Theorem uniioombllem6 25096

Description: Lemma for uniioombl 25097. (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 12861	. . . 4 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 12589	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
3		uniioombl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
4		eqidd 2733	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = (𝑇‘𝑚))
5		uniioombl.t	. . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
6		eqidd 2733	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) = (((abs ∘ − ) ∘ 𝐺)‘𝑎))
7		uniioombl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8		eqid 2732	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
9	8	ovolfsf 24979	. . . . . . . . . 10 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
10	7, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
11	10	ffvelcdmda 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞))
12		elrege0 13427	. . . . . . . 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 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ)
15	13	simprd 496	. . . . . 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 25089	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
24	8, 5	ovolsf 24980	. . . . . . . . . . . . 13 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
25	7, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞))
26	25	frnd 6722	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
27		icossxr 13405	. . . . . . . . . . 11 ⊢ (0[,)+∞) ⊆ ℝ*
28	26, 27	sstrdi 3993	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*)
29		supxrub 13299	. . . . . . . . . 10 ⊢ ((ran 𝑇 ⊆ ℝ* ∧ 𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
30	28, 29	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
31	30	ralrimiva 3146	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
32	25	ffnd 6715	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 Fn ℕ)
33		breq1 5150	. . . . . . . . . 10 ⊢ (𝑥 = (𝑇‘𝑚) → (𝑥 ≤ sup(ran 𝑇, ℝ*, < ) ↔ (𝑇‘𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
34	33	ralrn 7086	. . . . . . . . 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 5207	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇‘𝑚) ≤ 𝑥)
39	1, 5, 2, 6, 14, 15, 38	isumsup2 15788	. . . . 5 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
40		rge0ssre 13429	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
41	26, 40	sstrdi 3993	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
42		1nn 12219	. . . . . . . . 9 ⊢ 1 ∈ ℕ
43	25	fdmd 6725	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑇 = ℕ)
44	42, 43	eleqtrrid 2840	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ dom 𝑇)
45	44	ne0d 4334	. . . . . . 7 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
46		dm0rn0 5922	. . . . . . . 8 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
47	46	necon3bii 2993	. . . . . . 7 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
48	45, 47	sylib 217	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
49		brralrspcev 5207	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ 𝑦)
51		supxrre 13302	. . . . . 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 1371	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
53	39, 52	breqtrrd 5175	. . . 4 ⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ*, < ))
54	1, 2, 3, 4, 53	climi2 15451	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
55	1	r19.2uz 15294	. . 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 12589	. . . . . . . . 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 13010	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℝ+)
61	58, 60	rpdivcld 13029	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐶 / 𝑚) ∈ ℝ+)
62		fvex 6901	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑧)) ∈ V
63	62	inex1 5316	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
64	63	rgenw 3065	. . . . . . . . . . . . . 14 ⊢ ∀𝑧 ∈ ℕ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
65		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
66	65	fnmpt 6687	. . . . . . . . . . . . . 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 13526	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
69		fvco2 6985	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 482	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
72		2fveq3 6893	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑖 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑖)))
73	72	ineq1d 4210	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑖 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))))
74		fvex 6901	. . . . . . . . . . . . . . . 16 ⊢ ((,)‘(𝐹‘𝑖)) ∈ V
75	74	inex1 5316	. . . . . . . . . . . . . . 15 ⊢ (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ∈ V
76	73, 65, 75	fvmpt 6995	. . . . . . . . . . . . . 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 6892	. . . . . . . . . . . 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 2772	. . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
81	80, 1	eleqtrdi 2843	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ≥‘1))
82		inss2 4228	. . . . . . . . . . . . 13 ⊢ (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) ⊆ ((,)‘(𝐺‘𝑗))
83	7	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
84		elfznn 13526	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
85		ffvelcdm 7080	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
86	83, 84, 85	syl2an 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
87	86	elin2d 4198	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) ∈ (ℝ × ℝ))
88		1st2nd2 8010	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑗) ∈ (ℝ × ℝ) → (𝐺‘𝑗) = ⟨(1st ‘(𝐺‘𝑗)), (2nd ‘(𝐺‘𝑗))⟩)
89	87, 88	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺‘𝑗) = ⟨(1st ‘(𝐺‘𝑗)), (2nd ‘(𝐺‘𝑗))⟩)
90	89	fveq2d 6892	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺‘𝑗)) = ((,)‘⟨(1st ‘(𝐺‘𝑗)), (2nd ‘(𝐺‘𝑗))⟩))
91		df-ov 7408	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗))) = ((,)‘⟨(1st ‘(𝐺‘𝑗)), (2nd ‘(𝐺‘𝑗))⟩)
92	90, 91	eqtr4di 2790	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺‘𝑗)) = ((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗))))
93		ioossre 13381	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗))) ⊆ ℝ
94	92, 93	eqsstrdi 4035	. . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺‘𝑗))) = (vol*‘((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗)))))
97		ovolfcl 24974	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺‘𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺‘𝑗)) ∈ ℝ ∧ (1st ‘(𝐺‘𝑗)) ≤ (2nd ‘(𝐺‘𝑗))))
98	83, 84, 97	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((1st ‘(𝐺‘𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺‘𝑗)) ∈ ℝ ∧ (1st ‘(𝐺‘𝑗)) ≤ (2nd ‘(𝐺‘𝑗))))
99		ovolioo 25076	. . . . . . . . . . . . . . . . 17 ⊢ (((1st ‘(𝐺‘𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺‘𝑗)) ∈ ℝ ∧ (1st ‘(𝐺‘𝑗)) ≤ (2nd ‘(𝐺‘𝑗))) → (vol*‘((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗)))) = ((2nd ‘(𝐺‘𝑗)) − (1st ‘(𝐺‘𝑗))))
100	98, 99	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗)))) = ((2nd ‘(𝐺‘𝑗)) − (1st ‘(𝐺‘𝑗))))
101	96, 100	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺‘𝑗))) = ((2nd ‘(𝐺‘𝑗)) − (1st ‘(𝐺‘𝑗))))
102	98	simp2d 1143	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (2nd ‘(𝐺‘𝑗)) ∈ ℝ)
103	98	simp1d 1142	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (1st ‘(𝐺‘𝑗)) ∈ ℝ)
104	102, 103	resubcld 11638	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇‘𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((2nd ‘(𝐺‘𝑗)) − (1st ‘(𝐺‘𝑗))) ∈ ℝ)
105	101, 104	eqeltrd 2833	. . . . . . . . . . . . . 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 24994	. . . . . . . . . . . . 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 1466	. . . . . . . . . . . 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 11238	. . . . . . . . . . 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 15672	. . . . . . . . . 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 2738	. . . . . . . . 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 6893	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑘)))
113	112	ineq1d 4210	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
114	113	cbvmptv 5260	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (𝑘 ∈ ℕ ↦ (((,)‘(𝐹‘𝑘)) ∩ ((,)‘(𝐺‘𝑗))))
115		eqeq1 2736	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑧 = ∅ ↔ 𝑥 = ∅))
116		infeq1 9467	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → inf(𝑧, ℝ*, < ) = inf(𝑥, ℝ*, < ))
117		supeq1 9436	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → sup(𝑧, ℝ*, < ) = sup(𝑥, ℝ*, < ))
118	116, 117	opeq12d 4880	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ⟨inf(𝑧, ℝ*, < ), sup(𝑧, ℝ*, < )⟩ = ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩)
119	115, 118	ifbieq2d 4553	. . . . . . . . . . . . 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 5260	. . . . . . . . . . . 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 25091	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)))
122	84, 121	sylan2 593	. . . . . . . . . 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 15451	. . . . . . . 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 12588	. . . . . . . . 9 ⊢ 1 ∈ ℤ
126	1	rexuz3 15291	. . . . . . . . 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 3146	. . . . . 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 13933	. . . . . . 7 ⊢ (1...𝑚) ∈ Fin
131		rexfiuz 15290	. . . . . . 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 15291	. . . . . 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 15294	. . . 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 481	. . . . 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 481	. . . . 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 481	. . . . 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 481	. . . . 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 481	. . . . 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 481	. . . . 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 481	. . . . 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 2732	. . . . 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 6893	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑧 → ((,)‘(𝐹‘𝑖)) = ((,)‘(𝐹‘𝑧)))
152	151	ineq1d 4210	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑧 → (((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
153	152	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑧 → (vol*‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))))
154	153	cbvsumv 15638	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))))
155		2fveq3 6893	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((,)‘(𝐺‘𝑗)) = ((,)‘(𝐺‘𝑘)))
156	155	ineq2d 4211	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘))))
157	156	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = (vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
158	157	sumeq2sdv 15646	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
159	154, 158	eqtrid 2784	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹‘𝑖)) ∩ ((,)‘(𝐺‘𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝑘)))))
160	155	ineq1d 4210	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (((,)‘(𝐺‘𝑗)) ∩ 𝐴) = (((,)‘(𝐺‘𝑘)) ∩ 𝐴))
161	160	fveq2d 6892	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐺‘𝑗)) ∩ 𝐴)) = (vol*‘(((,)‘(𝐺‘𝑘)) ∩ 𝐴)))
162	159, 161	oveq12d 7423	. . . . . . . . 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 6892	. . . . . . . 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 5157	. . . . . . 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 3234	. . . . . 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 2732	. . . . 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 25095	. . . 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 468	. . 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 3161	. 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 3161	1 ⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐴)) + (vol*‘(𝐸 ∖ 𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  ifcif 4527  ⟨cop 4633  ∪ cuni 4907  Disj wdisj 5112   class class class wbr 5147   ↦ cmpt 5230   × cxp 5673  dom cdm 5675  ran crn 5676   “ cima 5678   ∘ ccom 5679   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405  1^st c1st 7969  2^nd c2nd 7970  Fincfn 8935  supcsup 9431  infcinf 9432  ℝcr 11105  0cc0 11106  1c1 11107   + caddc 11109   · cmul 11111  +∞cpnf 11241  ℝ^*cxr 11243   < clt 11244   ≤ cle 11245   − cmin 11440   / cdiv 11867  ℕcn 12208  4c4 12265  ℤcz 12554  ℤ_≥cuz 12818  ℝ⁺crp 12970  (,)cioo 13320  [,)cico 13322  ...cfz 13480  seqcseq 13962  abscabs 15177   ⇝ cli 15424  Σcsu 15628  vol*covol 24970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-acn 9933  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-rlim 15429  df-sum 15629  df-rest 17364  df-topgen 17385  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-top 22387  df-topon 22404  df-bases 22440  df-cmp 22882  df-ovol 24972  df-vol 24973

This theorem is referenced by:  uniioombl  25097