uniioombllem6 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > uniioombllem6		Structured version Visualization version GIF version

Theorem uniioombllem6 25510

Description: Lemma for uniioombl 25511. (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 12889	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12617	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
3		uniioombl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
4		eqidd 2729	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = (𝑇‘𝑚))
5		uniioombl.t	. . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
6		eqidd 2729	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) = (((abs ∘ − ) ∘ 𝐺)‘𝑎))
7		uniioombl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8		eqid 2728	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
9	8	ovolfsf 25393	. . . . . . . . . 10 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
10	7, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
11	10	ffvelcdmda 7088	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞))
12		elrege0 13457	. . . . . . . 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 25503	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
24	8, 5	ovolsf 25394	. . . . . . . . . . . . 13 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
25	7, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞))
26	25	frnd 6724	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
27		icossxr 13435	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ^*
28	26, 27	sstrdi 3990	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
29		supxrub 13329	. . . . . . . . . 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 3142	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ^*, < ))
32	25	ffnd 6717	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 Fn ℕ)
33		breq1 5145	. . . . . . . . . 10 ⊢ (𝑥 = (𝑇‘𝑚) → (𝑥 ≤ sup(ran 𝑇, ℝ^, < ) ↔ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ^, < )))
34	33	ralrn 7092	. . . . . . . . 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 5202	. . . . . . 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 15818	. . . . 5 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
40		rge0ssre 13459	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
41	26, 40	sstrdi 3990	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
42		1nn 12247	. . . . . . . . 9 ⊢ 1 ∈ ℕ
43	25	fdmd 6727	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑇 = ℕ)
44	42, 43	eleqtrrid 2836	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ dom 𝑇)
45	44	ne0d 4331	. . . . . . 7 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
46		dm0rn0 5921	. . . . . . . 8 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
47	46	necon3bii 2989	. . . . . . 7 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
48	45, 47	sylib 217	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
49		brralrspcev 5202	. . . . . . 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 13332	. . . . . 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 5170	. . . 4 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ^*, < ))
54	1, 2, 3, 4, 53	climi2 15481	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)
55	1	r19.2uz 15324	. . 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 12617	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
58	3	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝐶 ∈ ℝ⁺)
59		simplrl 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℕ)
60	59	nnrpd 13040	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℝ⁺)
61	58, 60	rpdivcld 13059	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐶 / 𝑚) ∈ ℝ⁺)
62		fvex 6904	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑧)) ∈ V
63	62	inex1 5311	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
64	63	rgenw 3061	. . . . . . . . . . . . . 14 ⊢ ∀𝑧 ∈ ℕ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
65		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
66	65	fnmpt 6689	. . . . . . . . . . . . . 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 13556	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
69		fvco2 6989	. . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑖 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑖)))
73	72	ineq1d 4207	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑖 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))))
74		fvex 6904	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑖)) ∈ V
75	74	inex1 5311	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
76	73, 65, 75	fvmpt 6999	. . . . . . . . . . . . . 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 6895	. . . . . . . . . . . 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 2768	. . . . . . . . . . 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 2839	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
82		inss2 4225	. . . . . . . . . . . . 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 13556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
85		ffvelcdm 7085	. . . . . . . . . . . . . . . . . . . 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 4195	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) ∈ (ℝ × ℝ))
88		1st2nd2 8026	. . . . . . . . . . . . . . . . . 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 6895	. . . . . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗))) = ((,)‘⟨(1^st ‘(𝐺‘𝑗)), (2^nd ‘(𝐺‘𝑗))⟩)
92	90, 91	eqtr4di 2786	. . . . . . . . . . . . . . 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 13411	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝐺‘𝑗))(,)(2^nd ‘(𝐺‘𝑗))) ⊆ ℝ
94	92, 93	eqsstrdi 4032	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺‘𝑗)) ⊆ ℝ)
95	94	ad2antrr 725	. . . . . . . . . . . . 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 6895	. . . . . . . . . . . . . . . 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 25388	. . . . . . . . . . . . . . . . . 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 25490	. . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . 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 11666	. . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol‘((,)‘(𝐺‘𝑗))) ∈ ℝ)
106	105	ad2antrr 725	. . . . . . . . . . . . 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 25408	. . . . . . . . . . . . 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 1463	. . . . . . . . . . . 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 11266	. . . . . . . . . . 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 15702	. . . . . . . . . 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 2734	. . . . . . . . 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 6896	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑘)))
113	112	ineq1d 4207	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
114	113	cbvmptv 5255	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑘 ∈ ℕ ↦ (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
115		eqeq1 2732	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 = ∅ ↔ 𝑥 = ∅))
116		infeq1 9493	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → inf(𝑧, ℝ^, < ) = inf(𝑥, ℝ^, < ))
117		supeq1 9462	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → sup(𝑧, ℝ^, < ) = sup(𝑥, ℝ^, < ))
118	116, 117	opeq12d 4877	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ⟨inf(𝑧, ℝ^, < ), sup(𝑧, ℝ^, < )⟩ = ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩)
119	115, 118	ifbieq2d 4550	. . . . . . . . . . . . 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 5255	. . . . . . . . . . . 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 25505	. . . . . . . . . . 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 714	. . . . . . . . 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 15481	. . . . . . . 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 12616	. . . . . . . . 9 ⊢ 1 ∈ ℤ
126	1	rexuz3 15321	. . . . . . . . 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 3142	. . . . . 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 13963	. . . . . . 7 ⊢ (1...𝑚) ∈ Fin
131		rexfiuz 15320	. . . . . . 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 15321	. . . . . 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 15324	. . . 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 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)))) → 𝑚 ∈ ℕ)
147		simprlr 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)))) → (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ^, < ))) < 𝐶)
148		eqid 2728	. . . . 5 ⊢ ∪ (((,) ∘ 𝐺) “ (1...𝑚)) = ∪ (((,) ∘ 𝐺) “ (1...𝑚))
149		simprrl 780	. . . . 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 781	. . . . . 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 6896	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑧 → ((,)‘(𝐹‘𝑖)) = ((,)‘(𝐹‘𝑧)))
152	151	ineq1d 4207	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑧 → (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
153	152	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑧 → (vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))
154	153	cbvsumv 15668	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
155		2fveq3 6896	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((,)‘(𝐺‘𝑗)) = ((,)‘(𝐺‘𝑘)))
156	155	ineq2d 4208	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘))))
157	156	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
158	157	sumeq2sdv 15676	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
159	154, 158	eqtrid 2780	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → Σ𝑖 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
160	155	ineq1d 4207	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐺‘𝑗)) ∩ 𝐴) = (((,)‘(𝐺‘𝑘)) ∩ 𝐴))
161	160	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (vol‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)) = (vol‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴)))
162	159, 161	oveq12d 7432	. . . . . . . . 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 6895	. . . . . . . 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 5152	. . . . . . 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 3230	. . . . . 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 2728	. . . . 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 25509	. . . 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 3157	. 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 3157	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 1534 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 ∃wrex 3066 Vcvv 3470 ∖ cdif 3942 ∩ cin 3944 ⊆ wss 3945 ∅c0 4318 ifcif 4524 ⟨cop 4630 ∪ cuni 4903 Disj wdisj 5107 class class class wbr 5142 ↦ cmpt 5225 × cxp 5670 dom cdm 5672 ran crn 5673 “ cima 5675 ∘ ccom 5676 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 Fincfn 8957 supcsup 9457 infcinf 9458 ℝcr 11131 0cc0 11132 1c1 11133 + caddc 11135 · cmul 11137 +∞cpnf 11269 ℝ^*cxr 11271 < clt 11272 ≤ cle 11273 − cmin 11468 / cdiv 11895 ℕcn 12236 4c4 12293 ℤcz 12582 ℤ_≥cuz 12846 ℝ⁺crp 13000 (,)cioo 13350 [,)cico 13352 ...cfz 13510 seqcseq 13992 abscabs 15207 ⇝ cli 15454 Σcsu 15658 vol*covol 25384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-dju 9918 df-card 9956 df-acn 9959 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-fl 13783 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-rlim 15459 df-sum 15659 df-rest 17397 df-topgen 17418 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-top 22789 df-topon 22806 df-bases 22842 df-cmp 23284 df-ovol 25386 df-vol 25387

This theorem is referenced by: uniioombl 25511

W3C validator