Theorem ovolunlem1 24092

Description: Lemma for ovolun 24094. (Contributed by Mario Carneiro, 12-Jun-2014.)

Hypotheses

Ref	Expression
ovolun.a	⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
ovolun.b	⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
ovolun.c	⊢ (𝜑 → 𝐶 ∈ ℝ+)
ovolun.s	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolun.t	⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ovolun.u	⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ovolun.f1	⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
ovolun.f2	⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))
ovolun.f3	⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
ovolun.g1	⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
ovolun.g2	⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺))
ovolun.g3	⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
ovolun.h	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))

Assertion

Ref	Expression
ovolunlem1	⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))

Distinct variable groups:   𝐶,𝑛   𝑛,𝐹   𝐴,𝑛   𝐵,𝑛   𝑛,𝐺   𝜑,𝑛

Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑈(𝑛)   𝐻(𝑛)

Proof of Theorem ovolunlem1

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

Step	Hyp	Ref	Expression
1		ovolun.a	. . . . 5 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
2	1	simpld 497	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		ovolun.b	. . . . 5 ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
4	3	simpld 497	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
5	2, 4	unssd 4161	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ℝ)
6		ovolun.g1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
7		elovolmlem 24069	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8	6, 7	sylib 220	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
9	8	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
10	9	ffvelrnda 6845	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑛 / 2) ∈ ℕ) → (𝐺‘(𝑛 / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
11		nneo 12060	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
12	11	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
13	12	con2bid 357	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑛 + 1) / 2) ∈ ℕ ↔ ¬ (𝑛 / 2) ∈ ℕ))
14	13	biimpar 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → ((𝑛 + 1) / 2) ∈ ℕ)
15		ovolun.f1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
16		elovolmlem 24069	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
17	15, 16	sylib 220	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18	17	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19	18	ffvelrnda 6845	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ((𝑛 + 1) / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
20	14, 19	syldan 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	10, 20	ifclda 4500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) ∈ ( ≤ ∩ (ℝ × ℝ)))
22		ovolun.h	. . . . . . . 8 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))
23	21, 22	fmptd 6872	. . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24		eqid 2821	. . . . . . . 8 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
25		ovolun.u	. . . . . . . 8 ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
26	24, 25	ovolsf 24067	. . . . . . 7 ⊢ (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
27	23, 26	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈:ℕ⟶(0[,)+∞))
28		rge0ssre 12838	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
29		fss 6521	. . . . . 6 ⊢ ((𝑈:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑈:ℕ⟶ℝ)
30	27, 28, 29	sylancl 588	. . . . 5 ⊢ (𝜑 → 𝑈:ℕ⟶ℝ)
31	30	frnd 6515	. . . 4 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ)
32		1nn 11643	. . . . . . 7 ⊢ 1 ∈ ℕ
33		1z 12006	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
34		seqfn 13375	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ≥‘1))
35	33, 34	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ≥‘1))
36	25	fneq1i 6444	. . . . . . . . . 10 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ)
37		nnuz 12275	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
38	37	fneq2i 6445	. . . . . . . . . 10 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ≥‘1))
39	36, 38	bitri 277	. . . . . . . . 9 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ≥‘1))
40	35, 39	sylibr 236	. . . . . . . 8 ⊢ (𝜑 → 𝑈 Fn ℕ)
41		fndm 6449	. . . . . . . 8 ⊢ (𝑈 Fn ℕ → dom 𝑈 = ℕ)
42	40, 41	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑈 = ℕ)
43	32, 42	eleqtrrid 2920	. . . . . 6 ⊢ (𝜑 → 1 ∈ dom 𝑈)
44	43	ne0d 4300	. . . . 5 ⊢ (𝜑 → dom 𝑈 ≠ ∅)
45		dm0rn0 5789	. . . . . 6 ⊢ (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
46	45	necon3bii 3068	. . . . 5 ⊢ (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
47	44, 46	sylib 220	. . . 4 ⊢ (𝜑 → ran 𝑈 ≠ ∅)
48	1	simprd 498	. . . . . . . 8 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
49	3	simprd 498	. . . . . . . 8 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ)
50	48, 49	readdcld 10664	. . . . . . 7 ⊢ (𝜑 → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
51		ovolun.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
52	51	rpred 12425	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
53	50, 52	readdcld 10664	. . . . . 6 ⊢ (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ)
54		ovolun.s	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
55		ovolun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
56		ovolun.f2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))
57		ovolun.f3	. . . . . . . . 9 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
58		ovolun.g2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺))
59		ovolun.g3	. . . . . . . . 9 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
60	1, 3, 51, 54, 55, 25, 15, 56, 57, 6, 58, 59, 22	ovolunlem1a 24091	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
61	60	ralrimiva 3182	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
62		breq1 5061	. . . . . . . . 9 ⊢ (𝑧 = (𝑈‘𝑘) → (𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
63	62	ralrn 6848	. . . . . . . 8 ⊢ (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
64	40, 63	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
65	61, 64	mpbird 259	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
66		brralrspcev 5118	. . . . . 6 ⊢ (((((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ ∧ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)) → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘)
67	53, 65, 66	syl2anc 586	. . . . 5 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘)
68		ressxr 10679	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
69	31, 68	sstrdi 3978	. . . . . 6 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ*)
70		supxrbnd2 12709	. . . . . 6 ⊢ (ran 𝑈 ⊆ ℝ* → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
71	69, 70	syl 17	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
72	67, 71	mpbid 234	. . . 4 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) < +∞)
73		supxrbnd 12715	. . . 4 ⊢ ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ sup(ran 𝑈, ℝ*, < ) < +∞) → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
74	31, 47, 72, 73	syl3anc 1367	. . 3 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
75		nncn 11640	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
76	75	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
77		1cnd 10630	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℂ)
78	76	2timesd 11874	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) = (𝑚 + 𝑚))
79	78	oveq1d 7165	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = ((𝑚 + 𝑚) − 1))
80	76, 76, 77, 79	assraddsubd 11048	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
81		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
82		nnm1nn0 11932	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ0)
83		nnnn0addcl 11921	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 − 1) ∈ ℕ0) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
84	81, 82, 83	syl2anc2 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
85	80, 84	eqeltrd 2913	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) ∈ ℕ)
86		oveq1 7157	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 / 2) = (((2 · 𝑚) − 1) / 2))
87	86	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 / 2) ∈ ℕ ↔ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
88	86	fveq2d 6668	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐺‘(𝑛 / 2)) = (𝐺‘(((2 · 𝑚) − 1) / 2)))
89		oveq1 7157	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 + 1) = (((2 · 𝑚) − 1) + 1))
90	89	fvoveq1d 7172	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
91	87, 88, 90	ifbieq12d 4493	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ((2 · 𝑚) − 1) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) = if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))))
92		fvex 6677	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘(((2 · 𝑚) − 1) / 2)) ∈ V
93		fvex 6677	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) ∈ V
94	92, 93	ifex 4514	. . . . . . . . . . . . . 14 ⊢ if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) ∈ V
95	91, 22, 94	fvmpt 6762	. . . . . . . . . . . . 13 ⊢ (((2 · 𝑚) − 1) ∈ ℕ → (𝐻‘((2 · 𝑚) − 1)) = if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))))
96	85, 95	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))))
97		2nn 11704	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
98		nnmulcl 11655	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
99	97, 81, 98	sylancr 589	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
100	99	nncnd 11648	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℂ)
101		ax-1cn 10589	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
102		npcan 10889	. . . . . . . . . . . . . . . . . 18 ⊢ (((2 · 𝑚) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
103	100, 101, 102	sylancl 588	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
104	103	oveq1d 7165	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = ((2 · 𝑚) / 2))
105		2cn 11706	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℂ
106		2ne0 11735	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
107		divcan3 11318	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑚) / 2) = 𝑚)
108	105, 106, 107	mp3an23 1449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℂ → ((2 · 𝑚) / 2) = 𝑚)
109	76, 108	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) = 𝑚)
110	104, 109	eqtrd 2856	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = 𝑚)
111	110, 81	eqeltrd 2913	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ)
112		nneo 12060	. . . . . . . . . . . . . . . 16 ⊢ (((2 · 𝑚) − 1) ∈ ℕ → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
113	85, 112	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
114	113	con2bid 357	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ ↔ ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
115	111, 114	mpbid 234	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ)
116	115	iffalsed 4477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
117	110	fveq2d 6668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) = (𝐹‘𝑚))
118	96, 116, 117	3eqtrd 2860	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚))
119		fveqeq2 6673	. . . . . . . . . . . 12 ⊢ (𝑘 = ((2 · 𝑚) − 1) → ((𝐻‘𝑘) = (𝐹‘𝑚) ↔ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)))
120	119	rspcev 3622	. . . . . . . . . . 11 ⊢ ((((2 · 𝑚) − 1) ∈ ℕ ∧ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
121	85, 118, 120	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
122		fveq2 6664	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (1st ‘(𝐻‘𝑘)) = (1st ‘(𝐹‘𝑚)))
123	122	breq1d 5068	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1st ‘(𝐹‘𝑚)) < 𝑧))
124		fveq2 6664	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (2nd ‘(𝐻‘𝑘)) = (2nd ‘(𝐹‘𝑚)))
125	124	breq2d 5070	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (𝑧 < (2nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2nd ‘(𝐹‘𝑚))))
126	123, 125	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘))) ↔ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚)))))
127	126	biimprcd 252	. . . . . . . . . . 11 ⊢ (((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚))) → ((𝐻‘𝑘) = (𝐹‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
128	127	reximdv 3273	. . . . . . . . . 10 ⊢ (((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚))) → (∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
129	121, 128	syl5com 31	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
130	129	rexlimdva 3284	. . . . . . . 8 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
131	130	ralimdv 3178	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚))) → ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
132		ovolfioo 24062	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚)))))
133	2, 17, 132	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚)))))
134		ovolfioo 24062	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
135	2, 23, 134	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
136	131, 133, 135	3imtr4d 296	. . . . . 6 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻)))
137	56, 136	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻))
138		oveq1 7157	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 / 2) = ((2 · 𝑚) / 2))
139	138	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → ((𝑛 / 2) ∈ ℕ ↔ ((2 · 𝑚) / 2) ∈ ℕ))
140	138	fveq2d 6668	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐺‘(𝑛 / 2)) = (𝐺‘((2 · 𝑚) / 2)))
141		oveq1 7157	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 + 1) = ((2 · 𝑚) + 1))
142	141	fvoveq1d 7172	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘(((2 · 𝑚) + 1) / 2)))
143	139, 140, 142	ifbieq12d 4493	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (2 · 𝑚) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
144		fvex 6677	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘((2 · 𝑚) / 2)) ∈ V
145		fvex 6677	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘(((2 · 𝑚) + 1) / 2)) ∈ V
146	144, 145	ifex 4514	. . . . . . . . . . . . . 14 ⊢ if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) ∈ V
147	143, 22, 146	fvmpt 6762	. . . . . . . . . . . . 13 ⊢ ((2 · 𝑚) ∈ ℕ → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
148	99, 147	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
149	109, 81	eqeltrd 2913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) ∈ ℕ)
150	149	iftrued 4474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) = (𝐺‘((2 · 𝑚) / 2)))
151	109	fveq2d 6668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘((2 · 𝑚) / 2)) = (𝐺‘𝑚))
152	148, 150, 151	3eqtrd 2860	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚))
153		fveqeq2 6673	. . . . . . . . . . . 12 ⊢ (𝑘 = (2 · 𝑚) → ((𝐻‘𝑘) = (𝐺‘𝑚) ↔ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)))
154	153	rspcev 3622	. . . . . . . . . . 11 ⊢ (((2 · 𝑚) ∈ ℕ ∧ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
155	99, 152, 154	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
156		fveq2 6664	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (1st ‘(𝐻‘𝑘)) = (1st ‘(𝐺‘𝑚)))
157	156	breq1d 5068	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1st ‘(𝐺‘𝑚)) < 𝑧))
158		fveq2 6664	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (2nd ‘(𝐻‘𝑘)) = (2nd ‘(𝐺‘𝑚)))
159	158	breq2d 5070	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (𝑧 < (2nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2nd ‘(𝐺‘𝑚))))
160	157, 159	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘))) ↔ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
161	160	biimprcd 252	. . . . . . . . . . 11 ⊢ (((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))) → ((𝐻‘𝑘) = (𝐺‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
162	161	reximdv 3273	. . . . . . . . . 10 ⊢ (((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))) → (∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
163	155, 162	syl5com 31	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
164	163	rexlimdva 3284	. . . . . . . 8 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
165	164	ralimdv 3178	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))) → ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
166		ovolfioo 24062	. . . . . . . 8 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
167	4, 8, 166	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
168		ovolfioo 24062	. . . . . . . 8 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
169	4, 23, 168	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
170	165, 167, 169	3imtr4d 296	. . . . . 6 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻)))
171	58, 170	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻))
172	137, 171	unssd 4161	. . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻))
173	25	ovollb 24074	. . . 4 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
174	23, 172, 173	syl2anc 586	. . 3 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
175		ovollecl 24078	. . 3 ⊢ (((𝐴 ∪ 𝐵) ⊆ ℝ ∧ sup(ran 𝑈, ℝ*, < ) ∈ ℝ ∧ (vol*‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ*, < )) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
176	5, 74, 174, 175	syl3anc 1367	. 2 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
177	53	rexrd 10685	. . . 4 ⊢ (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*)
178		supxrleub 12713	. . . 4 ⊢ ((ran 𝑈 ⊆ ℝ* ∧ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*) → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
179	69, 177, 178	syl2anc 586	. . 3 ⊢ (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
180	65, 179	mpbird 259	. 2 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
181	176, 74, 53, 174, 180	letrd 10791	1 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  ifcif 4466  ∪ cuni 4831   class class class wbr 5058   ↦ cmpt 5138   × cxp 5547  dom cdm 5549  ran crn 5550   ∘ ccom 5553   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  1^st c1st 7681  2^nd c2nd 7682   ↑_m cmap 8400  supcsup 8898  ℂcc 10529  ℝcr 10530  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536  +∞cpnf 10666  ℝ^*cxr 10668   < clt 10669   ≤ cle 10670   − cmin 10864   / cdiv 11291  ℕcn 11632  2c2 11686  ℕ₀cn0 11891  ℤcz 11975  ℤ_≥cuz 12237  ℝ⁺crp 12383  (,)cioo 12732  [,)cico 12734  seqcseq 13363  abscabs 14587  vol*covol 24057

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-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  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-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-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-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-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-inf 8901  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-rp 12384  df-ioo 12736  df-ico 12738  df-fz 12887  df-fl 13156  df-seq 13364  df-exp 13424  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-ovol 24059

This theorem is referenced by:  ovolunlem2  24093