Theorem ovolval5lem2 47771

Description: ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1^st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2^nd ‘(𝐹‘𝑛))⟩ ∈ (ℝ × ℝ)). (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
ovolval5lem2.q	⊢ 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
ovolval5lem2.y	⊢ (𝜑 → 𝑌 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
ovolval5lem2.z	⊢ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))
ovolval5lem2.f	⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
ovolval5lem2.s	⊢ (𝜑 → 𝐴 ⊆ ∪ ran ([,) ∘ 𝐹))
ovolval5lem2.w	⊢ (𝜑 → 𝑊 ∈ ℝ+)
ovolval5lem2.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩)

Assertion

Ref	Expression
ovolval5lem2	⊢ (𝜑 → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑌 +e 𝑊))

Distinct variable groups:   𝐴,𝑓,𝑧   𝑛,𝐹   𝑓,𝐺   𝑛,𝐺   𝑧,𝑄   𝑛,𝑊   𝑧,𝑊   𝑧,𝑌   𝑓,𝑍,𝑧   𝜑,𝑛

Allowed substitution hints:   𝜑(𝑧, 𝑓)   𝐴(𝑛)   𝑄(𝑓, 𝑛)   𝐹(𝑧, 𝑓)   𝐺(𝑧)   𝑊(𝑓)   𝑌(𝑓, 𝑛)   𝑍(𝑛)

Proof of Theorem ovolval5lem2

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolval5lem2.z	. . . . . 6 ⊢ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
3		nnex 12348	. . . . . . 7 ⊢ ℕ ∈ V
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
5		volioof 47105	. . . . . . . 8 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞))
7		rexpssxrxp 11361	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
8	7	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
9		ovolval5lem2.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
10	9	ffvelcdmda 7089	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ × ℝ))
11		xp1st 8037	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
12	10, 11	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
13		ovolval5lem2.w	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊 ∈ ℝ+)
14	13	rpred 13171	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ ℝ)
15	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
16		2nn 12423	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
17	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
18		nnnn0 12620	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
19	17, 18	nnexpcld 14396	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
20	19	nnred 12357	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
21	20	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
22	19	nnne0d 12395	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
23	22	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
24	15, 21, 23	redivcld 12152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ)
25	12, 24	resubcld 11751	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))) ∈ ℝ)
26		xp2nd 8038	. . . . . . . . . 10 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
27	10, 26	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
28	25, 27	opelxpd 5690	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩ ∈ (ℝ × ℝ))
29		ovolval5lem2.g	. . . . . . . 8 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩)
30	28, 29	fmptd 7119	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ⟶(ℝ × ℝ))
31	6, 8, 30	fcoss 46331	. . . . . 6 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐺):ℕ⟶(0[,]+∞))
32	4, 31	sge0xrcl 47503	. . . . 5 ⊢ (𝜑 → (Σ^‘((vol ∘ (,)) ∘ 𝐺)) ∈ ℝ*)
33	2, 32	eqeltrd 2860	. . . 4 ⊢ (𝜑 → 𝑍 ∈ ℝ*)
34		reex 11298	. . . . . . . . 9 ⊢ ℝ ∈ V
35	34, 34	xpex 7771	. . . . . . . 8 ⊢ (ℝ × ℝ) ∈ V
36	35	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
37	36, 4	elmapd 8867	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ ((ℝ × ℝ) ↑m ℕ) ↔ 𝐺:ℕ⟶(ℝ × ℝ)))
38	30, 37	mpbird 260	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ((ℝ × ℝ) ↑m ℕ))
39		ovolval5lem2.s	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ([,) ∘ 𝐹))
40	30	ffvelcdmda 7089	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
41		xp1st 8037	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ)
42	40, 41	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ)
43	42	rexrd 11366	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ*)
44		xp2nd 8038	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘𝑛)) ∈ ℝ)
45	40, 44	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) ∈ ℝ)
46	45	rexrd 11366	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) ∈ ℝ*)
47	13	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ+)
48	19	nnrpd 13169	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
49	48	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
50	47, 49	rpdivcld 13188	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ+)
51	12, 50	ltsubrpd 13203	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))) < (1st ‘(𝐹‘𝑛)))
52		id 23	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
53		opex 5432	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩ ∈ V
54	53	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩ ∈ V)
55	29	fvmpt2 7010	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩ ∈ V) → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩)
56	52, 54, 55	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩)
57	56	fveq2d 6894	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩))
58		ovex 7457	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))) ∈ V
59		fvex 6903	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘(𝐹‘𝑛)) ∈ V
60		op1stg 8017	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))) ∈ V ∧ (2nd ‘(𝐹‘𝑛)) ∈ V) → (1st ‘⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩) = ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))))
61	58, 59, 60	mp2an 705	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩) = ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))
62	61	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1st ‘⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩) = ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))))
63	57, 62	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))))
64	63	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))))
65	64	breq1d 5112	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < (1st ‘(𝐹‘𝑛)) ↔ ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))) < (1st ‘(𝐹‘𝑛))))
66	51, 65	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) < (1st ‘(𝐹‘𝑛)))
67	56	fveq2d 6894	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩))
68	58, 59	op2nd 8014	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩) = (2nd ‘(𝐹‘𝑛))
69	68	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (2nd ‘⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩) = (2nd ‘(𝐹‘𝑛)))
70	67, 69	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘(𝐹‘𝑛)))
71	70	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = (2nd ‘(𝐹‘𝑛)))
72	71	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) = (2nd ‘(𝐺‘𝑛)))
73	27, 72	eqled 11420	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐺‘𝑛)))
74		icossioo 13578	. . . . . . . . . . . 12 ⊢ ((((1st ‘(𝐺‘𝑛)) ∈ ℝ* ∧ (2nd ‘(𝐺‘𝑛)) ∈ ℝ*) ∧ ((1st ‘(𝐺‘𝑛)) < (1st ‘(𝐹‘𝑛)) ∧ (2nd ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐺‘𝑛)))) → ((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))) ⊆ ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))))
75	43, 46, 66, 73, 74	syl22anc 852	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))) ⊆ ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))))
76		1st2nd2 8044	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
77	10, 76	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
78	77	fveq2d 6894	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ([,)‘(𝐹‘𝑛)) = ([,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩))
79		df-ov 7427	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))) = ([,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
80	79	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))) = ([,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩))
81	78, 80	eqtr4d 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ([,)‘(𝐹‘𝑛)) = ((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))))
82		1st2nd2 8044	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (𝐺‘𝑛) = ⟨(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))⟩)
83	40, 82	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ⟨(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))⟩)
84	83	fveq2d 6894	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐺‘𝑛)) = ((,)‘⟨(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))⟩))
85		df-ov 7427	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))) = ((,)‘⟨(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))⟩)
86	85	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))) = ((,)‘⟨(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))⟩))
87	84, 86	eqtr4d 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐺‘𝑛)) = ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))))
88	81, 87	sseq12d 3942	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (([,)‘(𝐹‘𝑛)) ⊆ ((,)‘(𝐺‘𝑛)) ↔ ((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))) ⊆ ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛)))))
89	75, 88	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ([,)‘(𝐹‘𝑛)) ⊆ ((,)‘(𝐺‘𝑛)))
90	89	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ((,)‘(𝐺‘𝑛)))
91		ss2iun 4969	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ((,)‘(𝐺‘𝑛)) → ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)))
92	90, 91	syl 18	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)))
93		fvex 6903	. . . . . . . . . . . . 13 ⊢ ([,)‘(𝐹‘𝑛)) ∈ V
94	93	rgenw 3080	. . . . . . . . . . . 12 ⊢ ∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ∈ V
95	94	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ∈ V)
96		dfiun3g 5951	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ∈ V → ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) = ∪ ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))))
97	95, 96	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) = ∪ ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))))
98		icof 46340	. . . . . . . . . . . . . . 15 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
99	98	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
100	9, 8, 99	fcomptss 46325	. . . . . . . . . . . . 13 ⊢ (𝜑 → ([,) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))))
101	100	eqcomd 2766	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))) = ([,) ∘ 𝐹))
102	101	rneqd 5921	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))) = ran ([,) ∘ 𝐹))
103	102	unieqd 4879	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))) = ∪ ran ([,) ∘ 𝐹))
104	97, 103	eqtr2d 2796	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ([,) ∘ 𝐹) = ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)))
105		fvex 6903	. . . . . . . . . . . . 13 ⊢ ((,)‘(𝐺‘𝑛)) ∈ V
106	105	rgenw 3080	. . . . . . . . . . . 12 ⊢ ∀𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) ∈ V
107	106	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) ∈ V)
108		dfiun3g 5951	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) ∈ V → ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) = ∪ ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))))
109	107, 108	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) = ∪ ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))))
110		ioof 13585	. . . . . . . . . . . . . . 15 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
111	110	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
112	30, 8, 111	fcomptss 46325	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((,) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))))
113	112	eqcomd 2766	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))) = ((,) ∘ 𝐺))
114	113	rneqd 5921	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))) = ran ((,) ∘ 𝐺))
115	114	unieqd 4879	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))) = ∪ ran ((,) ∘ 𝐺))
116	109, 115	eqtr2d 2796	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐺) = ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)))
117	104, 116	sseq12d 3942	. . . . . . . 8 ⊢ (𝜑 → (∪ ran ([,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛))))
118	92, 117	mpbird 260	. . . . . . 7 ⊢ (𝜑 → ∪ ran ([,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐺))
119	39, 118	sstrd 3919	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))
120	119, 2	jca 521	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
121		coeq2 5837	. . . . . . . . . 10 ⊢ (𝑓 = 𝐺 → ((,) ∘ 𝑓) = ((,) ∘ 𝐺))
122	121	rneqd 5921	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
123	122	unieqd 4879	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺))
124	123	sseq2d 3941	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺)))
125		coeq2 5837	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
126	125	fveq2d 6894	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
127	126	eqeq2d 2771	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
128	124, 127	anbi12d 644	. . . . . 6 ⊢ (𝑓 = 𝐺 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))))
129	128	rspcev 3576	. . . . 5 ⊢ ((𝐺 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
130	38, 120, 129	syl2anc 596	. . . 4 ⊢ (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
131	33, 130	jca 521	. . 3 ⊢ (𝜑 → (𝑍 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
132		eqeq1 2764	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
133	132	anbi2d 642	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
134	133	rexbidv 3186	. . . 4 ⊢ (𝑧 = 𝑍 → (∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
135		ovolval5lem2.q	. . . 4 ⊢ 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
136	134, 135	elrab2 3648	. . 3 ⊢ (𝑍 ∈ 𝑄 ↔ (𝑍 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
137	131, 136	sylibr 237	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑄)
138		2fveq3 6895	. . . . . 6 ⊢ (𝑚 = 𝑛 → (1st ‘(𝐹‘𝑚)) = (1st ‘(𝐹‘𝑛)))
139		2fveq3 6895	. . . . . 6 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝐹‘𝑚)) = (2nd ‘(𝐹‘𝑛)))
140	138, 139	breq12d 5115	. . . . 5 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝐹‘𝑚)) < (2nd ‘(𝐹‘𝑚)) ↔ (1st ‘(𝐹‘𝑛)) < (2nd ‘(𝐹‘𝑛))))
141	140	cbvrabv 3422	. . . 4 ⊢ {𝑚 ∈ ℕ ∣ (1st ‘(𝐹‘𝑚)) < (2nd ‘(𝐹‘𝑚))} = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) < (2nd ‘(𝐹‘𝑛))}
142	12, 27, 13, 141	ovolval5lem1 47770	. . 3 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛)))))) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛)))))) +e 𝑊))
143		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑛𝐺
144	30, 8	fssd 6732	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶(ℝ* × ℝ*))
145	143, 144	volioofmpt 47112	. . . . . . 7 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))))))
146	64, 71	oveq12d 7442	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))) = (((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛))))
147	146	fveq2d 6894	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛)))) = (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛)))))
148	147	mpteq2dva 5197	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))))) = (𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛))))))
149	145, 148	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛))))))
150	149	fveq2d 6894	. . . . 5 ⊢ (𝜑 → (Σ^‘((vol ∘ (,)) ∘ 𝐺)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛)))))))
151	2, 150	eqtrd 2795	. . . 4 ⊢ (𝜑 → 𝑍 = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛)))))))
152		ovolval5lem2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
153		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑛𝐹
154		ressxr 11360	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ*
155		xpss2 5671	. . . . . . . . . . 11 ⊢ (ℝ ⊆ ℝ* → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
156	154, 155	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ × ℝ) ⊆ (ℝ × ℝ*)
157	156	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
158	9, 157	fssd 6732	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ*))
159	153, 158	volicofmpt 47115	. . . . . . 7 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))))))
160	159	fveq2d 6894	. . . . . 6 ⊢ (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛)))))))
161	152, 160	eqtrd 2795	. . . . 5 ⊢ (𝜑 → 𝑌 = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛)))))))
162	161	oveq1d 7439	. . . 4 ⊢ (𝜑 → (𝑌 +e 𝑊) = ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛)))))) +e 𝑊))
163	151, 162	breq12d 5115	. . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑌 +e 𝑊) ↔ (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛)))))) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛)))))) +e 𝑊)))
164	142, 163	mpbird 260	. 2 ⊢ (𝜑 → 𝑍 ≤ (𝑌 +e 𝑊))
165		breq1 5105	. . 3 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ (𝑌 +e 𝑊) ↔ 𝑍 ≤ (𝑌 +e 𝑊)))
166	165	rspcev 3576	. 2 ⊢ ((𝑍 ∈ 𝑄 ∧ 𝑍 ≤ (𝑌 +e 𝑊)) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑌 +e 𝑊))
167	137, 164, 166	syl2anc 596	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑌 +e 𝑊))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3896  𝒫 cpw 4556  ⟨cop 4834  ∪ cuni 4866  ∪ ciun 4950   class class class wbr 5102   ↦ cmpt 5185   × cxp 5649  ran crn 5652   ∘ ccom 5655  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  1^st c1st 8003  2^nd c2nd 8004   ↑_m cmap 8854  ℝcr 11206  0cc0 11207  +∞cpnf 11347  ℝ^*cxr 11349   < clt 11350   ≤ cle 11351   − cmin 11548   / cdiv 11980  ℕcn 12342  2c2 12404  ℝ⁺crp 13127   +_e cxad 13246  (,)cioo 13483  [,)cico 13485  [,]cicc 13486  ↑cexp 14211  volcvol 25793  Σ^{^}csumge0 47480

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7697  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-er 8724  df-map 8856  df-pm 8857  df-en 8981  df-dom 8982  df-sdom 8983  df-fin 8984  df-fi 9410  df-sup 9441  df-inf 9442  df-oi 9511  df-dju 9989  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-div 11981  df-nn 12343  df-2 12412  df-3 12413  df-n0 12614  df-z 12701  df-uz 12973  df-q 13083  df-rp 13128  df-xneg 13248  df-xadd 13249  df-xmul 13250  df-ioo 13487  df-ico 13489  df-icc 13490  df-fz 13647  df-fzo 13796  df-fl 13939  df-seq 14152  df-exp 14212  df-hash 14482  df-cj 15275  df-re 15276  df-im 15277  df-sqrt 15411  df-abs 15412  df-clim 15664  df-rlim 15665  df-sum 15863  df-rest 17602  df-topgen 17623  df-psmet 21679  df-xmet 21680  df-met 21681  df-bl 21682  df-mopn 21683  df-top 23221  df-topon 23238  df-bases 23273  df-cmp 23714  df-ovol 25794  df-vol 25795  df-sumge0 47481

This theorem is used by:  ovolval5lem3  47772