12 ⊢ (𝑤 = 1 → (1^st ‘(𝐽‘𝑤)) = (1^st ‘(𝐽‘1)))
307	306	breq1d 5163	. . . . . . . . . . 11 ⊢ (𝑤 = 1 → ((1^st ‘(𝐽‘𝑤)) ≤ 𝐿 ↔ (1^st ‘(𝐽‘1)) ≤ 𝐿))
308		eluzfz1 13562	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝐾))
309	70, 308	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ (1...𝐾))
310	307, 133, 309	rspcdva 3609	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(𝐽‘1)) ≤ 𝐿)
311	298, 303, 304, 305, 310	letrd 11421	. . . . . . . . 9 ⊢ (𝜑 → 1 ≤ 𝐿)
312		elnnz1 12640	. . . . . . . . 9 ⊢ (𝐿 ∈ ℕ ↔ (𝐿 ∈ ℤ ∧ 1 ≤ 𝐿))
313	142, 311, 312	sylanbrc 581	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℕ)
314	313, 69	eleqtrdi 2836	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (ℤ_≥‘1))
315	297, 314, 280	fsumser 15734	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) = (seq1( + , 𝐺)‘𝐿))
316		seqfn 14033	. . . . . . . . 9 ⊢ (1 ∈ ℤ → seq1( + , 𝐺) Fn (ℤ_≥‘1))
317	284, 316	syl 17	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺) Fn (ℤ_≥‘1))
318		fnfvelrn 7094	. . . . . . . 8 ⊢ ((seq1( + , 𝐺) Fn (ℤ_≥‘1) ∧ 𝐿 ∈ (ℤ_≥‘1)) → (seq1( + , 𝐺)‘𝐿) ∈ ran seq1( + , 𝐺))
319	317, 314, 318	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝐿) ∈ ran seq1( + , 𝐺))
320	291	rneqi 5943	. . . . . . 7 ⊢ ran 𝑇 = ran seq1( + , 𝐺)
321	319, 320	eleqtrrdi 2837	. . . . . 6 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝐿) ∈ ran 𝑇)
322	315, 321	eqeltrd 2826	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol*‘𝐴) ∈ ran 𝑇)
323		supxrub 13357	. . . . 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 582	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(vol‘𝐴) ≤ sup(ran 𝑇, ℝ^, < ))
325	98	recnd 11292	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
326		geo2sum 15877	. . . . . 6 ⊢ ((𝐿 ∈ ℕ ∧ 𝐵 ∈ ℂ) → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) = (𝐵 − (𝐵 / (2↑𝐿))))
327	313, 325, 326	syl2anc 582	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) = (𝐵 − (𝐵 / (2↑𝐿))))
328	313	nnnn0d 12584	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℕ₀)
329		nnexpcl 14094	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝐿 ∈ ℕ₀) → (2↑𝐿) ∈ ℕ)
330	99, 328, 329	sylancr 585	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐿) ∈ ℕ)
331	330	nnrpd 13068	. . . . . . . 8 ⊢ (𝜑 → (2↑𝐿) ∈ ℝ⁺)
332	97, 331	rpdivcld 13087	. . . . . . 7 ⊢ (𝜑 → (𝐵 / (2↑𝐿)) ∈ ℝ⁺)
333	332	rpge0d 13074	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐵 / (2↑𝐿)))
334	98, 330	nndivred 12318	. . . . . . 7 ⊢ (𝜑 → (𝐵 / (2↑𝐿)) ∈ ℝ)
335	98, 334	subge02d 11856	. . . . . 6 ⊢ (𝜑 → (0 ≤ (𝐵 / (2↑𝐿)) ↔ (𝐵 − (𝐵 / (2↑𝐿))) ≤ 𝐵))
336	333, 335	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝐵 − (𝐵 / (2↑𝐿))) ≤ 𝐵)
337	327, 336	eqbrtrd 5175	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...𝐿)(𝐵 / (2↑𝑛)) ≤ 𝐵)
338	96, 106, 108, 98, 324, 337	le2addd 11883	. . 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 11421	. 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 5177	1 ⊢ (𝜑 → (𝑈‘𝐾) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 Vcvv 3462 ∩ cin 3946 ⊆ wss 3947 ∅c0 4325 {csn 4633 ⟨cop 4639 ∪ cuni 4913 ∪ ciun 5001 class class class wbr 5153 ↦ cmpt 5236 × cxp 5680 dom cdm 5682 ran crn 5683 ↾ cres 5684 “ cima 5685 ∘ ccom 5686 Rel wrel 5687 Fn wfn 6549 ⟶wf 6550 –1-1→wf1 6551 –1-1-onto→wf1o 6553 ‘cfv 6554 (class class class)co 7424 1^st c1st 8001 2^nd c2nd 8002 ↑_m cmap 8855 ≈ cen 8971 Fincfn 8974 supcsup 9483 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 +∞cpnf 11295 -∞cmnf 11296 ℝ^*cxr 11297 < clt 11298 ≤ cle 11299 − cmin 11494 / cdiv 11921 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 ℤcz 12610 ℤ_≥cuz 12874 ℝ⁺crp 13028 (,)cioo 13378 [,)cico 13380 ...cfz 13538 seqcseq 14021 ↑cexp 14081 abscabs 15239 ⇝ cli 15486 Σcsu 15690 vol*covol 25482

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-ioo 13382 df-ico 13384 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-rlim 15491 df-sum 15691 df-ovol 25484

This theorem is referenced by: ovoliunlem2 25523

W3C validator