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

Theorem uniioombllem6 25637
Description: Lemma for uniioombl 25638. (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 12919 . . . 4 ℕ = (ℤ‘1)
2 1zzd 12646 . . . 4 (𝜑 → 1 ∈ ℤ)
3 uniioombl.c . . . 4 (𝜑𝐶 ∈ ℝ+)
4 eqidd 2736 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) = (𝑇𝑚))
5 uniioombl.t . . . . . 6 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
6 eqidd 2736 . . . . . 6 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) = (((abs ∘ − ) ∘ 𝐺)‘𝑎))
7 uniioombl.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8 eqid 2735 . . . . . . . . . . 11 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
98ovolfsf 25520 . . . . . . . . . 10 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
107, 9syl 17 . . . . . . . . 9 (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
1110ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞))
12 elrege0 13491 . . . . . . . 8 ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
1311, 12sylib 218 . . . . . . 7 ((𝜑𝑎 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
1413simpld 494 . . . . . 6 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ)
1513simprd 495 . . . . . 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 25630 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
248, 5ovolsf 25521 . . . . . . . . . . . . 13 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
257, 24syl 17 . . . . . . . . . . . 12 (𝜑𝑇:ℕ⟶(0[,)+∞))
2625frnd 6745 . . . . . . . . . . 11 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
27 icossxr 13469 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℝ*
2826, 27sstrdi 4008 . . . . . . . . . 10 (𝜑 → ran 𝑇 ⊆ ℝ*)
29 supxrub 13363 . . . . . . . . . 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 3144 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
3225ffnd 6738 . . . . . . . . 9 (𝜑𝑇 Fn ℕ)
33 breq1 5151 . . . . . . . . . 10 (𝑥 = (𝑇𝑚) → (𝑥 ≤ sup(ran 𝑇, ℝ*, < ) ↔ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
3433ralrn 7108 . . . . . . . . 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 232 . . . . . . 7 (𝜑 → ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < ))
37 brralrspcev 5208 . . . . . . 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 15879 . . . . 5 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
40 rge0ssre 13493 . . . . . . 7 (0[,)+∞) ⊆ ℝ
4126, 40sstrdi 4008 . . . . . 6 (𝜑 → ran 𝑇 ⊆ ℝ)
42 1nn 12275 . . . . . . . . 9 1 ∈ ℕ
4325fdmd 6747 . . . . . . . . 9 (𝜑 → dom 𝑇 = ℕ)
4442, 43eleqtrrid 2846 . . . . . . . 8 (𝜑 → 1 ∈ dom 𝑇)
4544ne0d 4348 . . . . . . 7 (𝜑 → dom 𝑇 ≠ ∅)
46 dm0rn0 5938 . . . . . . . 8 (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
4746necon3bii 2991 . . . . . . 7 (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
4845, 47sylib 218 . . . . . 6 (𝜑 → ran 𝑇 ≠ ∅)
49 brralrspcev 5208 . . . . . . 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 13366 . . . . . 6 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
5241, 48, 50, 51syl3anc 1370 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
5339, 52breqtrrd 5176 . . . 4 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ*, < ))
541, 2, 3, 4, 53climi2 15544 . . 3 (𝜑 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
551r19.2uz 15387 . . 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 12646 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
583ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝐶 ∈ ℝ+)
59 simplrl 777 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℕ)
6059nnrpd 13073 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℝ+)
6158, 60rpdivcld 13092 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐶 / 𝑚) ∈ ℝ+)
62 fvex 6920 . . . . . . . . . . . . . . . 16 ((,)‘(𝐹𝑧)) ∈ V
6362inex1 5323 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V
6463rgenw 3063 . . . . . . . . . . . . . 14 𝑧 ∈ ℕ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V
65 eqid 2735 . . . . . . . . . . . . . . 15 (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
6665fnmpt 6709 . . . . . . . . . . . . . 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 13590 . . . . . . . . . . . . 13 (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
69 fvco2 7006 . . . . . . . . . . . . 13 (((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) Fn ℕ ∧ 𝑖 ∈ ℕ) → ((vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))‘𝑖) = (vol*‘((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))‘𝑖)))
7067, 68, 69syl2an 596 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
72 2fveq3 6912 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑖 → ((,)‘(𝐹𝑧)) = ((,)‘(𝐹𝑖)))
7372ineq1d 4227 . . . . . . . . . . . . . . 15 (𝑧 = 𝑖 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
74 fvex 6920 . . . . . . . . . . . . . . . 16 ((,)‘(𝐹𝑖)) ∈ V
7574inex1 5323 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ V
7673, 65, 75fvmpt 7016 . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 484 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
8180, 1eleqtrdi 2849 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
82 inss2 4246 . . . . . . . . . . . . 13 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗))
837adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
84 elfznn 13590 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
85 ffvelcdm 7101 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
8683, 84, 85syl2an 596 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
8786elin2d 4215 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) ∈ (ℝ × ℝ))
88 1st2nd2 8052 . . . . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
91 df-ov 7434 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
9290, 91eqtr4di 2793 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
93 ioossre 13445 . . . . . . . . . . . . . . 15 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
9492, 93eqsstrdi 4050 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
9594ad2antrr 726 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
9692fveq2d 6911 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) = (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))))
97 ovolfcl 25515 . . . . . . . . . . . . . . . . . 18 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
9883, 84, 97syl2an 596 . . . . . . . . . . . . . . . . 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 25617 . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
10298simp2d 1142 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
10398simp1d 1141 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
104102, 103resubcld 11689 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
105101, 104eqeltrd 2839 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
106105ad2antrr 726 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
107 ovolsscl 25535 . . . . . . . . . . . . 13 (((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
10882, 95, 106, 107mp3an2i 1465 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
109108recnd 11287 . . . . . . . . . . 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 15763 . . . . . . . . . 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 2741 . . . . . . . . 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 6912 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((,)‘(𝐹𝑧)) = ((,)‘(𝐹𝑘)))
113112ineq1d 4227 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑘)) ∩ ((,)‘(𝐺𝑗))))
114113cbvmptv 5261 . . . . . . . . . . . 12 (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (𝑘 ∈ ℕ ↦ (((,)‘(𝐹𝑘)) ∩ ((,)‘(𝐺𝑗))))
115 eqeq1 2739 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑧 = ∅ ↔ 𝑥 = ∅))
116 infeq1 9514 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → inf(𝑧, ℝ*, < ) = inf(𝑥, ℝ*, < ))
117 supeq1 9483 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → sup(𝑧, ℝ*, < ) = sup(𝑥, ℝ*, < ))
118116, 117opeq12d 4886 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → ⟨inf(𝑧, ℝ*, < ), sup(𝑧, ℝ*, < )⟩ = ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩)
119115, 118ifbieq2d 4557 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → if(𝑧 = ∅, ⟨0, 0⟩, ⟨inf(𝑧, ℝ*, < ), sup(𝑧, ℝ*, < )⟩) = if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))
120119cbvmptv 5261 . . . . . . . . . . . 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 25632 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
12284, 121sylan2 593 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑚)) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
123122adantlr 715 . . . . . . . . 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 15544 . . . . . . . 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 12645 . . . . . . . . 9 1 ∈ ℤ
1261rexuz3 15384 . . . . . . . . 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 218 . . . . . . 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 3144 . . . . . 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 14010 . . . . . . 7 (1...𝑚) ∈ Fin
131 rexfiuz 15383 . . . . . . 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 234 . . . . 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 15384 . . . . . 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 234 . . . 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 15387 . . . 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 480 . . . . 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 480 . . . . 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 480 . . . . 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 480 . . . . 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 480 . . . . 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 480 . . . . 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 480 . . . . 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 779 . . . . 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 780 . . . . 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 2735 . . . . 5 (((,) ∘ 𝐺) “ (1...𝑚)) = (((,) ∘ 𝐺) “ (1...𝑚))
149 simprrl 781 . . . . 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 782 . . . . . 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 6912 . . . . . . . . . . . . . 14 (𝑖 = 𝑧 → ((,)‘(𝐹𝑖)) = ((,)‘(𝐹𝑧)))
152151ineq1d 4227 . . . . . . . . . . . . 13 (𝑖 = 𝑧 → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
153152fveq2d 6911 . . . . . . . . . . . 12 (𝑖 = 𝑧 → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))
154153cbvsumv 15729 . . . . . . . . . . 11 Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
155 2fveq3 6912 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → ((,)‘(𝐺𝑗)) = ((,)‘(𝐺𝑘)))
156155ineq2d 4228 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘))))
157156fveq2d 6911 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
158157sumeq2sdv 15736 . . . . . . . . . . 11 (𝑗 = 𝑘 → Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
159154, 158eqtrid 2787 . . . . . . . . . 10 (𝑗 = 𝑘 → Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
160155ineq1d 4227 . . . . . . . . . . 11 (𝑗 = 𝑘 → (((,)‘(𝐺𝑗)) ∩ 𝐴) = (((,)‘(𝐺𝑘)) ∩ 𝐴))
161160fveq2d 6911 . . . . . . . . . 10 (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))
162159, 161oveq12d 7449 . . . . . . . . 9 (𝑗 = 𝑘 → (Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴))) = (Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴))))
163162fveq2d 6911 . . . . . . . 8 (𝑗 = 𝑘 → (abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) = (abs‘(Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))))
164163breq1d 5158 . . . . . . 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)))
165164cbvralvw 3235 . . . . . 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 218 . . . . 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 2735 . . . . 5 (((,) ∘ 𝐹) “ (1...𝑛)) = (((,) ∘ 𝐹) “ (1...𝑛))
168139, 140, 18, 19, 141, 142, 143, 144, 5, 145, 146, 147, 148, 149, 166, 167uniioombllem5 25636 . . . 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 467 . . 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 3159 . 2 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
17156, 170rexlimddv 3159 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cin 3962  wss 3963  c0 4339  ifcif 4531  cop 4637   cuni 4912  Disj wdisj 5115   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  cima 5692  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Fincfn 8984  supcsup 9478  infcinf 9479  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  +∞cpnf 11290  *cxr 11292   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  4c4 12321  cz 12611  cuz 12876  +crp 13032  (,)cioo 13384  [,)cico 13386  ...cfz 13544  seqcseq 14039  abscabs 15270  cli 15517  Σcsu 15719  vol*covol 25511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-acn 9980  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-rest 17469  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-top 22916  df-topon 22933  df-bases 22969  df-cmp 23411  df-ovol 25513  df-vol 25514
This theorem is referenced by:  uniioombl  25638
  Copyright terms: Public domain W3C validator