Theorem ovolunlem1 25396

Description: Lemma for ovolun 25398. (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 4143	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ℝ)
6		ovolun.g1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
7		elovolmlem 25373	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8	6, 7	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
9	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
10	9	ffvelcdmda 7018	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑛 / 2) ∈ ℕ) → (𝐺‘(𝑛 / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
11		nneo 12560	. . . . . . . . . . . . 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 25373	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
17	15, 16	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18	17	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
19	18	ffvelcdmda 7018	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ((𝑛 + 1) / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
20	14, 19	syldan 591	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	10, 20	ifclda 4512	. . . . . . . 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 7048	. . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24		eqid 2729	. . . . . . . 8 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
25		ovolun.u	. . . . . . . 8 ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
26	24, 25	ovolsf 25371	. . . . . . 7 ⊢ (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
27	23, 26	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈:ℕ⟶(0[,)+∞))
28		rge0ssre 13359	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
29		fss 6668	. . . . . 6 ⊢ ((𝑈:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑈:ℕ⟶ℝ)
30	27, 28, 29	sylancl 586	. . . . 5 ⊢ (𝜑 → 𝑈:ℕ⟶ℝ)
31	30	frnd 6660	. . . 4 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ)
32		1nn 12139	. . . . . . 7 ⊢ 1 ∈ ℕ
33		1z 12505	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
34		seqfn 13920	. . . . . . . . . 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 6579	. . . . . . . . . 10 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ)
37		nnuz 12778	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
38	37	fneq2i 6580	. . . . . . . . . 10 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ≥‘1))
39	36, 38	bitri 275	. . . . . . . . 9 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ≥‘1))
40	35, 39	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → 𝑈 Fn ℕ)
41	40	fndmd 6587	. . . . . . 7 ⊢ (𝜑 → dom 𝑈 = ℕ)
42	32, 41	eleqtrrid 2835	. . . . . 6 ⊢ (𝜑 → 1 ∈ dom 𝑈)
43	42	ne0d 4293	. . . . 5 ⊢ (𝜑 → dom 𝑈 ≠ ∅)
44		dm0rn0 5867	. . . . . 6 ⊢ (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
45	44	necon3bii 2977	. . . . 5 ⊢ (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
46	43, 45	sylib 218	. . . 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 11144	. . . . . . 7 ⊢ (𝜑 → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
50		ovolun.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
51	50	rpred 12937	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
52	49, 51	readdcld 11144	. . . . . 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 25395	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
60	59	ralrimiva 3121	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
61		breq1 5095	. . . . . . . . 9 ⊢ (𝑧 = (𝑈‘𝑘) → (𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ (𝑈‘𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
62	61	ralrn 7022	. . . . . . . 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 257	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
65		brralrspcev 5152	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘)
67		ressxr 11159	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
68	31, 67	sstrdi 3948	. . . . . 6 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ*)
69		supxrbnd2 13224	. . . . . 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 232	. . . 4 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) < +∞)
72		supxrbnd 13230	. . . 4 ⊢ ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ sup(ran 𝑈, ℝ*, < ) < +∞) → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
73	31, 46, 71, 72	syl3anc 1373	. . 3 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
74		nncn 12136	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
75	74	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
76		1cnd 11110	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℂ)
77	75	2timesd 12367	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) = (𝑚 + 𝑚))
78	77	oveq1d 7364	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = ((𝑚 + 𝑚) − 1))
79	75, 75, 76, 78	assraddsubd 11534	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
80		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
81		nnm1nn0 12425	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ0)
82		nnnn0addcl 12414	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 − 1) ∈ ℕ0) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
83	80, 81, 82	syl2anc2 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
84	79, 83	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) ∈ ℕ)
85		oveq1 7356	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 / 2) = (((2 · 𝑚) − 1) / 2))
86	85	eleq1d 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 / 2) ∈ ℕ ↔ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
87	85	fveq2d 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐺‘(𝑛 / 2)) = (𝐺‘(((2 · 𝑚) − 1) / 2)))
88		oveq1 7356	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 + 1) = (((2 · 𝑚) − 1) + 1))
89	88	fvoveq1d 7371	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
90	86, 87, 89	ifbieq12d 4505	. . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘(((2 · 𝑚) − 1) / 2)) ∈ V
92		fvex 6835	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) ∈ V
93	91, 92	ifex 4527	. . . . . . . . . . . . . 14 ⊢ if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) ∈ V
94	90, 22, 93	fvmpt 6930	. . . . . . . . . . . . 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 12201	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
97		nnmulcl 12152	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
98	96, 80, 97	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
99	98	nncnd 12144	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℂ)
100		ax-1cn 11067	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
101		npcan 11372	. . . . . . . . . . . . . . . . . 18 ⊢ (((2 · 𝑚) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
102	99, 100, 101	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
103	102	oveq1d 7364	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = ((2 · 𝑚) / 2))
104		2cn 12203	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℂ
105		2ne0 12232	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
106		divcan3 11805	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑚) / 2) = 𝑚)
107	104, 105, 106	mp3an23 1455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℂ → ((2 · 𝑚) / 2) = 𝑚)
108	75, 107	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) = 𝑚)
109	103, 108	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = 𝑚)
110	109, 80	eqeltrd 2828	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ)
111		nneo 12560	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ)
115	114	iffalsed 4487	. . . . . . . . . . . 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 6826	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) = (𝐹‘𝑚))
117	95, 115, 116	3eqtrd 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚))
118		fveqeq2 6831	. . . . . . . . . . . 12 ⊢ (𝑘 = ((2 · 𝑚) − 1) → ((𝐻‘𝑘) = (𝐹‘𝑚) ↔ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)))
119	118	rspcev 3577	. . . . . . . . . . 11 ⊢ ((((2 · 𝑚) − 1) ∈ ℕ ∧ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
120	84, 117, 119	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
121		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (1st ‘(𝐻‘𝑘)) = (1st ‘(𝐹‘𝑚)))
122	121	breq1d 5102	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1st ‘(𝐹‘𝑚)) < 𝑧))
123		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (2nd ‘(𝐻‘𝑘)) = (2nd ‘(𝐹‘𝑚)))
124	123	breq2d 5104	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (𝑧 < (2nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2nd ‘(𝐹‘𝑚))))
125	122, 124	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘))) ↔ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚)))))
126	125	biimprcd 250	. . . . . . . . . . 11 ⊢ (((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚))) → ((𝐻‘𝑘) = (𝐹‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
127	126	reximdv 3144	. . . . . . . . . 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 3130	. . . . . . . 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 3143	. . . . . . 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 25366	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐹‘𝑚)))))
133		ovolfioo 25366	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
135	130, 132, 134	3imtr4d 294	. . . . . 6 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻)))
136	55, 135	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻))
137		oveq1 7356	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 / 2) = ((2 · 𝑚) / 2))
138	137	eleq1d 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → ((𝑛 / 2) ∈ ℕ ↔ ((2 · 𝑚) / 2) ∈ ℕ))
139	137	fveq2d 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐺‘(𝑛 / 2)) = (𝐺‘((2 · 𝑚) / 2)))
140		oveq1 7356	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 + 1) = ((2 · 𝑚) + 1))
141	140	fvoveq1d 7371	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘(((2 · 𝑚) + 1) / 2)))
142	138, 139, 141	ifbieq12d 4505	. . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘((2 · 𝑚) / 2)) ∈ V
144		fvex 6835	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘(((2 · 𝑚) + 1) / 2)) ∈ V
145	143, 144	ifex 4527	. . . . . . . . . . . . . 14 ⊢ if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) ∈ V
146	142, 22, 145	fvmpt 6930	. . . . . . . . . . . . 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 2828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) ∈ ℕ)
149	148	iftrued 4484	. . . . . . . . . . . 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 6826	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘((2 · 𝑚) / 2)) = (𝐺‘𝑚))
151	147, 149, 150	3eqtrd 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚))
152		fveqeq2 6831	. . . . . . . . . . . 12 ⊢ (𝑘 = (2 · 𝑚) → ((𝐻‘𝑘) = (𝐺‘𝑚) ↔ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)))
153	152	rspcev 3577	. . . . . . . . . . 11 ⊢ (((2 · 𝑚) ∈ ℕ ∧ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
154	98, 151, 153	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
155		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (1st ‘(𝐻‘𝑘)) = (1st ‘(𝐺‘𝑚)))
156	155	breq1d 5102	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1st ‘(𝐺‘𝑚)) < 𝑧))
157		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (2nd ‘(𝐻‘𝑘)) = (2nd ‘(𝐺‘𝑚)))
158	157	breq2d 5104	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (𝑧 < (2nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2nd ‘(𝐺‘𝑚))))
159	156, 158	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘))) ↔ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
160	159	biimprcd 250	. . . . . . . . . . 11 ⊢ (((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))) → ((𝐻‘𝑘) = (𝐺‘𝑚) → ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
161	160	reximdv 3144	. . . . . . . . . 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 3130	. . . . . . . 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 3143	. . . . . . 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 25366	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚)))))
167		ovolfioo 25366	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑘)))))
169	164, 166, 168	3imtr4d 294	. . . . . 6 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻)))
170	57, 169	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻))
171	136, 170	unssd 4143	. . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻))
172	25	ovollb 25378	. . . 4 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
173	23, 171, 172	syl2anc 584	. . 3 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
174		ovollecl 25382	. . 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 1373	. 2 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
176	52	rexrd 11165	. . . 4 ⊢ (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*)
177		supxrleub 13228	. . . 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 584	. . 3 ⊢ (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
179	64, 178	mpbird 257	. 2 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
180	175, 73, 52, 173, 179	letrd 11273	1 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  ifcif 4476  ∪ cuni 4858   class class class wbr 5092   ↦ cmpt 5173   × cxp 5617  dom cdm 5619  ran crn 5620   ∘ ccom 5623   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923   ↑_m cmap 8753  supcsup 9330  ℂcc 11007  ℝcr 11008  0cc0 11009  1c1 11010   + caddc 11012   · cmul 11014  +∞cpnf 11146  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150   − cmin 11347   / cdiv 11777  ℕcn 12128  2c2 12183  ℕ₀cn0 12384  ℤcz 12471  ℤ_≥cuz 12735  ℝ⁺crp 12893  (,)cioo 13248  [,)cico 13250  seqcseq 13908  abscabs 15141  vol*covol 25361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-ioo 13252  df-ico 13254  df-fz 13411  df-fl 13696  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-ovol 25363

This theorem is referenced by:  ovolunlem2  25397