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Theorem uniioombllem6 24657

Description: Lemma for uniioombl 24658. (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 12550	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12281	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
3		uniioombl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
4		eqidd 2739	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = (𝑇‘𝑚))
5		uniioombl.t	. . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
6		eqidd 2739	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) = (((abs ∘ − ) ∘ 𝐺)‘𝑎))
7		uniioombl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8		eqid 2738	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
9	8	ovolfsf 24540	. . . . . . . . . 10 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
10	7, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
11	10	ffvelrnda 6943	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞))
12		elrege0 13115	. . . . . . . 8 ⊢ ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
13	11, 12	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
14	13	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ)
15	13	simprd 495	. . . . . 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 24650	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
24	8, 5	ovolsf 24541	. . . . . . . . . . . . 13 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
25	7, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞))
26	25	frnd 6592	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
27		icossxr 13093	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ^*
28	26, 27	sstrdi 3929	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
29		supxrub 12987	. . . . . . . . . 10 ⊢ ((ran 𝑇 ⊆ ℝ^* ∧ 𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
30	28, 29	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
31	30	ralrimiva 3107	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
32	25	ffnd 6585	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 Fn ℕ)
33		breq1 5073	. . . . . . . . . 10 ⊢ (𝑥 = (𝑇‘𝑚) → (𝑥 ≤ sup(ran 𝑇, ℝ^, < ) ↔ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^, < )))
34	33	ralrn 6946	. . . . . . . . 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 231	. . . . . . 7 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^*, < ))
37		brralrspcev 5130	. . . . . . 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 583	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ 𝑥)
39	1, 5, 2, 6, 14, 15, 38	isumsup2 15486	. . . . 5 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
40		rge0ssre 13117	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
41	26, 40	sstrdi 3929	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
42		1nn 11914	. . . . . . . . 9 ⊢ 1 ∈ ℕ
43	25	fdmd 6595	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑇 = ℕ)
44	42, 43	eleqtrrid 2846	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ dom 𝑇)
45	44	ne0d 4266	. . . . . . 7 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
46		dm0rn0 5823	. . . . . . . 8 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
47	46	necon3bii 2995	. . . . . . 7 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
48	45, 47	sylib 217	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
49		brralrspcev 5130	. . . . . . 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 583	. . . . . 6 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ 𝑦)
51		supxrre 12990	. . . . . 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 1369	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) = sup(ran 𝑇, ℝ, < ))
53	39, 52	breqtrrd 5098	. . . 4 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ^*, < ))
54	1, 2, 3, 4, 53	climi2 15148	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)
55	1	r19.2uz 14991	. . 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 12281	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
58	3	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝐶 ∈ ℝ⁺)
59		simplrl 773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℕ)
60	59	nnrpd 12699	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℝ⁺)
61	58, 60	rpdivcld 12718	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐶 / 𝑚) ∈ ℝ⁺)
62		fvex 6769	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑧)) ∈ V
63	62	inex1 5236	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
64	63	rgenw 3075	. . . . . . . . . . . . . 14 ⊢ ∀𝑧 ∈ ℕ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
65		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
66	65	fnmpt 6557	. . . . . . . . . . . . . 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 13214	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
69		fvco2 6847	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
72		2fveq3 6761	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑖 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑖)))
73	72	ineq1d 4142	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑖 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))))
74		fvex 6769	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑖)) ∈ V
75	74	inex1 5236	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
76	73, 65, 75	fvmpt 6857	. . . . . . . . . . . . . 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 6760	. . . . . . . . . . . 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 2778	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
81	80, 1	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
82		inss2 4160	. . . . . . . . . . . . 13 ⊢ (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ⊆ ((,)‘(𝐺‘𝑗))
83	7	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
84		elfznn 13214	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
85		ffvelrn 6941	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
86	83, 84, 85	syl2an 595	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
87	86	elin2d 4129	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) ∈ (ℝ × ℝ))
88		1st2nd2 7843	. . . . . . . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . . . 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 7258	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗))) = ((,)‘⟨(1^st ‘(𝐺‘𝑗)), (2^nd ‘(𝐺‘𝑗))⟩)
92	90, 91	eqtr4di 2797	. . . . . . . . . . . . . . 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 13069	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗))) ⊆ ℝ
94	92, 93	eqsstrdi 3971	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺‘𝑗)) ⊆ ℝ)
95	94	ad2antrr 722	. . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . . . 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 24535	. . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . 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 24637	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 1141	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (2^nd ‘(𝐺‘𝑗)) ∈ ℝ)
103	98	simp1d 1140	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (1^st ‘(𝐺‘𝑗)) ∈ ℝ)
104	102, 103	resubcld 11333	. . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol‘((,)‘(𝐺‘𝑗))) ∈ ℝ)
106	105	ad2antrr 722	. . . . . . . . . . . . 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 24555	. . . . . . . . . . . . 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 1464	. . . . . . . . . . . 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 10934	. . . . . . . . . . 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 15370	. . . . . . . . . 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 2744	. . . . . . . . 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 6761	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑘)))
113	112	ineq1d 4142	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
114	113	cbvmptv 5183	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑘 ∈ ℕ ↦ (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
115		eqeq1 2742	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 = ∅ ↔ 𝑥 = ∅))
116		infeq1 9165	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → inf(𝑧, ℝ^, < ) = inf(𝑥, ℝ^, < ))
117		supeq1 9134	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → sup(𝑧, ℝ^, < ) = sup(𝑥, ℝ^, < ))
118	116, 117	opeq12d 4809	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ⟨inf(𝑧, ℝ^, < ), sup(𝑧, ℝ^, < )⟩ = ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩)
119	115, 118	ifbieq2d 4482	. . . . . . . . . . . . 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 5183	. . . . . . . . . . . 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 24652	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))
122	84, 121	sylan2 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))
123	122	adantlr 711	. . . . . . . . 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 15148	. . . . . . . 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 12280	. . . . . . . . 9 ⊢ 1 ∈ ℤ
126	1	rexuz3 14988	. . . . . . . . 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 217	. . . . . . 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 3107	. . . . . 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 13620	. . . . . . 7 ⊢ (1...𝑚) ∈ Fin
131		rexfiuz 14987	. . . . . . 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 233	. . . . 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 14988	. . . . . 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 233	. . . 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 14991	. . . 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 480	. . . . 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 480	. . . . 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 480	. . . . 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 480	. . . . 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 480	. . . . 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 480	. . . . 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 480	. . . . 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 775	. . . . 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 776	. . . . 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 2738	. . . . 5 ⊢ ∪ (((,) ∘ 𝐺) “ (1...𝑚)) = ∪ (((,) ∘ 𝐺) “ (1...𝑚))
149		simprrl 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)))) → 𝑛 ∈ ℕ)
150		simprrr 778	. . . . . 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 6761	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑧 → ((,)‘(𝐹‘𝑖)) = ((,)‘(𝐹‘𝑧)))
152	151	ineq1d 4142	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑧 → (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
153	152	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑧 → (vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))
154	153	cbvsumv 15336	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
155		2fveq3 6761	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((,)‘(𝐺‘𝑗)) = ((,)‘(𝐺‘𝑘)))
156	155	ineq2d 4143	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘))))
157	156	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
158	157	sumeq2sdv 15344	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
159	154, 158	syl5eq 2791	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
160	155	ineq1d 4142	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐺‘𝑗)) ∩ 𝐴) = (((,)‘(𝐺‘𝑘)) ∩ 𝐴))
161	160	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)) = (vol‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴)))
162	159, 161	oveq12d 7273	. . . . . . . . 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 6760	. . . . . . . 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 5080	. . . . . . 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 3372	. . . . . 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 217	. . . . 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 2738	. . . . 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 24656	. . . 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 467	. . 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 3219	. 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 3219	1 ⊢ (𝜑 → ((vol‘(𝐸 ∩ 𝐴)) + (vol‘(𝐸 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 ifcif 4456 ⟨cop 4564 ∪ cuni 4836 Disj wdisj 5035 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 dom cdm 5580 ran crn 5581 “ cima 5583 ∘ ccom 5584 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 Fincfn 8691 supcsup 9129 infcinf 9130 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 4c4 11960 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 (,)cioo 13008 [,)cico 13010 ...cfz 13168 seqcseq 13649 abscabs 14873 ⇝ cli 15121 Σcsu 15325 vol*covol 24531

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-rest 17050 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-top 21951 df-topon 21968 df-bases 22004 df-cmp 22446 df-ovol 24533 df-vol 24534

This theorem is referenced by: uniioombl 24658

W3C validator