Theorem ovollb2lem 25647

Description: Lemma for ovollb2 25648. (Contributed by Mario Carneiro, 24-Mar-2015.)

Hypotheses

Ref	Expression
ovollb2.1	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovollb2.2	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)
ovollb2.3	⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ovollb2.4	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovollb2.5	⊢ (𝜑 → 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹))
ovollb2.6	⊢ (𝜑 → 𝐵 ∈ ℝ+)
ovollb2.7	⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)

Assertion

Ref	Expression
ovollb2lem	⊢ (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))

Distinct variable groups:   𝐴,𝑛   𝑛,𝐹   𝐵,𝑛   𝜑,𝑛   𝑆,𝑛

Allowed substitution hints:   𝑇(𝑛)   𝐺(𝑛)

Proof of Theorem ovollb2lem

Dummy variables 𝑚 𝑦 𝑧 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovollb2.5	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹))
2		ovollb2.4	. . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		ovolficcss 25628	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
4	2, 3	syl 18	. . . 4 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
5	1, 4	sstrd 3947	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
6		ovolcl 25637	. . 3 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
7	5, 6	syl 18	. 2 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ*)
8		ovolfcl 25625	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
9	2, 8	sylan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
10	9	simp1d 1160	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
11		ovollb2.6	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ+)
12	11	rphalfcld 13067	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 / 2) ∈ ℝ+)
13	12	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
14		2nn 12309	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
15		nnnn0 12506	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
16	15	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17		nnexpcl 14106	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
18	14, 16, 17	sylancr 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℕ)
19	18	nnrpd 13053	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
20	13, 19	rpdivcld 13072	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ+)
21	20	rpred 13055	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ)
22	10, 21	resubcld 11637	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
23	9	simp2d 1161	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
24	23, 21	readdcld 11233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ)
25	10, 20	ltsubrpd 13087	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < (1st ‘(𝐹‘𝑛)))
26	9	simp3d 1162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
27	23, 20	ltaddrpd 13088	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) < ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
28	10, 23, 24, 26, 27	lelttrd 11363	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) < ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
29	22, 10, 24, 25, 28	lttrd 11366	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
30	22, 24, 29	ltled 11353	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))))
31		df-br 5110	. . . . . . . . 9 ⊢ (((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ↔ ⟨((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ≤ )
32	30, 31	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ≤ )
33	22, 24	opelxpd 5700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ (ℝ × ℝ))
34	32, 33	elind 4153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
35		ovollb2.2	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩)
36	34, 35	fmptd 7109	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
37		eqid 2763	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
38		ovollb2.3	. . . . . . 7 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
39	37, 38	ovolsf 25631	. . . . . 6 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
40	36, 39	syl 18	. . . . 5 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞))
41	40	frnd 6714	. . . 4 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
42		icossxr 13454	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
43	41, 42	sstrdi 3949	. . 3 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*)
44		supxrcl 13336	. . 3 ⊢ (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
45	43, 44	syl 18	. 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
46		ovollb2.7	. . . 4 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
47	11	rpred 13055	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
48	46, 47	readdcld 11233	. . 3 ⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ)
49	48	rexrd 11254	. 2 ⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*)
50		2fveq3 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (1st ‘(𝐹‘𝑛)) = (1st ‘(𝐹‘𝑚)))
51		oveq2 7418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
52	51	oveq2d 7426	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((𝐵 / 2) / (2↑𝑛)) = ((𝐵 / 2) / (2↑𝑚)))
53	50, 52	oveq12d 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) = ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
54		2fveq3 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (2nd ‘(𝐹‘𝑛)) = (2nd ‘(𝐹‘𝑚)))
55	54, 52	oveq12d 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))) = ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
56	53, 55	opeq12d 4846	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ⟨((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))⟩ = ⟨((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
57		opex 5445	. . . . . . . . . . . . . . 15 ⊢ ⟨((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩ ∈ V
58	56, 35, 57	fvmpt 6989	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = ⟨((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
59	58	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = ⟨((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩)
60	59	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺‘𝑚)) = (1st ‘⟨((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
61		ovex 7443	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ V
62		ovex 7443	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))) ∈ V
63	61, 62	op1st 7990	. . . . . . . . . . . 12 ⊢ (1st ‘⟨((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))
64	60, 63	eqtrdi 2814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺‘𝑚)) = ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))))
65		ovolfcl 25625	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹‘𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑚)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑚)) ≤ (2nd ‘(𝐹‘𝑚))))
66	2, 65	sylan 591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹‘𝑚)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑚)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑚)) ≤ (2nd ‘(𝐹‘𝑚))))
67	66	simp1d 1160	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹‘𝑚)) ∈ ℝ)
68	12	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℝ+)
69		nnnn0 12506	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
70	69	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ0)
71		nnexpcl 14106	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ)
72	14, 70, 71	sylancr 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℕ)
73	72	nnrpd 13053	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℝ+)
74	68, 73	rpdivcld 13072	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℝ+)
75	67, 74	ltsubrpd 13087	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))) < (1st ‘(𝐹‘𝑚)))
76	64, 75	eqbrtrd 5133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺‘𝑚)) < (1st ‘(𝐹‘𝑚)))
77	76	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺‘𝑚)) < (1st ‘(𝐹‘𝑚)))
78		ovolfcl 25625	. . . . . . . . . . . . 13 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐺‘𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺‘𝑚)) ∈ ℝ ∧ (1st ‘(𝐺‘𝑚)) ≤ (2nd ‘(𝐺‘𝑚))))
79	36, 78	sylan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐺‘𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺‘𝑚)) ∈ ℝ ∧ (1st ‘(𝐺‘𝑚)) ≤ (2nd ‘(𝐺‘𝑚))))
80	79	simp1d 1160	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺‘𝑚)) ∈ ℝ)
81	80	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐺‘𝑚)) ∈ ℝ)
82	67	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹‘𝑚)) ∈ ℝ)
83	5	sselda 3937	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
84	83	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → 𝑧 ∈ ℝ)
85		ltletr 11297	. . . . . . . . . 10 ⊢ (((1st ‘(𝐺‘𝑚)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑚)) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((1st ‘(𝐺‘𝑚)) < (1st ‘(𝐹‘𝑚)) ∧ (1st ‘(𝐹‘𝑚)) ≤ 𝑧) → (1st ‘(𝐺‘𝑚)) < 𝑧))
86	81, 82, 84, 85	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐺‘𝑚)) < (1st ‘(𝐹‘𝑚)) ∧ (1st ‘(𝐹‘𝑚)) ≤ 𝑧) → (1st ‘(𝐺‘𝑚)) < 𝑧))
87	77, 86	mpand 707	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹‘𝑚)) ≤ 𝑧 → (1st ‘(𝐺‘𝑚)) < 𝑧))
88	66	simp2d 1161	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹‘𝑚)) ∈ ℝ)
89	88, 74	ltaddrpd 13088	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹‘𝑚)) < ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
90	59	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐺‘𝑚)) = (2nd ‘⟨((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩))
91	61, 62	op2nd 7991	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))⟩) = ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))
92	90, 91	eqtrdi 2814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐺‘𝑚)) = ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
93	89, 92	breqtrrd 5139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹‘𝑚)) < (2nd ‘(𝐺‘𝑚)))
94	93	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹‘𝑚)) < (2nd ‘(𝐺‘𝑚)))
95	88	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹‘𝑚)) ∈ ℝ)
96	79	simp2d 1161	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐺‘𝑚)) ∈ ℝ)
97	96	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐺‘𝑚)) ∈ ℝ)
98		lelttr 11295	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ ∧ (2nd ‘(𝐹‘𝑚)) ∈ ℝ ∧ (2nd ‘(𝐺‘𝑚)) ∈ ℝ) → ((𝑧 ≤ (2nd ‘(𝐹‘𝑚)) ∧ (2nd ‘(𝐹‘𝑚)) < (2nd ‘(𝐺‘𝑚))) → 𝑧 < (2nd ‘(𝐺‘𝑚))))
99	84, 95, 97, 98	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → ((𝑧 ≤ (2nd ‘(𝐹‘𝑚)) ∧ (2nd ‘(𝐹‘𝑚)) < (2nd ‘(𝐺‘𝑚))) → 𝑧 < (2nd ‘(𝐺‘𝑚))))
100	94, 99	mpan2d 706	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 ≤ (2nd ‘(𝐹‘𝑚)) → 𝑧 < (2nd ‘(𝐺‘𝑚))))
101	87, 100	anim12d 620	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚))) → ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
102	101	reximdva 3178	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
103	102	ralimdva 3177	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚))) → ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
104		ovolficc 25627	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ([,] ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚)))))
105	5, 2, 104	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ([,] ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚)))))
106		ovolfioo 25626	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
107	5, 36, 106	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
108	103, 105, 107	3imtr4d 297	. . . 4 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ([,] ∘ 𝐹) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺)))
109	1, 108	mpd 16	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))
110	38	ovollb 25638	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺)) → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
111	36, 109, 110	syl2anc 595	. 2 ⊢ (𝜑 → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
112	38	fveq1i 6882	. . . . . . 7 ⊢ (𝑇‘𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘)
113		fzfid 14005	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
114		rge0ssre 13478	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ
115		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
116	115	ovolfsf 25630	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
117	2, 116	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
118	117	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
119		elfznn 13577	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
120		ffvelcdm 7076	. . . . . . . . . . . 12 ⊢ ((((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
121	118, 119, 120	syl2an 607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞))
122	114, 121	sselid 3935	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℝ)
123	122	recnd 11232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) ∈ ℂ)
124	11	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℝ+)
125	124, 73	rpdivcld 13072	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℝ+)
126	125	rpcnd 13057	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℂ)
127	119, 126	sylan2 604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
128	127	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ)
129	113, 123, 128	fsumadd 15787	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))))
130	37	ovolfsval 25629	. . . . . . . . . . . . 13 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺‘𝑚)) − (1st ‘(𝐺‘𝑚))))
131	36, 130	sylan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺‘𝑚)) − (1st ‘(𝐺‘𝑚))))
132	88	recnd 11232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝐹‘𝑚)) ∈ ℂ)
133	74	rpcnd 13057	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℂ)
134	67	recnd 11232	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝐹‘𝑚)) ∈ ℂ)
135	134, 133	subcld 11564	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ ℂ)
136	132, 133, 135	addsubassd 11584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((2nd ‘(𝐹‘𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))))))
137	92, 64	oveq12d 7428	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2nd ‘(𝐺‘𝑚)) − (1st ‘(𝐺‘𝑚))) = (((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))))
138	132, 134, 126	subadd23d 11586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((2nd ‘(𝐹‘𝑚)) − (1st ‘(𝐹‘𝑚))) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹‘𝑚)))))
139	115	ovolfsval 25629	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹‘𝑚)) − (1st ‘(𝐹‘𝑚))))
140	2, 139	sylan 591	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹‘𝑚)) − (1st ‘(𝐹‘𝑚))))
141	140	oveq1d 7425	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (((2nd ‘(𝐹‘𝑚)) − (1st ‘(𝐹‘𝑚))) + (𝐵 / (2↑𝑚))))
142	133, 134, 133	subsub3d 11594	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹‘𝑚))))
143	68	rpcnd 13057	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℂ)
144	72	nncnd 12244	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑𝑚) ∈ ℂ)
145	72	nnne0d 12281	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑𝑚) ≠ 0)
146	143, 143, 144, 145	divdird 12024	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))))
147	124	rpcnd 13057	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℂ)
148	147	2halvesd 12485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐵 / 2) + (𝐵 / 2)) = 𝐵)
149	148	oveq1d 7425	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (𝐵 / (2↑𝑚)))
150	146, 149	eqtr3d 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) = (𝐵 / (2↑𝑚)))
151	150	oveq1d 7425	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹‘𝑚))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹‘𝑚))))
152	142, 151	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹‘𝑚))))
153	152	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2nd ‘(𝐹‘𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))))) = ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹‘𝑚)))))
154	138, 141, 153	3eqtr4d 2808	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹‘𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))))))
155	136, 137, 154	3eqtr4d 2808	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2nd ‘(𝐺‘𝑚)) − (1st ‘(𝐺‘𝑚))) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
156	131, 155	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
157	119, 156	sylan2 604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
158	157	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))))
159		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
160		nnuz 12896	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
161	159, 160	eleqtrdi 2873	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
162	123, 128	addcld 11223	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) ∈ ℂ)
163	158, 161, 162	fsumser 15777	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
164		eqidd 2764	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑚) = (((abs ∘ − ) ∘ 𝐹)‘𝑚))
165	164, 161, 123	fsumser 15777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
166		ovollb2.1	. . . . . . . . . . 11 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
167	166	fveq1i 6882	. . . . . . . . . 10 ⊢ (𝑆‘𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
168	165, 167	eqtr4di 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (𝑆‘𝑘))
169	11	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℝ+)
170	169	rpcnd 13057	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
171		geo2sum 15923	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ 𝐵 ∈ ℂ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
172	159, 170, 171	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘))))
173	168, 172	oveq12d 7428	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))) = ((𝑆‘𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
174	129, 163, 173	3eqtr3d 2806	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) = ((𝑆‘𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
175	112, 174	eqtrid 2810	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑇‘𝑘) = ((𝑆‘𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))))
176	115, 166	ovolsf 25631	. . . . . . . . . 10 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
177	2, 176	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
178	177	ffvelcdmda 7079	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ (0[,)+∞))
179	114, 178	sselid 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ℝ)
180	169	rpred 13055	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℝ)
181		nnnn0 12506	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
182	181	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
183		nnexpcl 14106	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
184	14, 182, 183	sylancr 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℕ)
185	184	nnrpd 13053	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2↑𝑘) ∈ ℝ+)
186	169, 185	rpdivcld 13072	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ+)
187	186	rpred 13055	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ)
188	180, 187	resubcld 11637	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ∈ ℝ)
189	46	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
190	177	frnd 6714	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
191	190, 42	sstrdi 3949	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
192	191	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran 𝑆 ⊆ ℝ*)
193	177	ffnd 6706	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 Fn ℕ)
194		fnfvelrn 7075	. . . . . . . . 9 ⊢ ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆)
195	193, 194	sylan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆)
196		supxrub 13345	. . . . . . . 8 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ (𝑆‘𝑘) ∈ ran 𝑆) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
197	192, 195, 196	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, < ))
198	180, 186	ltsubrpd 13087	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) < 𝐵)
199	188, 180, 198	ltled 11353	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ≤ 𝐵)
200	179, 188, 189, 180, 197, 199	le2addd 11828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑆‘𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
201	175, 200	eqbrtrd 5133	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
202	201	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
203		ffn 6705	. . . . 5 ⊢ (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
204		breq1 5112	. . . . . 6 ⊢ (𝑦 = (𝑇‘𝑘) → (𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
205	204	ralrn 7083	. . . . 5 ⊢ (𝑇 Fn ℕ → (∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
206	40, 203, 205	3syl 19	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
207	202, 206	mpbird 260	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
208		supxrleub 13347	. . . 4 ⊢ ((ran 𝑇 ⊆ ℝ* ∧ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*) → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
209	43, 49, 208	syl2anc 595	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)))
210	207, 209	mpbird 260	. 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))
211	7, 45, 49, 111, 210	xrletrd 13182	1 ⊢ (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∩ cin 3904   ⊆ wss 3905  ⟨cop 4595  ∪ cuni 4872   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659  ran crn 5662   ∘ ccom 5665   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  supcsup 9396  ℂcc 11093  ℝcr 11094  0cc0 11095  1c1 11096   + caddc 11098  +∞cpnf 11235  ℝ^*cxr 11237   < clt 11238   ≤ cle 11239   − cmin 11436   / cdiv 11866  ℕcn 12228  2c2 12290  ℕ₀cn0 12499  ℤ_≥cuz 12857  ℝ⁺crp 13011  (,)cioo 13367  [,)cico 13369  [,]cicc 13370  ...cfz 13530  seqcseq 14033  ↑cexp 14093  abscabs 15281  Σcsu 15733  vol*covol 25621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-rp 13012  df-ioo 13371  df-ico 13373  df-icc 13374  df-fz 13531  df-fzo 13679  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-sum 15734  df-ovol 25623

This theorem is referenced by:  ovollb2  25648