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Theorem ovolunlem1 23265

Description: Lemma for ovolun 23267. (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 475	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		ovolun.b	. . . . 5 ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
4	3	simpld 475	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
5	2, 4	unssd 3789	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ℝ)
6		ovolun.g1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ))
7		reex 10027	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
8	7, 7	xpex 6962	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ∈ V
9	8	inex2 4800	. . . . . . . . . . . . 13 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
10		nnex 11026	. . . . . . . . . . . . 13 ⊢ ℕ ∈ V
11	9, 10	elmap 7886	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
12	6, 11	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
13	12	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
14	13	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑛 / 2) ∈ ℕ) → (𝐺‘(𝑛 / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
15		nneo 11461	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
16	15	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
17	16	con2bid 344	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑛 + 1) / 2) ∈ ℕ ↔ ¬ (𝑛 / 2) ∈ ℕ))
18	17	biimpar 502	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → ((𝑛 + 1) / 2) ∈ ℕ)
19		ovolun.f1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ))
20	9, 10	elmap 7886	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
21	19, 20	sylib 208	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22	21	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
23	22	ffvelrnda 6359	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ((𝑛 + 1) / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
24	18, 23	syldan 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
25	14, 24	ifclda 4120	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) ∈ ( ≤ ∩ (ℝ × ℝ)))
26		ovolun.h	. . . . . . . 8 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))
27	25, 26	fmptd 6385	. . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28		eqid 2622	. . . . . . . 8 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
29		ovolun.u	. . . . . . . 8 ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
30	28, 29	ovolsf 23241	. . . . . . 7 ⊢ (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
31	27, 30	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈:ℕ⟶(0[,)+∞))
32		rge0ssre 12280	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
33		fss 6056	. . . . . 6 ⊢ ((𝑈:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑈:ℕ⟶ℝ)
34	31, 32, 33	sylancl 694	. . . . 5 ⊢ (𝜑 → 𝑈:ℕ⟶ℝ)
35		frn 6053	. . . . 5 ⊢ (𝑈:ℕ⟶ℝ → ran 𝑈 ⊆ ℝ)
36	34, 35	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ)
37		1nn 11031	. . . . . . 7 ⊢ 1 ∈ ℕ
38		1z 11407	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
39		seqfn 12813	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ_≥‘1))
40	38, 39	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ_≥‘1))
41	29	fneq1i 5985	. . . . . . . . . 10 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ)
42		nnuz 11723	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
43	42	fneq2i 5986	. . . . . . . . . 10 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ_≥‘1))
44	41, 43	bitri 264	. . . . . . . . 9 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ_≥‘1))
45	40, 44	sylibr 224	. . . . . . . 8 ⊢ (𝜑 → 𝑈 Fn ℕ)
46		fndm 5990	. . . . . . . 8 ⊢ (𝑈 Fn ℕ → dom 𝑈 = ℕ)
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑈 = ℕ)
48	37, 47	syl5eleqr 2708	. . . . . 6 ⊢ (𝜑 → 1 ∈ dom 𝑈)
49		ne0i 3921	. . . . . 6 ⊢ (1 ∈ dom 𝑈 → dom 𝑈 ≠ ∅)
50	48, 49	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝑈 ≠ ∅)
51		dm0rn0 5342	. . . . . 6 ⊢ (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
52	51	necon3bii 2846	. . . . 5 ⊢ (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
53	50, 52	sylib 208	. . . 4 ⊢ (𝜑 → ran 𝑈 ≠ ∅)
54	1	simprd 479	. . . . . . . 8 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
55	3	simprd 479	. . . . . . . 8 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ)
56	54, 55	readdcld 10069	. . . . . . 7 ⊢ (𝜑 → ((vol‘𝐴) + (vol‘𝐵)) ∈ ℝ)
57		ovolun.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
58	57	rpred 11872	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
59	56, 58	readdcld 10069	. . . . . 6 ⊢ (𝜑 → (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ∈ ℝ)
60		ovolun.s	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
61		ovolun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
62		ovolun.f2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))
63		ovolun.f3	. . . . . . . . 9 ⊢ (𝜑 → sup(ran 𝑆, ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2)))
64		ovolun.g2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺))
65		ovolun.g3	. . . . . . . . 9 ⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2)))
66	1, 3, 57, 60, 61, 29, 19, 62, 63, 6, 64, 65, 26	ovolunlem1a 23264	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑈‘𝑘) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶))
67	66	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶))
68		breq1 4656	. . . . . . . . 9 ⊢ (𝑧 = (𝑈‘𝑘) → (𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ↔ (𝑈‘𝑘) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶)))
69	68	ralrn 6362	. . . . . . . 8 ⊢ (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶)))
70	45, 69	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶)))
71	67, 70	mpbird 247	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶))
72		breq2 4657	. . . . . . . 8 ⊢ (𝑘 = (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) → (𝑧 ≤ 𝑘 ↔ 𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶)))
73	72	ralbidv 2986	. . . . . . 7 ⊢ (𝑘 = (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘 ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶)))
74	73	rspcev 3309	. . . . . 6 ⊢ (((((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ∈ ℝ ∧ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶)) → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘)
75	59, 71, 74	syl2anc 693	. . . . 5 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘)
76		ressxr 10083	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
77	36, 76	syl6ss 3615	. . . . . 6 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ^*)
78		supxrbnd2 12152	. . . . . 6 ⊢ (ran 𝑈 ⊆ ℝ^* → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘 ↔ sup(ran 𝑈, ℝ^*, < ) < +∞))
79	77, 78	syl 17	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘 ↔ sup(ran 𝑈, ℝ^*, < ) < +∞))
80	75, 79	mpbid 222	. . . 4 ⊢ (𝜑 → sup(ran 𝑈, ℝ^*, < ) < +∞)
81		supxrbnd 12158	. . . 4 ⊢ ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ sup(ran 𝑈, ℝ^, < ) < +∞) → sup(ran 𝑈, ℝ^, < ) ∈ ℝ)
82	36, 53, 80, 81	syl3anc 1326	. . 3 ⊢ (𝜑 → sup(ran 𝑈, ℝ^*, < ) ∈ ℝ)
83		nncn 11028	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
84	83	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
85	84	2timesd 11275	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) = (𝑚 + 𝑚))
86	85	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = ((𝑚 + 𝑚) − 1))
87		1cnd 10056	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℂ)
88	84, 84, 87	addsubassd 10412	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
89	86, 88	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
90		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
91		nnm1nn0 11334	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ₀)
92	91	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 − 1) ∈ ℕ₀)
93		nnnn0addcl 11323	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 − 1) ∈ ℕ₀) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
94	90, 92, 93	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
95	89, 94	eqeltrd 2701	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) ∈ ℕ)
96		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 / 2) = (((2 · 𝑚) − 1) / 2))
97	96	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 / 2) ∈ ℕ ↔ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
98	96	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐺‘(𝑛 / 2)) = (𝐺‘(((2 · 𝑚) − 1) / 2)))
99		oveq1 6657	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 + 1) = (((2 · 𝑚) − 1) + 1))
100	99	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 + 1) / 2) = ((((2 · 𝑚) − 1) + 1) / 2))
101	100	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
102	97, 98, 101	ifbieq12d 4113	. . . . . . . . . . . . . 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))))
103		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘(((2 · 𝑚) − 1) / 2)) ∈ V
104		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) ∈ V
105	103, 104	ifex 4156	. . . . . . . . . . . . . 14 ⊢ if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) ∈ V
106	102, 26, 105	fvmpt 6282	. . . . . . . . . . . . 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))))
107	95, 106	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))))
108		2nn 11185	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
109		nnmulcl 11043	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
110	108, 90, 109	sylancr 695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
111	110	nncnd 11036	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℂ)
112		ax-1cn 9994	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
113		npcan 10290	. . . . . . . . . . . . . . . . . 18 ⊢ (((2 · 𝑚) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
114	111, 112, 113	sylancl 694	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
115	114	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = ((2 · 𝑚) / 2))
116		2cn 11091	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℂ
117		2ne0 11113	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
118		divcan3 10711	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑚) / 2) = 𝑚)
119	116, 117, 118	mp3an23 1416	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℂ → ((2 · 𝑚) / 2) = 𝑚)
120	84, 119	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) = 𝑚)
121	115, 120	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = 𝑚)
122	121, 90	eqeltrd 2701	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ)
123		nneo 11461	. . . . . . . . . . . . . . . 16 ⊢ (((2 · 𝑚) − 1) ∈ ℕ → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
124	95, 123	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
125	124	con2bid 344	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ ↔ ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
126	122, 125	mpbid 222	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ)
127	126	iffalsed 4097	. . . . . . . . . . . 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)))
128	121	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) = (𝐹‘𝑚))
129	107, 127, 128	3eqtrd 2660	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚))
130		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((2 · 𝑚) − 1) → (𝐻‘𝑘) = (𝐻‘((2 · 𝑚) − 1)))
131	130	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑘 = ((2 · 𝑚) − 1) → ((𝐻‘𝑘) = (𝐹‘𝑚) ↔ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)))
132	131	rspcev 3309	. . . . . . . . . . 11 ⊢ ((((2 · 𝑚) − 1) ∈ ℕ ∧ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
133	95, 129, 132	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
134		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (1^st ‘(𝐻‘𝑘)) = (1^st ‘(𝐹‘𝑚)))
135	134	breq1d 4663	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → ((1^st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1^st ‘(𝐹‘𝑚)) < 𝑧))
136		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (2^nd ‘(𝐻‘𝑘)) = (2^nd ‘(𝐹‘𝑚)))
137	136	breq2d 4665	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (𝑧 < (2^nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2^nd ‘(𝐹‘𝑚))))
138	135, 137	anbi12d 747	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘))) ↔ ((1^st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑚)))))
139	138	biimprcd 240	. . . . . . . . . . 11 ⊢ (((1^st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑚))) → ((𝐻‘𝑘) = (𝐹‘𝑚) → ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
140	139	reximdv 3016	. . . . . . . . . 10 ⊢ (((1^st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑚))) → (∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚) → ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
141	133, 140	syl5com 31	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((1^st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑚))) → ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
142	141	rexlimdva 3031	. . . . . . . 8 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ((1^st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑚))) → ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
143	142	ralimdv 2963	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1^st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑚))) → ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
144		ovolfioo 23236	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1^st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑚)))))
145	2, 21, 144	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1^st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑚)))))
146		ovolfioo 23236	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
147	2, 27, 146	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
148	143, 145, 147	3imtr4d 283	. . . . . 6 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻)))
149	62, 148	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻))
150		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 / 2) = ((2 · 𝑚) / 2))
151	150	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → ((𝑛 / 2) ∈ ℕ ↔ ((2 · 𝑚) / 2) ∈ ℕ))
152	150	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐺‘(𝑛 / 2)) = (𝐺‘((2 · 𝑚) / 2)))
153		oveq1 6657	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 + 1) = ((2 · 𝑚) + 1))
154	153	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → ((𝑛 + 1) / 2) = (((2 · 𝑚) + 1) / 2))
155	154	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘(((2 · 𝑚) + 1) / 2)))
156	151, 152, 155	ifbieq12d 4113	. . . . . . . . . . . . . 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))))
157		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘((2 · 𝑚) / 2)) ∈ V
158		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘(((2 · 𝑚) + 1) / 2)) ∈ V
159	157, 158	ifex 4156	. . . . . . . . . . . . . 14 ⊢ if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) ∈ V
160	156, 26, 159	fvmpt 6282	. . . . . . . . . . . . 13 ⊢ ((2 · 𝑚) ∈ ℕ → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
161	110, 160	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
162	120, 90	eqeltrd 2701	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) ∈ ℕ)
163	162	iftrued 4094	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) = (𝐺‘((2 · 𝑚) / 2)))
164	120	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘((2 · 𝑚) / 2)) = (𝐺‘𝑚))
165	161, 163, 164	3eqtrd 2660	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚))
166		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = (2 · 𝑚) → (𝐻‘𝑘) = (𝐻‘(2 · 𝑚)))
167	166	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑘 = (2 · 𝑚) → ((𝐻‘𝑘) = (𝐺‘𝑚) ↔ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)))
168	167	rspcev 3309	. . . . . . . . . . 11 ⊢ (((2 · 𝑚) ∈ ℕ ∧ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
169	110, 165, 168	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
170		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (1^st ‘(𝐻‘𝑘)) = (1^st ‘(𝐺‘𝑚)))
171	170	breq1d 4663	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → ((1^st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1^st ‘(𝐺‘𝑚)) < 𝑧))
172		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (2^nd ‘(𝐻‘𝑘)) = (2^nd ‘(𝐺‘𝑚)))
173	172	breq2d 4665	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (𝑧 < (2^nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2^nd ‘(𝐺‘𝑚))))
174	171, 173	anbi12d 747	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘))) ↔ ((1^st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐺‘𝑚)))))
175	174	biimprcd 240	. . . . . . . . . . 11 ⊢ (((1^st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐺‘𝑚))) → ((𝐻‘𝑘) = (𝐺‘𝑚) → ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
176	175	reximdv 3016	. . . . . . . . . 10 ⊢ (((1^st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐺‘𝑚))) → (∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚) → ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
177	169, 176	syl5com 31	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((1^st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐺‘𝑚))) → ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
178	177	rexlimdva 3031	. . . . . . . 8 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ((1^st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐺‘𝑚))) → ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
179	178	ralimdv 2963	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1^st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐺‘𝑚))) → ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
180		ovolfioo 23236	. . . . . . . 8 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1^st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐺‘𝑚)))))
181	4, 12, 180	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1^st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐺‘𝑚)))))
182		ovolfioo 23236	. . . . . . . 8 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
183	4, 27, 182	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
184	179, 181, 183	3imtr4d 283	. . . . . 6 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻)))
185	64, 184	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻))
186	149, 185	unssd 3789	. . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻))
187	29	ovollb 23247	. . . 4 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻)) → (vol‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ^, < ))
188	27, 186, 187	syl2anc 693	. . 3 ⊢ (𝜑 → (vol‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ^, < ))
189		ovollecl 23251	. . 3 ⊢ (((𝐴 ∪ 𝐵) ⊆ ℝ ∧ sup(ran 𝑈, ℝ^, < ) ∈ ℝ ∧ (vol‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ^, < )) → (vol‘(𝐴 ∪ 𝐵)) ∈ ℝ)
190	5, 82, 188, 189	syl3anc 1326	. 2 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
191	59	rexrd 10089	. . . 4 ⊢ (𝜑 → (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ∈ ℝ^*)
192		supxrleub 12156	. . . 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)))
193	77, 191, 192	syl2anc 693	. . 3 ⊢ (𝜑 → (sup(ran 𝑈, ℝ^, < ) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶)))
194	71, 193	mpbird 247	. 2 ⊢ (𝜑 → sup(ran 𝑈, ℝ^, < ) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))
195	190, 82, 59, 188, 194	letrd 10194	1 ⊢ (𝜑 → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ifcif 4086 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ran crn 5115 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 ↑_𝑚 cmap 7857 supcsup 8346 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 (,)cioo 12175 [,)cico 12177 seqcseq 12801 abscabs 13974 vol*covol 23231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ioo 12179 df-ico 12181 df-fz 12327 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-ovol 23233

This theorem is referenced by: ovolunlem2 23266

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