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Theorem uniioombllem6 25909

Description: Lemma for uniioombl 25910. (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 13004	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12727	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
3		uniioombl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
4		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = (𝑇‘𝑚))
5		uniioombl.t	. . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
6		eqidd 2761	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) = (((abs ∘ − ) ∘ 𝐺)‘𝑎))
7		uniioombl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8		eqid 2760	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
9	8	ovolfsf 25792	. . . . . . . . . 10 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
10	7, 9	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
11	10	ffvelcdmda 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞))
12		elrege0 13585	. . . . . . . 8 ⊢ ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
13	11, 12	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
14	13	simpld 500	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ)
15	13	simprd 501	. . . . . 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 25902	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
24	8, 5	ovolsf 25793	. . . . . . . . . . . . 13 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
25	7, 24	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞))
26	25	frnd 6718	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
27		icossxr 13563	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ^*
28	26, 27	sstrdi 3920	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
29		supxrub 13454	. . . . . . . . . 10 ⊢ ((ran 𝑇 ⊆ ℝ^* ∧ 𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
30	28, 29	sylan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
31	30	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
32	25	ffnd 6710	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 Fn ℕ)
33		breq1 5105	. . . . . . . . . 10 ⊢ (𝑥 = (𝑇‘𝑚) → (𝑥 ≤ sup(ran 𝑇, ℝ^, < ) ↔ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^, < )))
34	33	ralrn 7088	. . . . . . . . 9 ⊢ (𝑇 Fn ℕ → (∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ^, < ) ↔ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^, < )))
35	32, 34	syl 18	. . . . . . . 8 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ^, < ) ↔ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^, < )))
36	31, 35	mpbid 235	. . . . . . 7 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^*, < ))
37		brralrspcev 5164	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ 𝑥)
39	1, 5, 2, 6, 14, 15, 38	isumsup2 16015	. . . . 5 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
40		rge0ssre 13587	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
41	26, 40	sstrdi 3920	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
42		1nn 12346	. . . . . . . . 9 ⊢ 1 ∈ ℕ
43	25	fdmd 6720	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑇 = ℕ)
44	42, 43	eleqtrrid 2867	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ dom 𝑇)
45	44	ne0d 4287	. . . . . . 7 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
46		dm0rn0 5905	. . . . . . . 8 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
47	46	necon3bii 3007	. . . . . . 7 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
48	45, 47	sylib 221	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
49		brralrspcev 5164	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ 𝑦)
51		supxrre 13457	. . . . . 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 1398	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) = sup(ran 𝑇, ℝ, < ))
53	39, 52	breqtrrd 5132	. . . 4 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ^*, < ))
54	1, 2, 3, 4, 53	climi2 15678	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)
55	1	r19.2uz 15519	. . 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 18	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℕ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)
57		1zzd 12727	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
58	3	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝐶 ∈ ℝ⁺)
59		simplrl 789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℕ)
60	59	nnrpd 13162	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℝ⁺)
61	58, 60	rpdivcld 13181	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐶 / 𝑚) ∈ ℝ⁺)
62		fvex 6898	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑧)) ∈ V
63	62	inex1 5276	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
64	63	rgenw 3080	. . . . . . . . . . . . . 14 ⊢ ∀𝑧 ∈ ℕ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
65		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
66	65	fnmpt 6679	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ℕ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V → (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) Fn ℕ)
67	64, 66	mp1i 14	. . . . . . . . . . . . 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 13687	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
69		fvco2 6982	. . . . . . . . . . . . 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 608	. . . . . . . . . . . 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 487	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
72		2fveq3 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑖 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑖)))
73	72	ineq1d 4116	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑖 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))))
74		fvex 6898	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑖)) ∈ V
75	74	inex1 5276	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
76	73, 65, 75	fvmpt 6993	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → ((𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))‘𝑖) = (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))))
77	71, 76	syl 18	. . . . . . . . . . . . 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 6889	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 490	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
81	80, 1	eleqtrdi 2870	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
82		inss2 4134	. . . . . . . . . . . . 13 ⊢ (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ⊆ ((,)‘(𝐺‘𝑗))
83	7	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
84		elfznn 13687	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
85		ffvelcdm 7081	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
86	83, 84, 85	syl2an 608	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
87	86	elin2d 4102	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) ∈ (ℝ × ℝ))
88		1st2nd2 8040	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑗) ∈ (ℝ × ℝ) → (𝐺‘𝑗) = ⟨(1^st ‘(𝐺‘𝑗)), (2^nd ‘(𝐺‘𝑗))⟩)
89	87, 88	syl 18	. . . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗))) = ((,)‘⟨(1^st ‘(𝐺‘𝑗)), (2^nd ‘(𝐺‘𝑗))⟩)
92	90, 91	eqtr4di 2813	. . . . . . . . . . . . . . 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 13538	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗))) ⊆ ℝ
94	92, 93	eqsstrdi 3952	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺‘𝑗)) ⊆ ℝ)
95	94	ad2antrr 739	. . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . 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 25787	. . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . 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 25889	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . 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 1161	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (2^nd ‘(𝐺‘𝑗)) ∈ ℝ)
103	98	simp1d 1160	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (1^st ‘(𝐺‘𝑗)) ∈ ℝ)
104	102, 103	resubcld 11744	. . . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol‘((,)‘(𝐺‘𝑗))) ∈ ℝ)
106	105	ad2antrr 739	. . . . . . . . . . . . 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 25807	. . . . . . . . . . . . 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 1495	. . . . . . . . . . . 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 11337	. . . . . . . . . . 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 15896	. . . . . . . . . 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 2766	. . . . . . . . 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 6890	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑘)))
113	112	ineq1d 4116	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
114	113	cbvmptv 5208	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑘 ∈ ℕ ↦ (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
115		eqeq1 2764	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 = ∅ ↔ 𝑥 = ∅))
116		infeq1 9469	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → inf(𝑧, ℝ^, < ) = inf(𝑥, ℝ^, < ))
117		supeq1 9437	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → sup(𝑧, ℝ^, < ) = sup(𝑥, ℝ^, < ))
118	116, 117	opeq12d 4840	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ⟨inf(𝑧, ℝ^, < ), sup(𝑧, ℝ^, < )⟩ = ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩)
119	115, 118	ifbieq2d 4508	. . . . . . . . . . . . 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 5208	. . . . . . . . . . . 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 25904	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))
122	84, 121	sylan2 605	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))
123	122	adantlr 728	. . . . . . . . 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 15678	. . . . . . . 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 12726	. . . . . . . . 9 ⊢ 1 ∈ ℤ
126	1	rexuz3 15516	. . . . . . . . 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 221	. . . . . . 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 3154	. . . . . 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 14115	. . . . . . 7 ⊢ (1...𝑚) ∈ Fin
131		rexfiuz 15515	. . . . . . 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 237	. . . . 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 15516	. . . . . 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 237	. . . 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 15519	. . . 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 18	. . 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 486	. . . . 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 486	. . . . 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 486	. . . . 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 486	. . . . 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 486	. . . . 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 486	. . . . 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 486	. . . . 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 791	. . . . 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 792	. . . . 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 2760	. . . . 5 ⊢ ∪ (((,) ∘ 𝐺) “ (1...𝑚)) = ∪ (((,) ∘ 𝐺) “ (1...𝑚))
149		simprrl 793	. . . . 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 794	. . . . . 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 6890	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑧 → ((,)‘(𝐹‘𝑖)) = ((,)‘(𝐹‘𝑧)))
152	151	ineq1d 4116	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑧 → (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
153	152	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑧 → (vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))
154	153	cbvsumv 15863	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
155		2fveq3 6890	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((,)‘(𝐺‘𝑗)) = ((,)‘(𝐺‘𝑘)))
156	155	ineq2d 4117	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘))))
157	156	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
158	157	sumeq2sdv 15870	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
159	154, 158	eqtrid 2807	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
160	155	ineq1d 4116	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐺‘𝑗)) ∩ 𝐴) = (((,)‘(𝐺‘𝑘)) ∩ 𝐴))
161	160	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)) = (vol‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴)))
162	159, 161	oveq12d 7438	. . . . . . . . 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 6889	. . . . . . . 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 5112	. . . . . . 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 3240	. . . . . 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 221	. . . . 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 2760	. . . . 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 25908	. . . 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 473	. . 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 3169	. 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 3169	1 ⊢ (𝜑 → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ∩ cin 4089 ∅c0 4278 ifcif 4481 ⟨cop 4589 ∪ cuni 4866 Disj wdisj 5069 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 dom cdm 5650 ran crn 5651 “ cima 5653 ∘ ccom 5654 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 Fincfn 8973 supcsup 9432 infcinf 9433 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 +∞cpnf 11340 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 − cmin 11541 / cdiv 11973 ℕcn 12335 4c4 12399 ℤcz 12693 ℤ_≥cuz 12965 ℝ⁺crp 13120 (,)cioo 13476 [,)cico 13478 ...cfz 13639 seqcseq 14144 abscabs 15401 ⇝ cli 15651 Σcsu 15853 vol*covol 25783

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-acn 10023 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-n0 12607 df-z 12694 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-rest 17593 df-topgen 17614 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-top 23212 df-topon 23229 df-bases 23264 df-cmp 23705 df-ovol 25785 df-vol 25786

This theorem is used by: uniioombl 25910

W3C validator