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Theorem ovolunlem1 23072

Description: Lemma for ovolun 23074. (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 474	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		ovolun.b	. . . . 5 ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
4	3	simpld 474	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
5	2, 4	unssd 3751	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ℝ)
6		ovolun.g1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ))
7		reex 9906	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
8	7, 7	xpex 6860	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ∈ V
9	8	inex2 4728	. . . . . . . . . . . . 13 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
10		nnex 10903	. . . . . . . . . . . . 13 ⊢ ℕ ∈ V
11	9, 10	elmap 7772	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
12	6, 11	sylib 207	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
13	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
14	13	ffvelrnda 6267	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑛 / 2) ∈ ℕ) → (𝐺‘(𝑛 / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
15		nneo 11337	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
16	15	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
17	16	con2bid 343	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑛 + 1) / 2) ∈ ℕ ↔ ¬ (𝑛 / 2) ∈ ℕ))
18	17	biimpar 501	. . . . . . . . . 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 7772	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
21	19, 20	sylib 207	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22	21	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
23	22	ffvelrnda 6267	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ((𝑛 + 1) / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
24	18, 23	syldan 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
25	14, 24	ifclda 4070	. . . . . . . 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 6292	. . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28		eqid 2610	. . . . . . . 8 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
29		ovolun.u	. . . . . . . 8 ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
30	28, 29	ovolsf 23048	. . . . . . 7 ⊢ (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
31	27, 30	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈:ℕ⟶(0[,)+∞))
32		rge0ssre 12151	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
33		fss 5969	. . . . . 6 ⊢ ((𝑈:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑈:ℕ⟶ℝ)
34	31, 32, 33	sylancl 693	. . . . 5 ⊢ (𝜑 → 𝑈:ℕ⟶ℝ)
35		frn 5966	. . . . 5 ⊢ (𝑈:ℕ⟶ℝ → ran 𝑈 ⊆ ℝ)
36	34, 35	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ)
37		1nn 10908	. . . . . . 7 ⊢ 1 ∈ ℕ
38		1z 11284	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
39		seqfn 12675	. . . . . . . . . 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 5899	. . . . . . . . . 10 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ)
42		nnuz 11599	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
43	42	fneq2i 5900	. . . . . . . . . 10 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ_≥‘1))
44	41, 43	bitri 263	. . . . . . . . 9 ⊢ (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ_≥‘1))
45	40, 44	sylibr 223	. . . . . . . 8 ⊢ (𝜑 → 𝑈 Fn ℕ)
46		fndm 5904	. . . . . . . 8 ⊢ (𝑈 Fn ℕ → dom 𝑈 = ℕ)
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑈 = ℕ)
48	37, 47	syl5eleqr 2695	. . . . . 6 ⊢ (𝜑 → 1 ∈ dom 𝑈)
49		ne0i 3880	. . . . . 6 ⊢ (1 ∈ dom 𝑈 → dom 𝑈 ≠ ∅)
50	48, 49	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝑈 ≠ ∅)
51		dm0rn0 5263	. . . . . 6 ⊢ (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
52	51	necon3bii 2834	. . . . 5 ⊢ (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
53	50, 52	sylib 207	. . . 4 ⊢ (𝜑 → ran 𝑈 ≠ ∅)
54	1	simprd 478	. . . . . . . 8 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
55	3	simprd 478	. . . . . . . 8 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ)
56	54, 55	readdcld 9948	. . . . . . 7 ⊢ (𝜑 → ((vol‘𝐴) + (vol‘𝐵)) ∈ ℝ)
57		ovolun.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
58	57	rpred 11748	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
59	56, 58	readdcld 9948	. . . . . 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 23071	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑈‘𝑘) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶))
67	66	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑈‘𝑘) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶))
68		breq1 4586	. . . . . . . . 9 ⊢ (𝑧 = (𝑈‘𝑘) → (𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ↔ (𝑈‘𝑘) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶)))
69	68	ralrn 6270	. . . . . . . 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 246	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶))
72		breq2 4587	. . . . . . . 8 ⊢ (𝑘 = (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) → (𝑧 ≤ 𝑘 ↔ 𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶)))
73	72	ralbidv 2969	. . . . . . 7 ⊢ (𝑘 = (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘 ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶)))
74	73	rspcev 3282	. . . . . 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 691	. . . . 5 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑘)
76		ressxr 9962	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
77	36, 76	syl6ss 3580	. . . . . 6 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ^*)
78		supxrbnd2 12024	. . . . . 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 221	. . . 4 ⊢ (𝜑 → sup(ran 𝑈, ℝ^*, < ) < +∞)
81		supxrbnd 12030	. . . 4 ⊢ ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ sup(ran 𝑈, ℝ^, < ) < +∞) → sup(ran 𝑈, ℝ^, < ) ∈ ℝ)
82	36, 53, 80, 81	syl3anc 1318	. . 3 ⊢ (𝜑 → sup(ran 𝑈, ℝ^*, < ) ∈ ℝ)
83		nncn 10905	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
84	83	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
85	84	2timesd 11152	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) = (𝑚 + 𝑚))
86	85	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = ((𝑚 + 𝑚) − 1))
87		1cnd 9935	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℂ)
88	84, 84, 87	addsubassd 10291	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
89	86, 88	eqtrd 2644	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
90		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
91		nnm1nn0 11211	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ₀)
92	91	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 − 1) ∈ ℕ₀)
93		nnnn0addcl 11200	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ ∧ (𝑚 − 1) ∈ ℕ₀) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
94	90, 92, 93	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
95	89, 94	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) ∈ ℕ)
96		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 / 2) = (((2 · 𝑚) − 1) / 2))
97	96	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 / 2) ∈ ℕ ↔ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
98	96	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐺‘(𝑛 / 2)) = (𝐺‘(((2 · 𝑚) − 1) / 2)))
99		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝑛 + 1) = (((2 · 𝑚) − 1) + 1))
100	99	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 + 1) / 2) = ((((2 · 𝑚) − 1) + 1) / 2))
101	100	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ((2 · 𝑚) − 1) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
102	97, 98, 101	ifbieq12d 4063	. . . . . . . . . . . . . 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 6113	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘(((2 · 𝑚) − 1) / 2)) ∈ V
104		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) ∈ V
105	103, 104	ifex 4106	. . . . . . . . . . . . . 14 ⊢ if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) ∈ V
106	102, 26, 105	fvmpt 6191	. . . . . . . . . . . . 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 11062	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
109		nnmulcl 10920	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
110	108, 90, 109	sylancr 694	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
111	110	nncnd 10913	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℂ)
112		ax-1cn 9873	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
113		npcan 10169	. . . . . . . . . . . . . . . . . 18 ⊢ (((2 · 𝑚) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
114	111, 112, 113	sylancl 693	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
115	114	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = ((2 · 𝑚) / 2))
116		2cn 10968	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℂ
117		2ne0 10990	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
118		divcan3 10590	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑚) / 2) = 𝑚)
119	116, 117, 118	mp3an23 1408	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℂ → ((2 · 𝑚) / 2) = 𝑚)
120	84, 119	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) = 𝑚)
121	115, 120	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = 𝑚)
122	121, 90	eqeltrd 2688	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ)
123		nneo 11337	. . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ ↔ ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
126	122, 125	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ)
127	126	iffalsed 4047	. . . . . . . . . . . 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 6107	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) = (𝐹‘𝑚))
129	107, 127, 128	3eqtrd 2648	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚))
130		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((2 · 𝑚) − 1) → (𝐻‘𝑘) = (𝐻‘((2 · 𝑚) − 1)))
131	130	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (𝑘 = ((2 · 𝑚) − 1) → ((𝐻‘𝑘) = (𝐹‘𝑚) ↔ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)))
132	131	rspcev 3282	. . . . . . . . . . 11 ⊢ ((((2 · 𝑚) − 1) ∈ ℕ ∧ (𝐻‘((2 · 𝑚) − 1)) = (𝐹‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
133	95, 129, 132	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐹‘𝑚))
134		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (1^st ‘(𝐻‘𝑘)) = (1^st ‘(𝐹‘𝑚)))
135	134	breq1d 4593	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → ((1^st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1^st ‘(𝐹‘𝑚)) < 𝑧))
136		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (2^nd ‘(𝐻‘𝑘)) = (2^nd ‘(𝐹‘𝑚)))
137	136	breq2d 4595	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐹‘𝑚) → (𝑧 < (2^nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2^nd ‘(𝐹‘𝑚))))
138	135, 137	anbi12d 743	. . . . . . . . . . . 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 239	. . . . . . . . . . 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 2999	. . . . . . . . . 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 3013	. . . . . . . 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 2946	. . . . . . 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 23043	. . . . . . . 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 691	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1^st ‘(𝐹‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑚)))))
146		ovolfioo 23043	. . . . . . . 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 691	. . . . . . 7 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐴 ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
148	143, 145, 147	3imtr4d 282	. . . . . 6 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻)))
149	62, 148	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻))
150		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 / 2) = ((2 · 𝑚) / 2))
151	150	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → ((𝑛 / 2) ∈ ℕ ↔ ((2 · 𝑚) / 2) ∈ ℕ))
152	150	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐺‘(𝑛 / 2)) = (𝐺‘((2 · 𝑚) / 2)))
153		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (2 · 𝑚) → (𝑛 + 1) = ((2 · 𝑚) + 1))
154	153	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 · 𝑚) → ((𝑛 + 1) / 2) = (((2 · 𝑚) + 1) / 2))
155	154	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 · 𝑚) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘(((2 · 𝑚) + 1) / 2)))
156	151, 152, 155	ifbieq12d 4063	. . . . . . . . . . . . . 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 6113	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘((2 · 𝑚) / 2)) ∈ V
158		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘(((2 · 𝑚) + 1) / 2)) ∈ V
159	157, 158	ifex 4106	. . . . . . . . . . . . . 14 ⊢ if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) ∈ V
160	156, 26, 159	fvmpt 6191	. . . . . . . . . . . . 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 2688	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) ∈ ℕ)
163	162	iftrued 4044	. . . . . . . . . . . 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 6107	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘((2 · 𝑚) / 2)) = (𝐺‘𝑚))
165	161, 163, 164	3eqtrd 2648	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚))
166		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑘 = (2 · 𝑚) → (𝐻‘𝑘) = (𝐻‘(2 · 𝑚)))
167	166	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (𝑘 = (2 · 𝑚) → ((𝐻‘𝑘) = (𝐺‘𝑚) ↔ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)))
168	167	rspcev 3282	. . . . . . . . . . 11 ⊢ (((2 · 𝑚) ∈ ℕ ∧ (𝐻‘(2 · 𝑚)) = (𝐺‘𝑚)) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
169	110, 165, 168	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻‘𝑘) = (𝐺‘𝑚))
170		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (1^st ‘(𝐻‘𝑘)) = (1^st ‘(𝐺‘𝑚)))
171	170	breq1d 4593	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → ((1^st ‘(𝐻‘𝑘)) < 𝑧 ↔ (1^st ‘(𝐺‘𝑚)) < 𝑧))
172		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (2^nd ‘(𝐻‘𝑘)) = (2^nd ‘(𝐺‘𝑚)))
173	172	breq2d 4595	. . . . . . . . . . . . 13 ⊢ ((𝐻‘𝑘) = (𝐺‘𝑚) → (𝑧 < (2^nd ‘(𝐻‘𝑘)) ↔ 𝑧 < (2^nd ‘(𝐺‘𝑚))))
174	171, 173	anbi12d 743	. . . . . . . . . . . 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 239	. . . . . . . . . . 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 2999	. . . . . . . . . 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 3013	. . . . . . . 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 2946	. . . . . . 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 23043	. . . . . . . 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 691	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑧 ∈ 𝐵 ∃𝑚 ∈ ℕ ((1^st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐺‘𝑚)))))
182		ovolfioo 23043	. . . . . . . 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 691	. . . . . . 7 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ 𝐵 ∃𝑘 ∈ ℕ ((1^st ‘(𝐻‘𝑘)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑘)))))
184	179, 181, 183	3imtr4d 282	. . . . . 6 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻)))
185	64, 184	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐻))
186	149, 185	unssd 3751	. . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻))
187	29	ovollb 23054	. . . 4 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴 ∪ 𝐵) ⊆ ∪ ran ((,) ∘ 𝐻)) → (vol‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ^, < ))
188	27, 186, 187	syl2anc 691	. . 3 ⊢ (𝜑 → (vol‘(𝐴 ∪ 𝐵)) ≤ sup(ran 𝑈, ℝ^, < ))
189		ovollecl 23058	. . 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 1318	. 2 ⊢ (𝜑 → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
191	59	rexrd 9968	. . . 4 ⊢ (𝜑 → (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ∈ ℝ^*)
192		supxrleub 12028	. . . 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 691	. . 3 ⊢ (𝜑 → (sup(ran 𝑈, ℝ^, < ) ≤ (((vol‘𝐴) + (vol‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶)))
194	71, 193	mpbird 246	. 2 ⊢ (𝜑 → sup(ran 𝑈, ℝ^, < ) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))
195	190, 82, 59, 188, 194	letrd 10073	1 ⊢ (𝜑 → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ran crn 5039 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ↑_𝑚 cmap 7744 supcsup 8229 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 (,)cioo 12046 [,)cico 12048 seqcseq 12663 abscabs 13822 vol*covol 23038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ioo 12050 df-ico 12052 df-fz 12198 df-fl 12455 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-ovol 23040

This theorem is referenced by: ovolunlem2 23073

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