Theorem ovolval5lem2 42929

Description: |- ( ( ph /\ n e. NN ) -> <. ( ( 1st (𝐹 n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd (𝐹 n ) ) >. e. ( RR X. RR ) ) (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 11638	. . . . . . 7 ⊢ ℕ ∈ V
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
5		volioof 42266	. . . . . . . 8 ⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞))
7		rexpssxrxp 10680	. . . . . . . 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	ffvelrnda 6845	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ × ℝ))
11		xp1st 7715	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
12	10, 11	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
13		ovolval5lem2.w	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊 ∈ ℝ+)
14	13	rpred 12425	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ ℝ)
15	14	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
16		2nn 11704	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
17	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
18		nnnn0 11898	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
19	17, 18	nnexpcld 13600	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
20	19	nnred 11647	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
21	20	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
22	19	nnne0d 11681	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
23	22	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
24	15, 21, 23	redivcld 11462	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ)
25	12, 24	resubcld 11062	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))) ∈ ℝ)
26		xp2nd 7716	. . . . . . . . . 10 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
27	10, 26	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
28	25, 27	opelxpd 5587	. . . . . . . 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 6872	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ⟶(ℝ × ℝ))
31	6, 8, 30	fcoss 41466	. . . . . 6 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐺):ℕ⟶(0[,]+∞))
32	4, 31	sge0xrcl 42661	. . . . 5 ⊢ (𝜑 → (Σ^‘((vol ∘ (,)) ∘ 𝐺)) ∈ ℝ*)
33	2, 32	eqeltrd 2913	. . . 4 ⊢ (𝜑 → 𝑍 ∈ ℝ*)
34		reex 10622	. . . . . . . . 9 ⊢ ℝ ∈ V
35	34, 34	xpex 7470	. . . . . . . 8 ⊢ (ℝ × ℝ) ∈ V
36	35	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
37	36, 4	elmapd 8414	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ ((ℝ × ℝ) ↑m ℕ) ↔ 𝐺:ℕ⟶(ℝ × ℝ)))
38	30, 37	mpbird 259	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ ((ℝ × ℝ) ↑m ℕ))
39		ovolval5lem2.s	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ([,) ∘ 𝐹))
40	30	ffvelrnda 6845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
41		xp1st 7715	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ)
42	40, 41	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ)
43	42	rexrd 10685	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ*)
44		xp2nd 7716	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘𝑛)) ∈ ℝ)
45	40, 44	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) ∈ ℝ)
46	45	rexrd 10685	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) ∈ ℝ*)
47	13	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ+)
48	19	nnrpd 12423	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
49	48	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
50	47, 49	rpdivcld 12442	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ+)
51	12, 50	ltsubrpd 12457	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))) < (1st ‘(𝐹‘𝑛)))
52		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
53		opex 5348	. . . . . . . . . . . . . . . . . . 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 6773	. . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩)
57	56	fveq2d 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩))
58		ovex 7183	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))) ∈ V
59		fvex 6677	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘(𝐹‘𝑛)) ∈ V
60		op1stg 7695	. . . . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))))
64	63	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))))
65	64	breq1d 5068	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < (1st ‘(𝐹‘𝑛)) ↔ ((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))) < (1st ‘(𝐹‘𝑛))))
66	51, 65	mpbird 259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) < (1st ‘(𝐹‘𝑛)))
67	56	fveq2d 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩))
68	58, 59	op2nd 7692	. . . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘(𝐹‘𝑛)))
71	70	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = (2nd ‘(𝐹‘𝑛)))
72	71	eqcomd 2827	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) = (2nd ‘(𝐺‘𝑛)))
73	27, 72	eqled 10737	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐺‘𝑛)))
74		icossioo 12822	. . . . . . . . . . . 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 836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))) ⊆ ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))))
76		1st2nd2 7722	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
77	10, 76	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
78	77	fveq2d 6668	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ([,)‘(𝐹‘𝑛)) = ([,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩))
79		df-ov 7153	. . . . . . . . . . . . . 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 2859	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ([,)‘(𝐹‘𝑛)) = ((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))))
82		1st2nd2 7722	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (𝐺‘𝑛) = ⟨(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))⟩)
83	40, 82	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ⟨(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))⟩)
84	83	fveq2d 6668	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐺‘𝑛)) = ((,)‘⟨(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))⟩))
85		df-ov 7153	. . . . . . . . . . . . . 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 2859	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐺‘𝑛)) = ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))))
88	81, 87	sseq12d 3999	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (([,)‘(𝐹‘𝑛)) ⊆ ((,)‘(𝐺‘𝑛)) ↔ ((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))) ⊆ ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛)))))
89	75, 88	mpbird 259	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ([,)‘(𝐹‘𝑛)) ⊆ ((,)‘(𝐺‘𝑛)))
90	89	ralrimiva 3182	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ((,)‘(𝐺‘𝑛)))
91		ss2iun 4929	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ((,)‘(𝐺‘𝑛)) → ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)))
92	90, 91	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)))
93		fvex 6677	. . . . . . . . . . . . 13 ⊢ ([,)‘(𝐹‘𝑛)) ∈ V
94	93	rgenw 3150	. . . . . . . . . . . 12 ⊢ ∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ∈ V
95	94	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ∈ V)
96		dfiun3g 5829	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ∈ V → ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) = ∪ ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))))
97	95, 96	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) = ∪ ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))))
98		icof 41475	. . . . . . . . . . . . . . 15 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
99	98	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
100	9, 8, 99	fcomptss 41459	. . . . . . . . . . . . 13 ⊢ (𝜑 → ([,) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))))
101	100	eqcomd 2827	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))) = ([,) ∘ 𝐹))
102	101	rneqd 5802	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))) = ran ([,) ∘ 𝐹))
103	102	unieqd 4841	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹‘𝑛))) = ∪ ran ([,) ∘ 𝐹))
104	97, 103	eqtr2d 2857	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ([,) ∘ 𝐹) = ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)))
105		fvex 6677	. . . . . . . . . . . . 13 ⊢ ((,)‘(𝐺‘𝑛)) ∈ V
106	105	rgenw 3150	. . . . . . . . . . . 12 ⊢ ∀𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) ∈ V
107	106	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) ∈ V)
108		dfiun3g 5829	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) ∈ V → ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) = ∪ ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))))
109	107, 108	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)) = ∪ ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))))
110		ioof 12829	. . . . . . . . . . . . . . 15 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
111	110	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
112	30, 8, 111	fcomptss 41459	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((,) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))))
113	112	eqcomd 2827	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))) = ((,) ∘ 𝐺))
114	113	rneqd 5802	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))) = ran ((,) ∘ 𝐺))
115	114	unieqd 4841	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺‘𝑛))) = ∪ ran ((,) ∘ 𝐺))
116	109, 115	eqtr2d 2857	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐺) = ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛)))
117	104, 116	sseq12d 3999	. . . . . . . 8 ⊢ (𝜑 → (∪ ran ([,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∪ 𝑛 ∈ ℕ ([,)‘(𝐹‘𝑛)) ⊆ ∪ 𝑛 ∈ ℕ ((,)‘(𝐺‘𝑛))))
118	92, 117	mpbird 259	. . . . . . 7 ⊢ (𝜑 → ∪ ran ([,) ∘ 𝐹) ⊆ ∪ ran ((,) ∘ 𝐺))
119	39, 118	sstrd 3976	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))
120	119, 2	jca 514	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
121		coeq2 5723	. . . . . . . . . 10 ⊢ (𝑓 = 𝐺 → ((,) ∘ 𝑓) = ((,) ∘ 𝐺))
122	121	rneqd 5802	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
123	122	unieqd 4841	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐺))
124	123	sseq2d 3998	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐺)))
125		coeq2 5723	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
126	125	fveq2d 6668	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
127	126	eqeq2d 2832	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
128	124, 127	anbi12d 632	. . . . . 6 ⊢ (𝑓 = 𝐺 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))))
129	128	rspcev 3622	. . . . 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 586	. . . 4 ⊢ (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
131	33, 130	jca 514	. . 3 ⊢ (𝜑 → (𝑍 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
132		eqeq1 2825	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
133	132	anbi2d 630	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
134	133	rexbidv 3297	. . . 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 3682	. . 3 ⊢ (𝑍 ∈ 𝑄 ↔ (𝑍 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
137	131, 136	sylibr 236	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑄)
138		2fveq3 6669	. . . . . 6 ⊢ (𝑚 = 𝑛 → (1st ‘(𝐹‘𝑚)) = (1st ‘(𝐹‘𝑛)))
139		2fveq3 6669	. . . . . 6 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝐹‘𝑚)) = (2nd ‘(𝐹‘𝑛)))
140	138, 139	breq12d 5071	. . . . 5 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝐹‘𝑚)) < (2nd ‘(𝐹‘𝑚)) ↔ (1st ‘(𝐹‘𝑛)) < (2nd ‘(𝐹‘𝑛))))
141	140	cbvrabv 3491	. . . 4 ⊢ {𝑚 ∈ ℕ ∣ (1st ‘(𝐹‘𝑚)) < (2nd ‘(𝐹‘𝑚))} = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) < (2nd ‘(𝐹‘𝑛))}
142	12, 27, 13, 141	ovolval5lem1 42928	. . 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 2977	. . . . . . . 8 ⊢ Ⅎ𝑛𝐺
144	30, 8	fssd 6522	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶(ℝ* × ℝ*))
145	143, 144	volioofmpt 42273	. . . . . . 7 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))))))
146	64, 71	oveq12d 7168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛))) = (((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛))))
147	146	fveq2d 6668	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((1st ‘(𝐺‘𝑛))(,)(2nd ‘(𝐺‘𝑛)))) = (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛)))))
148	147	mpteq2dva 5153	. . . . . . 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 2856	. . . . . 6 ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛))))))
150	149	fveq2d 6668	. . . . 5 ⊢ (𝜑 → (Σ^‘((vol ∘ (,)) ∘ 𝐺)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹‘𝑛)))))))
151	2, 150	eqtrd 2856	. . . 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 2977	. . . . . . . 8 ⊢ Ⅎ𝑛𝐹
154		ressxr 10679	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ*
155		xpss2 5569	. . . . . . . . . . 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 6522	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ*))
159	153, 158	volicofmpt 42276	. . . . . . 7 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛))))))
160	159	fveq2d 6668	. . . . . 6 ⊢ (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛)))))))
161	152, 160	eqtrd 2856	. . . . 5 ⊢ (𝜑 → 𝑌 = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛)))))))
162	161	oveq1d 7165	. . . 4 ⊢ (𝜑 → (𝑌 +𝑒 𝑊) = ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑛))[,)(2nd ‘(𝐹‘𝑛)))))) +𝑒 𝑊))
163	151, 162	breq12d 5071	. . 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 259	. 2 ⊢ (𝜑 → 𝑍 ≤ (𝑌 +𝑒 𝑊))
165		breq1 5061	. . 3 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ (𝑌 +𝑒 𝑊) ↔ 𝑍 ≤ (𝑌 +𝑒 𝑊)))
166	165	rspcev 3622	. 2 ⊢ ((𝑍 ∈ 𝑄 ∧ 𝑍 ≤ (𝑌 +𝑒 𝑊)) → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))
167	137, 164, 166	syl2anc 586	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ⊆ wss 3935  𝒫 cpw 4538  ⟨cop 4566  ∪ cuni 4831  ∪ ciun 4911   class class class wbr 5058   ↦ cmpt 5138   × cxp 5547  ran crn 5550   ∘ ccom 5553  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  1^st c1st 7681  2^nd c2nd 7682   ↑_m cmap 8400  ℝcr 10530  0cc0 10531  +∞cpnf 10666  ℝ^*cxr 10668   < clt 10669   ≤ cle 10670   − cmin 10864   / cdiv 11291  ℕcn 11632  2c2 11686  ℝ⁺crp 12383   +_𝑒 cxad 12499  (,)cioo 12732  [,)cico 12734  [,]cicc 12735  ↑cexp 13423  volcvol 24058  Σ^{^}csumge0 42638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-sup 8900  df-inf 8901  df-oi 8968  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-ioo 12736  df-ico 12738  df-icc 12739  df-fz 12887  df-fzo 13028  df-fl 13156  df-seq 13364  df-exp 13424  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-rlim 14840  df-sum 15037  df-rest 16690  df-topgen 16711  df-psmet 20531  df-xmet 20532  df-met 20533  df-bl 20534  df-mopn 20535  df-top 21496  df-topon 21513  df-bases 21548  df-cmp 21989  df-ovol 24059  df-vol 24060  df-sumge0 42639

This theorem is referenced by:  ovolval5lem3  42930