Theorem ovolunlem1 24566

Description: Lemma for ovolun 24568. (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 494	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		ovolun.b	. . . . 5 ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
4	3	simpld 494	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
5	2, 4	unssd 4116	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ℝ)
6		ovolun.g1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
7		elovolmlem 24543	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8	6, 7	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
9	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
10	9	ffvelrnda 6943	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑛 / 2) ∈ ℕ) → (𝐺‘(𝑛 / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
11		nneo 12334	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
12	11	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
13	12	con2bid 354	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑛 + 1) / 2) ∈ ℕ ↔ ¬ (𝑛 / 2) ∈ ℕ))
14	13	biimpar 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → ((𝑛 + 1) / 2) ∈ ℕ)
15		ovolun.f1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
16		elovolmlem 24543	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
17	15, 16	sylib 217	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18	17	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19	18	ffvelrnda 6943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ((𝑛 + 1) / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
20	14, 19	syldan 590	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	10, 20	ifclda 4491	. . . . . . . 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 6970	. . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24		eqid 2738	. . . . . . . 8 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
25		ovolun.u	. . . . . . . 8 ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
26	24, 25	ovolsf 24541	. . . . . . 7 ⊢ (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
27	23, 26	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈:ℕ⟶(0[,)+∞))
28		rge0ssre 13117	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
29		fss 6601	. . . . . 6 ⊢ ((𝑈:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑈:ℕ⟶ℝ)
30	27, 28, 29	sylancl 585	. . . . 5 ⊢ (𝜑 → 𝑈:ℕ⟶ℝ)
31	30	frnd 6592	. . . 4 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ)
32		1nn 11914	. . . . . . 7 ⊢ 1 ∈ ℕ
33		1z 12280	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
34		seqfn 13661	. . . . . . . . . 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 6514	. . . . . . . . . 10 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ)
37		nnuz 12550	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
38	37	fneq2i 6515	. . . . . . . . . 10 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ≥‘1))
39	36, 38	bitri 274	. . . . . . . . 9 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ≥‘1))
40	35, 39	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → 𝑈 Fn ℕ)
41	40	fndmd 6522	. . . . . . 7 ⊢ (𝜑 → dom 𝑈 = ℕ)
42	32, 41	eleqtrrid 2846	. . . . . 6 ⊢ (𝜑 → 1 ∈ dom 𝑈)
43	42	ne0d 4266	. . . . 5 ⊢ (𝜑 → dom 𝑈 ≠ ∅)
44		dm0rn0 5823	. . . . . 6 ⊢ (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
45	44	necon3bii 2995	. . . . 5 ⊢ (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
46	43, 45	sylib 217	. . . 4 ⊢ (𝜑 → ran 𝑈 ≠ ∅)
47	1	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
48	3	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ)
49	47, 48	readdcld 10935	. . . . . . 7 ⊢ (𝜑 → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
50		ovolun.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
51	50	rpred 12701	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
52	49, 51	readdcld 10935	. . . . . 6 ⊢ (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ)
53		ovolun.s	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
54		ovolun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
55		ovolun.f2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))
56		ovolun.f3	. . . . . . . . 9 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
57		ovolun.g2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺))
58		ovolun.g3	. . . . . . . . 9 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
59	1, 3, 50, 53, 54, 25, 15, 55, 56, 6, 57, 58, 22	ovolunlem1a 24565	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
60	59	ralrimiva 3107	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
61		breq1 5073	. . . . . . . . 9 ⊢ (𝑧 = (𝑈‘𝑘) → (𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
62	61	ralrn 6946	. . . . . . . 8 ⊢ (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
63	40, 62	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
64	60, 63	mpbird 256	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
65		brralrspcev 5130	. . . . . 6 ⊢ (((((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ ∧ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)) → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘)
66	52, 64, 65	syl2anc 583	. . . . 5 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘)
67		ressxr 10950	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
68	31, 67	sstrdi 3929	. . . . . 6 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ*)
69		supxrbnd2 12985	. . . . . 6 ⊢ (ran 𝑈 ⊆ ℝ* → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
70	68, 69	syl 17	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
71	66, 70	mpbid 231	. . . 4 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) < +∞)
72		supxrbnd 12991	. . . 4 ⊢ ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ sup(ran 𝑈, ℝ*, < ) < +∞) → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
73	31, 46, 71, 72	syl3anc 1369	. . 3 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
74		nncn 11911	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
75	74	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
76		1cnd 10901	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℂ)
77	75	2timesd 12146	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) = (𝑚 + 𝑚))
78	77	oveq1d 7270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = ((𝑚 + 𝑚) − 1))
79	75, 75, 76, 78	assraddsubd 11319	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
80		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
81		nnm1nn0 12204	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ0)
82		nnnn0addcl 12193	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 − 1) ∈ ℕ0) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
83	80, 81, 82	syl2anc2 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
84	79, 83	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) ∈ ℕ)
85		oveq1 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 / 2) = (((2 · 𝑚) − 1) / 2))
86	85	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 / 2) ∈ ℕ ↔ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
87	85	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐺‘(𝑛 / 2)) = (𝐺‘(((2 · 𝑚) − 1) / 2)))
88		oveq1 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 + 1) = (((2 · 𝑚) − 1) + 1))
89	88	fvoveq1d 7277	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
90	86, 87, 89	ifbieq12d 4484	. . . . . . . . . . . . . 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))))
91		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘(((2 · 𝑚) − 1) / 2)) ∈ V
92		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) ∈ V
93	91, 92	ifex 4506	. . . . . . . . . . . . . 14 ⊢ if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) ∈ V
94	90, 22, 93	fvmpt 6857	. . . . . . . . . . . . 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))))
95	84, 94	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))))
96		2nn 11976	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
97		nnmulcl 11927	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
98	96, 80, 97	sylancr 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
99	98	nncnd 11919	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℂ)
100		ax-1cn 10860	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
101		npcan 11160	. . . . . . . . . . . . . . . . . 18 ⊢ (((2 · 𝑚) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
102	99, 100, 101	sylancl 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
103	102	oveq1d 7270	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = ((2 · 𝑚) / 2))
104		2cn 11978	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℂ
105		2ne0 12007	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
106		divcan3 11589	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑚) / 2) = 𝑚)
107	104, 105, 106	mp3an23 1451	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℂ → ((2 · 𝑚) / 2) = 𝑚)
108	75, 107	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) = 𝑚)
109	103, 108	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = 𝑚)
110	109, 80	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ)
111		nneo 12334	. . . . . . . . . . . . . . . 16 ⊢ (((2 · 𝑚) − 1) ∈ ℕ → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
112	84, 111	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
113	112	con2bid 354	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ ↔ ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
114	110, 113	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ)
115	114	iffalsed 4467	. . . . . . . . . . . 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)))
116	109	fveq2d 6760	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) = (𝐹‘𝑚))
117	95, 115, 116	3eqtrd 2782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚))
118		fveqeq2 6765	. . . . . . . . . . . 12 ⊢ (𝑘 = ((2 · 𝑚) − 1) → ((𝐻‘𝑘) = (𝐹‘𝑚) ↔ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)))
119	118	rspcev 3552	. . . . . . . . . . 11 ⊢ ((((2 · 𝑚) − 1) ∈ ℕ ∧ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
120	84, 117, 119	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
121		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (1st ‘(𝐻‘𝑘)) = (1st ‘(𝐹‘𝑚)))
122	121	breq1d 5080	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1st ‘(𝐹‘𝑚)) < 𝑧))
123		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (2nd ‘(𝐻‘𝑘)) = (2nd ‘(𝐹‘𝑚)))
124	123	breq2d 5082	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (𝑧 < (2nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2nd ‘(𝐹‘𝑚))))
125	122, 124	anbi12d 630	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘))) ↔ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚)))))
126	125	biimprcd 249	. . . . . . . . . . 11 ⊢ (((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚))) → ((𝐻‘𝑘) = (𝐹‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
127	126	reximdv 3201	. . . . . . . . . 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)))))
128	120, 127	syl5com 31	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
129	128	rexlimdva 3212	. . . . . . . 8 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
130	129	ralimdv 3103	. . . . . . 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)))))
131		ovolfioo 24536	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚)))))
132	2, 17, 131	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚)))))
133		ovolfioo 24536	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
134	2, 23, 133	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
135	130, 132, 134	3imtr4d 293	. . . . . 6 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻)))
136	55, 135	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻))
137		oveq1 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 / 2) = ((2 · 𝑚) / 2))
138	137	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → ((𝑛 / 2) ∈ ℕ ↔ ((2 · 𝑚) / 2) ∈ ℕ))
139	137	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐺‘(𝑛 / 2)) = (𝐺‘((2 · 𝑚) / 2)))
140		oveq1 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 + 1) = ((2 · 𝑚) + 1))
141	140	fvoveq1d 7277	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘(((2 · 𝑚) + 1) / 2)))
142	138, 139, 141	ifbieq12d 4484	. . . . . . . . . . . . . 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))))
143		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘((2 · 𝑚) / 2)) ∈ V
144		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘(((2 · 𝑚) + 1) / 2)) ∈ V
145	143, 144	ifex 4506	. . . . . . . . . . . . . 14 ⊢ if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) ∈ V
146	142, 22, 145	fvmpt 6857	. . . . . . . . . . . . 13 ⊢ ((2 · 𝑚) ∈ ℕ → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
147	98, 146	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
148	108, 80	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) ∈ ℕ)
149	148	iftrued 4464	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) = (𝐺‘((2 · 𝑚) / 2)))
150	108	fveq2d 6760	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘((2 · 𝑚) / 2)) = (𝐺‘𝑚))
151	147, 149, 150	3eqtrd 2782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚))
152		fveqeq2 6765	. . . . . . . . . . . 12 ⊢ (𝑘 = (2 · 𝑚) → ((𝐻‘𝑘) = (𝐺‘𝑚) ↔ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)))
153	152	rspcev 3552	. . . . . . . . . . 11 ⊢ (((2 · 𝑚) ∈ ℕ ∧ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
154	98, 151, 153	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
155		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (1st ‘(𝐻‘𝑘)) = (1st ‘(𝐺‘𝑚)))
156	155	breq1d 5080	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1st ‘(𝐺‘𝑚)) < 𝑧))
157		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (2nd ‘(𝐻‘𝑘)) = (2nd ‘(𝐺‘𝑚)))
158	157	breq2d 5082	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (𝑧 < (2nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2nd ‘(𝐺‘𝑚))))
159	156, 158	anbi12d 630	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘))) ↔ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
160	159	biimprcd 249	. . . . . . . . . . 11 ⊢ (((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))) → ((𝐻‘𝑘) = (𝐺‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
161	160	reximdv 3201	. . . . . . . . . 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)))))
162	154, 161	syl5com 31	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
163	162	rexlimdva 3212	. . . . . . . 8 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
164	163	ralimdv 3103	. . . . . . 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)))))
165		ovolfioo 24536	. . . . . . . 8 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
166	4, 8, 165	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
167		ovolfioo 24536	. . . . . . . 8 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
168	4, 23, 167	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
169	164, 166, 168	3imtr4d 293	. . . . . 6 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻)))
170	57, 169	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻))
171	136, 170	unssd 4116	. . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻))
172	25	ovollb 24548	. . . 4 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
173	23, 171, 172	syl2anc 583	. . 3 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
174		ovollecl 24552	. . 3 ⊢ (((𝐴 ∪ 𝐵) ⊆ ℝ ∧ sup(ran 𝑈, ℝ*, < ) ∈ ℝ ∧ (vol*‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ*, < )) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
175	5, 73, 173, 174	syl3anc 1369	. 2 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
176	52	rexrd 10956	. . . 4 ⊢ (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*)
177		supxrleub 12989	. . . 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)))
178	68, 176, 177	syl2anc 583	. . 3 ⊢ (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
179	64, 178	mpbird 256	. 2 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
180	175, 73, 52, 173, 179	letrd 11062	1 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ifcif 4456  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  dom cdm 5580  ran crn 5581   ∘ ccom 5584   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803   ↑_m cmap 8573  supcsup 9129  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   · cmul 10807  +∞cpnf 10937  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   − cmin 11135   / cdiv 11562  ℕcn 11903  2c2 11958  ℕ₀cn0 12163  ℤcz 12249  ℤ_≥cuz 12511  ℝ⁺crp 12659  (,)cioo 13008  [,)cico 13010  seqcseq 13649  abscabs 14873  vol*covol 24531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-ioo 13012  df-ico 13014  df-fz 13169  df-fl 13440  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-ovol 24533

This theorem is referenced by:  ovolunlem2  24567