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Theorem uniioombllem6 24104
Description: Lemma for uniioombl 24105. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypotheses
Ref Expression
uniioombl.1 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
uniioombl.3 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
uniioombl.a 𝐴 = ran ((,) ∘ 𝐹)
uniioombl.e (𝜑 → (vol*‘𝐸) ∈ ℝ)
uniioombl.c (𝜑𝐶 ∈ ℝ+)
uniioombl.g (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.s (𝜑𝐸 ran ((,) ∘ 𝐺))
uniioombl.t 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
uniioombl.v (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
Assertion
Ref Expression
uniioombllem6 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((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.
StepHypRef Expression
1 nnuz 12273 . . . 4 ℕ = (ℤ‘1)
2 1zzd 12005 . . . 4 (𝜑 → 1 ∈ ℤ)
3 uniioombl.c . . . 4 (𝜑𝐶 ∈ ℝ+)
4 eqidd 2825 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) = (𝑇𝑚))
5 uniioombl.t . . . . . 6 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
6 eqidd 2825 . . . . . 6 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) = (((abs ∘ − ) ∘ 𝐺)‘𝑎))
7 uniioombl.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8 eqid 2824 . . . . . . . . . . 11 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
98ovolfsf 23987 . . . . . . . . . 10 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
107, 9syl 17 . . . . . . . . 9 (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
1110ffvelrnda 6846 . . . . . . . 8 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞))
12 elrege0 12835 . . . . . . . 8 ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
1311, 12sylib 219 . . . . . . 7 ((𝜑𝑎 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
1413simpld 495 . . . . . 6 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ)
1513simprd 496 . . . . . 6 ((𝜑𝑎 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎))
16 uniioombl.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
17 uniioombl.2 . . . . . . . 8 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
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 (𝜑𝐸 ran ((,) ∘ 𝐺))
22 uniioombl.v . . . . . . . 8 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
2316, 17, 18, 19, 20, 3, 7, 21, 5, 22uniioombllem1 24097 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
248, 5ovolsf 23988 . . . . . . . . . . . . 13 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
257, 24syl 17 . . . . . . . . . . . 12 (𝜑𝑇:ℕ⟶(0[,)+∞))
2625frnd 6517 . . . . . . . . . . 11 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
27 icossxr 12814 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℝ*
2826, 27syl6ss 3982 . . . . . . . . . 10 (𝜑 → ran 𝑇 ⊆ ℝ*)
29 supxrub 12710 . . . . . . . . . 10 ((ran 𝑇 ⊆ ℝ*𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
3028, 29sylan 580 . . . . . . . . 9 ((𝜑𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
3130ralrimiva 3186 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
3225ffnd 6511 . . . . . . . . 9 (𝜑𝑇 Fn ℕ)
33 breq1 5065 . . . . . . . . . 10 (𝑥 = (𝑇𝑚) → (𝑥 ≤ sup(ran 𝑇, ℝ*, < ) ↔ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
3433ralrn 6849 . . . . . . . . 9 (𝑇 Fn ℕ → (∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < ) ↔ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
3532, 34syl 17 . . . . . . . 8 (𝜑 → (∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < ) ↔ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
3631, 35mpbid 233 . . . . . . 7 (𝜑 → ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < ))
37 brralrspcev 5122 . . . . . . 7 ((sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ 𝑥)
3823, 36, 37syl2anc 584 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ 𝑥)
391, 5, 2, 6, 14, 15, 38isumsup2 15193 . . . . 5 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
40 rge0ssre 12837 . . . . . . 7 (0[,)+∞) ⊆ ℝ
4126, 40syl6ss 3982 . . . . . 6 (𝜑 → ran 𝑇 ⊆ ℝ)
42 1nn 11641 . . . . . . . . 9 1 ∈ ℕ
4325fdmd 6519 . . . . . . . . 9 (𝜑 → dom 𝑇 = ℕ)
4442, 43eleqtrrid 2924 . . . . . . . 8 (𝜑 → 1 ∈ dom 𝑇)
4544ne0d 4304 . . . . . . 7 (𝜑 → dom 𝑇 ≠ ∅)
46 dm0rn0 5793 . . . . . . . 8 (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
4746necon3bii 3072 . . . . . . 7 (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
4845, 47sylib 219 . . . . . 6 (𝜑 → ran 𝑇 ≠ ∅)
49 brralrspcev 5122 . . . . . . 7 ((sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < )) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦)
5023, 31, 49syl2anc 584 . . . . . 6 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦)
51 supxrre 12713 . . . . . 6 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
5241, 48, 50, 51syl3anc 1365 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
5339, 52breqtrrd 5090 . . . 4 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ*, < ))
541, 2, 3, 4, 53climi2 14861 . . 3 (𝜑 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
551r19.2uz 14704 . . 3 (∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶 → ∃𝑚 ∈ ℕ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
5654, 55syl 17 . 2 (𝜑 → ∃𝑚 ∈ ℕ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
57 1zzd 12005 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
583ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝐶 ∈ ℝ+)
59 simplrl 773 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℕ)
6059nnrpd 12422 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℝ+)
6158, 60rpdivcld 12441 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐶 / 𝑚) ∈ ℝ+)
62 fvex 6679 . . . . . . . . . . . . . . . 16 ((,)‘(𝐹𝑧)) ∈ V
6362inex1 5217 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V
6463rgenw 3154 . . . . . . . . . . . . . 14 𝑧 ∈ ℕ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V
65 eqid 2824 . . . . . . . . . . . . . . 15 (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
6665fnmpt 6484 . . . . . . . . . . . . . 14 (∀𝑧 ∈ ℕ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V → (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) Fn ℕ)
6764, 66mp1i 13 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) Fn ℕ)
68 elfznn 12929 . . . . . . . . . . . . 13 (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
69 fvco2 6754 . . . . . . . . . . . . 13 (((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) Fn ℕ ∧ 𝑖 ∈ ℕ) → ((vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))‘𝑖) = (vol*‘((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))‘𝑖)))
7067, 68, 69syl2an 595 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))‘𝑖) = (vol*‘((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))‘𝑖)))
7168adantl 482 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
72 2fveq3 6671 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑖 → ((,)‘(𝐹𝑧)) = ((,)‘(𝐹𝑖)))
7372ineq1d 4191 . . . . . . . . . . . . . . 15 (𝑧 = 𝑖 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
74 fvex 6679 . . . . . . . . . . . . . . . 16 ((,)‘(𝐹𝑖)) ∈ V
7574inex1 5217 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ V
7673, 65, 75fvmpt 6764 . . . . . . . . . . . . . 14 (𝑖 ∈ ℕ → ((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))‘𝑖) = (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
7771, 76syl 17 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))‘𝑖) = (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
7877fveq2d 6670 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))‘𝑖)) = (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
7970, 78eqtrd 2860 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))‘𝑖) = (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
80 simpr 485 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
8180, 1syl6eleq 2927 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
82 inss2 4209 . . . . . . . . . . . . 13 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗))
837adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
84 elfznn 12929 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
85 ffvelrn 6844 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
8683, 84, 85syl2an 595 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
8786elin2d 4179 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) ∈ (ℝ × ℝ))
88 1st2nd2 7722 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑗) ∈ (ℝ × ℝ) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
8987, 88syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
9089fveq2d 6670 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
91 df-ov 7154 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
9290, 91syl6eqr 2878 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
93 ioossre 12791 . . . . . . . . . . . . . . 15 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
9492, 93eqsstrdi 4024 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
9594ad2antrr 722 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
9692fveq2d 6670 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) = (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))))
97 ovolfcl 23982 . . . . . . . . . . . . . . . . . 18 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
9883, 84, 97syl2an 595 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
99 ovolioo 24084 . . . . . . . . . . . . . . . . 17 (((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))) → (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
10098, 99syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
10196, 100eqtrd 2860 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
10298simp2d 1137 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
10398simp1d 1136 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
104102, 103resubcld 11060 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
105101, 104eqeltrd 2917 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
106105ad2antrr 722 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
107 ovolsscl 24002 . . . . . . . . . . . . 13 (((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
10882, 95, 106, 107mp3an2i 1459 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
109108recnd 10661 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℂ)
11079, 81, 109fsumser 15079 . . . . . . . . . 10 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))))‘𝑛))
111110eqcomd 2830 . . . . . . . . 9 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → (seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))))‘𝑛) = Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
112 2fveq3 6671 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((,)‘(𝐹𝑧)) = ((,)‘(𝐹𝑘)))
113112ineq1d 4191 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑘)) ∩ ((,)‘(𝐺𝑗))))
114113cbvmptv 5165 . . . . . . . . . . . 12 (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (𝑘 ∈ ℕ ↦ (((,)‘(𝐹𝑘)) ∩ ((,)‘(𝐺𝑗))))
115 eqeq1 2828 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑧 = ∅ ↔ 𝑥 = ∅))
116 infeq1 8932 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → inf(𝑧, ℝ*, < ) = inf(𝑥, ℝ*, < ))
117 supeq1 8901 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → sup(𝑧, ℝ*, < ) = sup(𝑥, ℝ*, < ))
118116, 117opeq12d 4809 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → ⟨inf(𝑧, ℝ*, < ), sup(𝑧, ℝ*, < )⟩ = ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩)
119115, 118ifbieq2d 4494 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → if(𝑧 = ∅, ⟨0, 0⟩, ⟨inf(𝑧, ℝ*, < ), sup(𝑧, ℝ*, < )⟩) = if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))
120119cbvmptv 5165 . . . . . . . . . . . 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*, < )⟩))
12116, 17, 18, 19, 20, 3, 7, 21, 5, 22, 114, 120uniioombllem2 24099 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
12284, 121sylan2 592 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑚)) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
123122adantlr 711 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
1241, 57, 61, 111, 123climi2 14861 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
125 1z 12004 . . . . . . . . 9 1 ∈ ℤ
1261rexuz3 14701 . . . . . . . . 9 (1 ∈ ℤ → (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))
127125, 126ax-mp 5 . . . . . . . 8 (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
128124, 127sylib 219 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
129128ralrimiva 3186 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → ∀𝑗 ∈ (1...𝑚)∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
130 fzfi 13333 . . . . . . 7 (1...𝑚) ∈ Fin
131 rexfiuz 14700 . . . . . . 7 ((1...𝑚) ∈ Fin → (∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∀𝑗 ∈ (1...𝑚)∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))
132130, 131ax-mp 5 . . . . . 6 (∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∀𝑗 ∈ (1...𝑚)∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
133129, 132sylibr 235 . . . . 5 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
1341rexuz3 14701 . . . . . 6 (1 ∈ ℤ → (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))
135125, 134ax-mp 5 . . . . 5 (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∃𝑎 ∈ ℤ ∀𝑛 ∈ (ℤ𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
136133, 135sylibr 235 . . . 4 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → ∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
1371r19.2uz 14704 . . . 4 (∃𝑎 ∈ ℕ ∀𝑛 ∈ (ℤ𝑎)∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) → ∃𝑛 ∈ ℕ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
138136, 137syl 17 . . 3 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → ∃𝑛 ∈ ℕ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
13916adantr 481 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
14017adantr 481 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
14120adantr 481 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → (vol*‘𝐸) ∈ ℝ)
1423adantr 481 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝐶 ∈ ℝ+)
1437adantr 481 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
14421adantr 481 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝐸 ran ((,) ∘ 𝐺))
14522adantr 481 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
146 simprll 775 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝑚 ∈ ℕ)
147 simprlr 776 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
148 eqid 2824 . . . . 5 (((,) ∘ 𝐺) “ (1...𝑚)) = (((,) ∘ 𝐺) “ (1...𝑚))
149 simprrl 777 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → 𝑛 ∈ ℕ)
150 simprrr 778 . . . . . 6 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))
151 2fveq3 6671 . . . . . . . . . . . . . 14 (𝑖 = 𝑧 → ((,)‘(𝐹𝑖)) = ((,)‘(𝐹𝑧)))
152151ineq1d 4191 . . . . . . . . . . . . 13 (𝑖 = 𝑧 → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
153152fveq2d 6670 . . . . . . . . . . . 12 (𝑖 = 𝑧 → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))
154153cbvsumv 15045 . . . . . . . . . . 11 Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
155 2fveq3 6671 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → ((,)‘(𝐺𝑗)) = ((,)‘(𝐺𝑘)))
156155ineq2d 4192 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘))))
157156fveq2d 6670 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
158157sumeq2sdv 15053 . . . . . . . . . . 11 (𝑗 = 𝑘 → Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
159154, 158syl5eq 2872 . . . . . . . . . 10 (𝑗 = 𝑘 → Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
160155ineq1d 4191 . . . . . . . . . . 11 (𝑗 = 𝑘 → (((,)‘(𝐺𝑗)) ∩ 𝐴) = (((,)‘(𝐺𝑘)) ∩ 𝐴))
161160fveq2d 6670 . . . . . . . . . 10 (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))
162159, 161oveq12d 7169 . . . . . . . . 9 (𝑗 = 𝑘 → (Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴))) = (Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴))))
163162fveq2d 6670 . . . . . . . 8 (𝑗 = 𝑘 → (abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) = (abs‘(Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))))
164163breq1d 5072 . . . . . . 7 (𝑗 = 𝑘 → ((abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ (abs‘(Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))) < (𝐶 / 𝑚)))
165164cbvralv 3457 . . . . . 6 (∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚) ↔ ∀𝑘 ∈ (1...𝑚)(abs‘(Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))) < (𝐶 / 𝑚))
166150, 165sylib 219 . . . . 5 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → ∀𝑘 ∈ (1...𝑚)(abs‘(Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))) < (𝐶 / 𝑚))
167 eqid 2824 . . . . 5 (((,) ∘ 𝐹) “ (1...𝑛)) = (((,) ∘ 𝐹) “ (1...𝑛))
168139, 140, 18, 19, 141, 142, 143, 144, 5, 145, 146, 147, 148, 149, 166, 167uniioombllem5 24103 . . . 4 ((𝜑 ∧ ((𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚)))) → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
169168anassrs 468 . . 3 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ (𝑛 ∈ ℕ ∧ ∀𝑗 ∈ (1...𝑚)(abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑚))) → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
170138, 169rexlimddv 3295 . 2 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
17156, 170rexlimddv 3295 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2106  wne 3020  wral 3142  wrex 3143  Vcvv 3499  cdif 3936  cin 3938  wss 3939  c0 4294  ifcif 4469  cop 4569   cuni 4836  Disj wdisj 5027   class class class wbr 5062  cmpt 5142   × cxp 5551  dom cdm 5553  ran crn 5554  cima 5556  ccom 5557   Fn wfn 6346  wf 6347  cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682  Fincfn 8501  supcsup 8896  infcinf 8897  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534  +∞cpnf 10664  *cxr 10666   < clt 10667  cle 10668  cmin 10862   / cdiv 11289  cn 11630  4c4 11686  cz 11973  cuz 12235  +crp 12382  (,)cioo 12731  [,)cico 12733  ...cfz 12885  seqcseq 13362  abscabs 14586  cli 14834  Σcsu 15035  vol*covol 23978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-disj 5028  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-acn 9363  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12383  df-xneg 12500  df-xadd 12501  df-xmul 12502  df-ioo 12735  df-ico 12737  df-icc 12738  df-fz 12886  df-fzo 13027  df-fl 13155  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-clim 14838  df-rlim 14839  df-sum 15036  df-rest 16688  df-topgen 16709  df-psmet 20453  df-xmet 20454  df-met 20455  df-bl 20456  df-mopn 20457  df-top 21418  df-topon 21435  df-bases 21470  df-cmp 21911  df-ovol 23980  df-vol 23981
This theorem is referenced by:  uniioombl  24105
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