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Theorem uniioombllem6 24183

Description: Lemma for uniioombl 24184. (Contributed by Mario Carneiro, 26-Mar-2015.)

**Hypotheses**
Ref	Expression
uniioombl.1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2	⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
uniioombl.3	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
uniioombl.a	⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)
uniioombl.e	⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
uniioombl.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
uniioombl.g	⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.s	⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))
uniioombl.t	⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
uniioombl.v	⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ≤ ((vol‘𝐸) + 𝐶))

**Assertion**
Ref	Expression
uniioombllem6	⊢ (𝜑 → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐺 𝑥,𝐴 𝑥,𝐶 𝜑,𝑥 𝑥,𝑇 Allowed substitution hints: 𝑆(𝑥) 𝐸(𝑥)

Proof of Theorem uniioombllem6 Dummy variables 𝑎 𝑖 𝑗 𝑘 𝑛 𝑦 𝑧 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12275	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12007	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
3		uniioombl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
4		eqidd 2822	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = (𝑇‘𝑚))
5		uniioombl.t	. . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
6		eqidd 2822	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) = (((abs ∘ − ) ∘ 𝐺)‘𝑎))
7		uniioombl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8		eqid 2821	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
9	8	ovolfsf 24066	. . . . . . . . . 10 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
10	7, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
11	10	ffvelrnda 6845	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞))
12		elrege0 12836	. . . . . . . 8 ⊢ ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
13	11, 12	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
14	13	simpld 497	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ)
15	13	simprd 498	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎))
16		uniioombl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
17		uniioombl.2	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
18		uniioombl.3	. . . . . . . 8 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
19		uniioombl.a	. . . . . . . 8 ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)
20		uniioombl.e	. . . . . . . 8 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
21		uniioombl.s	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))
22		uniioombl.v	. . . . . . . 8 ⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ≤ ((vol‘𝐸) + 𝐶))
23	16, 17, 18, 19, 20, 3, 7, 21, 5, 22	uniioombllem1 24176	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
24	8, 5	ovolsf 24067	. . . . . . . . . . . . 13 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
25	7, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞))
26	25	frnd 6515	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
27		icossxr 12815	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ^*
28	26, 27	sstrdi 3978	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
29		supxrub 12711	. . . . . . . . . 10 ⊢ ((ran 𝑇 ⊆ ℝ^* ∧ 𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
30	28, 29	sylan 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
31	30	ralrimiva 3182	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
32	25	ffnd 6509	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 Fn ℕ)
33		breq1 5061	. . . . . . . . . 10 ⊢ (𝑥 = (𝑇‘𝑚) → (𝑥 ≤ sup(ran 𝑇, ℝ^, < ) ↔ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^, < )))
34	33	ralrn 6848	. . . . . . . . 9 ⊢ (𝑇 Fn ℕ → (∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ^, < ) ↔ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^, < )))
35	32, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ^, < ) ↔ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^, < )))
36	31, 35	mpbid 234	. . . . . . 7 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^*, < ))
37		brralrspcev 5118	. . . . . . 7 ⊢ ((sup(ran 𝑇, ℝ^, < ) ∈ ℝ ∧ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^, < )) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ 𝑥)
38	23, 36, 37	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ 𝑥)
39	1, 5, 2, 6, 14, 15, 38	isumsup2 15195	. . . . 5 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
40		rge0ssre 12838	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
41	26, 40	sstrdi 3978	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
42		1nn 11643	. . . . . . . . 9 ⊢ 1 ∈ ℕ
43	25	fdmd 6517	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑇 = ℕ)
44	42, 43	eleqtrrid 2920	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ dom 𝑇)
45	44	ne0d 4300	. . . . . . 7 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
46		dm0rn0 5789	. . . . . . . 8 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
47	46	necon3bii 3068	. . . . . . 7 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
48	45, 47	sylib 220	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
49		brralrspcev 5118	. . . . . . 7 ⊢ ((sup(ran 𝑇, ℝ^, < ) ∈ ℝ ∧ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ^, < )) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ 𝑦)
50	23, 31, 49	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ 𝑦)
51		supxrre 12714	. . . . . 6 ⊢ ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ 𝑦) → sup(ran 𝑇, ℝ^*, < ) = sup(ran 𝑇, ℝ, < ))
52	41, 48, 50, 51	syl3anc 1367	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) = sup(ran 𝑇, ℝ, < ))
53	39, 52	breqtrrd 5086	. . . 4 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ^*, < ))
54	1, 2, 3, 4, 53	climi2 14862	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)
55	1	r19.2uz 14705	. . 3 ⊢ (∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶 → ∃𝑚 ∈ ℕ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)
56	54, 55	syl 17	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℕ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)
57		1zzd 12007	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
58	3	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝐶 ∈ ℝ⁺)
59		simplrl 775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℕ)
60	59	nnrpd 12423	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℝ⁺)
61	58, 60	rpdivcld 12442	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐶 / 𝑚) ∈ ℝ⁺)
62		fvex 6677	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑧)) ∈ V
63	62	inex1 5213	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
64	63	rgenw 3150	. . . . . . . . . . . . . 14 ⊢ ∀𝑧 ∈ ℕ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
65		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
66	65	fnmpt 6482	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ℕ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V → (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) Fn ℕ)
67	64, 66	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) Fn ℕ)
68		elfznn 12930	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
69		fvco2 6752	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) Fn ℕ ∧ 𝑖 ∈ ℕ) → ((vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))‘𝑖) = (vol*‘((𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))‘𝑖)))
70	67, 68, 69	syl2an 597	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((vol ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))‘𝑖) = (vol*‘((𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))‘𝑖)))
71	68	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
72		2fveq3 6669	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑖 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑖)))
73	72	ineq1d 4187	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑖 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))))
74		fvex 6677	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑖)) ∈ V
75	74	inex1 5213	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
76	73, 65, 75	fvmpt 6762	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → ((𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))‘𝑖) = (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))))
77	71, 76	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))‘𝑖) = (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))))
78	77	fveq2d 6668	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol‘((𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))‘𝑖)) = (vol*‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))))
79	70, 78	eqtrd 2856	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((vol ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))‘𝑖) = (vol*‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))))
80		simpr 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
81	80, 1	eleqtrdi 2923	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
82		inss2 4205	. . . . . . . . . . . . 13 ⊢ (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ⊆ ((,)‘(𝐺‘𝑗))
83	7	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
84		elfznn 12930	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
85		ffvelrn 6843	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
86	83, 84, 85	syl2an 597	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
87	86	elin2d 4175	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) ∈ (ℝ × ℝ))
88		1st2nd2 7722	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑗) ∈ (ℝ × ℝ) → (𝐺‘𝑗) = ⟨(1^st ‘(𝐺‘𝑗)), (2^nd ‘(𝐺‘𝑗))⟩)
89	87, 88	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) = ⟨(1^st ‘(𝐺‘𝑗)), (2^nd ‘(𝐺‘𝑗))⟩)
90	89	fveq2d 6668	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺‘𝑗)) = ((,)‘⟨(1^st ‘(𝐺‘𝑗)), (2^nd ‘(𝐺‘𝑗))⟩))
91		df-ov 7153	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗))) = ((,)‘⟨(1^st ‘(𝐺‘𝑗)), (2^nd ‘(𝐺‘𝑗))⟩)
92	90, 91	syl6eqr 2874	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺‘𝑗)) = ((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗))))
93		ioossre 12792	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗))) ⊆ ℝ
94	92, 93	eqsstrdi 4020	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺‘𝑗)) ⊆ ℝ)
95	94	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((,)‘(𝐺‘𝑗)) ⊆ ℝ)
96	92	fveq2d 6668	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol‘((,)‘(𝐺‘𝑗))) = (vol*‘((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗)))))
97		ovolfcl 24061	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1^st ‘(𝐺‘𝑗)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝑗)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝑗)) ≤ (2^nd ‘(𝐺‘𝑗))))
98	83, 84, 97	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((1^st ‘(𝐺‘𝑗)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝑗)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝑗)) ≤ (2^nd ‘(𝐺‘𝑗))))
99		ovolioo 24163	. . . . . . . . . . . . . . . . 17 ⊢ (((1^st ‘(𝐺‘𝑗)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝑗)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝑗)) ≤ (2^nd ‘(𝐺‘𝑗))) → (vol*‘((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗)))) = ((2^nd ‘(𝐺‘𝑗)) − (1^st ‘(𝐺‘𝑗))))
100	98, 99	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol‘((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗)))) = ((2^nd ‘(𝐺‘𝑗)) − (1^st ‘(𝐺‘𝑗))))
101	96, 100	eqtrd 2856	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol‘((,)‘(𝐺‘𝑗))) = ((2^nd ‘(𝐺‘𝑗)) − (1^st ‘(𝐺‘𝑗))))
102	98	simp2d 1139	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (2^nd ‘(𝐺‘𝑗)) ∈ ℝ)
103	98	simp1d 1138	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (1^st ‘(𝐺‘𝑗)) ∈ ℝ)
104	102, 103	resubcld 11062	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((2^nd ‘(𝐺‘𝑗)) − (1^st ‘(𝐺‘𝑗))) ∈ ℝ)
105	101, 104	eqeltrd 2913	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol‘((,)‘(𝐺‘𝑗))) ∈ ℝ)
106	105	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol‘((,)‘(𝐺‘𝑗))) ∈ ℝ)
107		ovolsscl 24081	. . . . . . . . . . . . 13 ⊢ (((((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ⊆ ((,)‘(𝐺‘𝑗)) ∧ ((,)‘(𝐺‘𝑗)) ⊆ ℝ ∧ (vol‘((,)‘(𝐺‘𝑗))) ∈ ℝ) → (vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) ∈ ℝ)
108	82, 95, 106, 107	mp3an2i 1462	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) ∈ ℝ)
109	108	recnd 10663	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) ∈ ℂ)
110	79, 81, 109	fsumser 15081	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = (seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))))‘𝑛))
111	110	eqcomd 2827	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → (seq1( + , (vol ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))))‘𝑛) = Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))))
112		2fveq3 6669	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑘)))
113	112	ineq1d 4187	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
114	113	cbvmptv 5161	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑘 ∈ ℕ ↦ (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
115		eqeq1 2825	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 = ∅ ↔ 𝑥 = ∅))
116		infeq1 8934	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → inf(𝑧, ℝ^, < ) = inf(𝑥, ℝ^, < ))
117		supeq1 8903	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → sup(𝑧, ℝ^, < ) = sup(𝑥, ℝ^, < ))
118	116, 117	opeq12d 4804	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ⟨inf(𝑧, ℝ^, < ), sup(𝑧, ℝ^, < )⟩ = ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩)
119	115, 118	ifbieq2d 4491	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → if(𝑧 = ∅, ⟨0, 0⟩, ⟨inf(𝑧, ℝ^, < ), sup(𝑧, ℝ^, < )⟩) = if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))
120	119	cbvmptv 5161	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ran (,) ↦ if(𝑧 = ∅, ⟨0, 0⟩, ⟨inf(𝑧, ℝ^, < ), sup(𝑧, ℝ^, < )⟩)) = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))
121	16, 17, 18, 19, 20, 3, 7, 21, 5, 22, 114, 120	uniioombllem2 24178	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))
122	84, 121	sylan2 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))
123	122	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → seq1( + , (vol ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))
124	1, 57, 61, 111, 123	climi2 14862	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
125		1z 12006	. . . . . . . . 9 ⊢ 1 ∈ ℤ
126	1	rexuz3 14702	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))
127	125, 126	ax-mp 5	. . . . . . . 8 ⊢ (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
128	124, 127	sylib 220	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
129	128	ralrimiva 3182	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) → ∀𝑗 ∈ (1...𝑚)∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
130		fzfi 13334	. . . . . . 7 ⊢ (1...𝑚) ∈ Fin
131		rexfiuz 14701	. . . . . . 7 ⊢ ((1...𝑚) ∈ Fin → (∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∀𝑗 ∈ (1...𝑚)∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))
132	130, 131	ax-mp 5	. . . . . 6 ⊢ (∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∀𝑗 ∈ (1...𝑚)∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
133	129, 132	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) → ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
134	1	rexuz3 14702	. . . . . 6 ⊢ (1 ∈ ℤ → (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))
135	125, 134	ax-mp 5	. . . . 5 ⊢ (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ_≥‘𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
136	133, 135	sylibr 236	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) → ∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
137	1	r19.2uz 14705	. . . 4 ⊢ (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ_≥‘𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) → ∃𝑛 ∈ ℕ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
138	136, 137	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) → ∃𝑛 ∈ ℕ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
139	16	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
140	17	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
141	20	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → (vol‘𝐸) ∈ ℝ)
142	3	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝐶 ∈ ℝ⁺)
143	7	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
144	21	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))
145	22	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → sup(ran 𝑇, ℝ^, < ) ≤ ((vol*‘𝐸) + 𝐶))
146		simprll 777	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝑚 ∈ ℕ)
147		simprlr 778	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)
148		eqid 2821	. . . . 5 ⊢ ∪ (((,) ∘ 𝐺) “ (1...𝑚)) = ∪ (((,) ∘ 𝐺) “ (1...𝑚))
149		simprrl 779	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝑛 ∈ ℕ)
150		simprrr 780	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
151		2fveq3 6669	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑧 → ((,)‘(𝐹‘𝑖)) = ((,)‘(𝐹‘𝑧)))
152	151	ineq1d 4187	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑧 → (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
153	152	fveq2d 6668	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑧 → (vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))
154	153	cbvsumv 15047	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
155		2fveq3 6669	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((,)‘(𝐺‘𝑗)) = ((,)‘(𝐺‘𝑘)))
156	155	ineq2d 4188	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘))))
157	156	fveq2d 6668	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
158	157	sumeq2sdv 15055	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
159	154, 158	syl5eq 2868	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
160	155	ineq1d 4187	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐺‘𝑗)) ∩ 𝐴) = (((,)‘(𝐺‘𝑘)) ∩ 𝐴))
161	160	fveq2d 6668	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)) = (vol‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴)))
162	159, 161	oveq12d 7168	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴))) = (Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))) − (vol‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴))))
163	162	fveq2d 6668	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) = (abs‘(Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))) − (vol‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴)))))
164	163	breq1d 5068	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ (abs‘(Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))) − (vol‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴)))) < (𝐶 / 𝑚)))
165	164	cbvralvw 3449	. . . . . 6 ⊢ (∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∀𝑘 ∈ (1...𝑚)(abs‘(Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))) − (vol‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴)))) < (𝐶 / 𝑚))
166	150, 165	sylib 220	. . . . 5 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → ∀𝑘 ∈ (1...𝑚)(abs‘(Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))) − (vol*‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴)))) < (𝐶 / 𝑚))
167		eqid 2821	. . . . 5 ⊢ ∪ (((,) ∘ 𝐹) “ (1...𝑛)) = ∪ (((,) ∘ 𝐹) “ (1...𝑛))
168	139, 140, 18, 19, 141, 142, 143, 144, 5, 145, 146, 147, 148, 149, 166, 167	uniioombllem5 24182	. . . 4 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ ((vol‘𝐸) + (4 · 𝐶)))
169	168	anassrs 470	. . 3 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) − (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))) → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ ((vol‘𝐸) + (4 · 𝐶)))
170	138, 169	rexlimddv 3291	. 2 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ ((vol‘𝐸) + (4 · 𝐶)))
171	56, 170	rexlimddv 3291	1 ⊢ (𝜑 → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 ifcif 4466 ⟨cop 4566 ∪ cuni 4831 Disj wdisj 5023 class class class wbr 5058 ↦ cmpt 5138 × cxp 5547 dom cdm 5549 ran crn 5550 “ cima 5552 ∘ ccom 5553 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 1^st c1st 7681 2^nd c2nd 7682 Fincfn 8503 supcsup 8898 infcinf 8899 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 4c4 11688 ℤcz 11975 ℤ_≥cuz 12237 ℝ⁺crp 12383 (,)cioo 12732 [,)cico 12734 ...cfz 12886 seqcseq 13363 abscabs 14587 ⇝ cli 14835 Σcsu 15036 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-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 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-fal 1546 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-int 4869 df-iun 4913 df-disj 5024 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-se 5509 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-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 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-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-rest 16690 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-cmp 21989 df-ovol 24059 df-vol 24060

This theorem is referenced by: uniioombl 24184

W3C validator