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

Theorem uniioombllem6 23556
Description: Lemma for uniioombl 23557. (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 11916 . . . 4 ℕ = (ℤ‘1)
2 1zzd 11600 . . . 4 (𝜑 → 1 ∈ ℤ)
3 uniioombl.c . . . 4 (𝜑𝐶 ∈ ℝ+)
4 eqidd 2761 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) = (𝑇𝑚))
5 uniioombl.t . . . . . 6 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
6 eqidd 2761 . . . . . 6 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) = (((abs ∘ − ) ∘ 𝐺)‘𝑎))
7 uniioombl.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8 eqid 2760 . . . . . . . . . . 11 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
98ovolfsf 23440 . . . . . . . . . 10 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
107, 9syl 17 . . . . . . . . 9 (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
1110ffvelrnda 6522 . . . . . . . 8 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞))
12 elrege0 12471 . . . . . . . 8 ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
1311, 12sylib 208 . . . . . . 7 ((𝜑𝑎 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐺)‘𝑎)))
1413simpld 477 . . . . . 6 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ)
1513simprd 482 . . . . . 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 23549 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
248, 5ovolsf 23441 . . . . . . . . . . . . 13 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
257, 24syl 17 . . . . . . . . . . . 12 (𝜑𝑇:ℕ⟶(0[,)+∞))
26 frn 6214 . . . . . . . . . . . 12 (𝑇:ℕ⟶(0[,)+∞) → ran 𝑇 ⊆ (0[,)+∞))
2725, 26syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
28 icossxr 12451 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℝ*
2927, 28syl6ss 3756 . . . . . . . . . 10 (𝜑 → ran 𝑇 ⊆ ℝ*)
30 supxrub 12347 . . . . . . . . . 10 ((ran 𝑇 ⊆ ℝ*𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
3129, 30sylan 489 . . . . . . . . 9 ((𝜑𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
3231ralrimiva 3104 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
33 ffn 6206 . . . . . . . . . 10 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
3425, 33syl 17 . . . . . . . . 9 (𝜑𝑇 Fn ℕ)
35 breq1 4807 . . . . . . . . . 10 (𝑥 = (𝑇𝑚) → (𝑥 ≤ sup(ran 𝑇, ℝ*, < ) ↔ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
3635ralrn 6525 . . . . . . . . 9 (𝑇 Fn ℕ → (∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < ) ↔ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
3734, 36syl 17 . . . . . . . 8 (𝜑 → (∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < ) ↔ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
3832, 37mpbid 222 . . . . . . 7 (𝜑 → ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < ))
39 breq2 4808 . . . . . . . . 9 (𝑥 = sup(ran 𝑇, ℝ*, < ) → ((𝑇𝑚) ≤ 𝑥 ↔ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
4039ralbidv 3124 . . . . . . . 8 (𝑥 = sup(ran 𝑇, ℝ*, < ) → (∀𝑚 ∈ ℕ (𝑇𝑚) ≤ 𝑥 ↔ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
4140rspcev 3449 . . . . . . 7 ((sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ 𝑥)
4223, 38, 41syl2anc 696 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ 𝑥)
431, 5, 2, 6, 14, 15, 42isumsup2 14777 . . . . 5 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
44 rge0ssre 12473 . . . . . . 7 (0[,)+∞) ⊆ ℝ
4527, 44syl6ss 3756 . . . . . 6 (𝜑 → ran 𝑇 ⊆ ℝ)
46 1nn 11223 . . . . . . . . 9 1 ∈ ℕ
47 fdm 6212 . . . . . . . . . 10 (𝑇:ℕ⟶(0[,)+∞) → dom 𝑇 = ℕ)
4825, 47syl 17 . . . . . . . . 9 (𝜑 → dom 𝑇 = ℕ)
4946, 48syl5eleqr 2846 . . . . . . . 8 (𝜑 → 1 ∈ dom 𝑇)
50 ne0i 4064 . . . . . . . 8 (1 ∈ dom 𝑇 → dom 𝑇 ≠ ∅)
5149, 50syl 17 . . . . . . 7 (𝜑 → dom 𝑇 ≠ ∅)
52 dm0rn0 5497 . . . . . . . 8 (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
5352necon3bii 2984 . . . . . . 7 (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
5451, 53sylib 208 . . . . . 6 (𝜑 → ran 𝑇 ≠ ∅)
55 breq2 4808 . . . . . . . . 9 (𝑦 = sup(ran 𝑇, ℝ*, < ) → (𝑥𝑦𝑥 ≤ sup(ran 𝑇, ℝ*, < )))
5655ralbidv 3124 . . . . . . . 8 (𝑦 = sup(ran 𝑇, ℝ*, < ) → (∀𝑥 ∈ ran 𝑇 𝑥𝑦 ↔ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < )))
5756rspcev 3449 . . . . . . 7 ((sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < )) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦)
5823, 32, 57syl2anc 696 . . . . . 6 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦)
59 supxrre 12350 . . . . . 6 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
6045, 54, 58, 59syl3anc 1477 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
6143, 60breqtrrd 4832 . . . 4 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ*, < ))
621, 2, 3, 4, 61climi2 14441 . . 3 (𝜑 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
631r19.2uz 14290 . . 3 (∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶 → ∃𝑚 ∈ ℕ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
6462, 63syl 17 . 2 (𝜑 → ∃𝑚 ∈ ℕ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
65 1zzd 11600 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
663ad2antrr 764 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝐶 ∈ ℝ+)
67 simplrl 819 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℕ)
6867nnrpd 12063 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℝ+)
6966, 68rpdivcld 12082 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐶 / 𝑚) ∈ ℝ+)
70 fvex 6362 . . . . . . . . . . . . . . . 16 ((,)‘(𝐹𝑧)) ∈ V
7170inex1 4951 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V
7271rgenw 3062 . . . . . . . . . . . . . 14 𝑧 ∈ ℕ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V
73 eqid 2760 . . . . . . . . . . . . . . 15 (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
7473fnmpt 6181 . . . . . . . . . . . . . 14 (∀𝑧 ∈ ℕ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V → (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) Fn ℕ)
7572, 74mp1i 13 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) Fn ℕ)
76 elfznn 12563 . . . . . . . . . . . . 13 (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
77 fvco2 6435 . . . . . . . . . . . . 13 (((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) Fn ℕ ∧ 𝑖 ∈ ℕ) → ((vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))‘𝑖) = (vol*‘((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))‘𝑖)))
7875, 76, 77syl2an 495 . . . . . . . . . . . 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)))
7976adantl 473 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
80 fveq2 6352 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑖 → (𝐹𝑧) = (𝐹𝑖))
8180fveq2d 6356 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑖 → ((,)‘(𝐹𝑧)) = ((,)‘(𝐹𝑖)))
8281ineq1d 3956 . . . . . . . . . . . . . . 15 (𝑧 = 𝑖 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
83 fvex 6362 . . . . . . . . . . . . . . . 16 ((,)‘(𝐹𝑖)) ∈ V
8483inex1 4951 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ V
8582, 73, 84fvmpt 6444 . . . . . . . . . . . . . 14 (𝑖 ∈ ℕ → ((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))‘𝑖) = (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
8679, 85syl 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))))
8786fveq2d 6356 . . . . . . . . . . . 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)))))
8878, 87eqtrd 2794 . . . . . . . . . . 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)))))
89 simpr 479 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9089, 1syl6eleq 2849 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
91 inss2 3977 . . . . . . . . . . . . . 14 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗))
9291a1i 11 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)))
93 inss2 3977 . . . . . . . . . . . . . . . . . . 19 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
947adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
95 elfznn 12563 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
96 ffvelrn 6520 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
9794, 95, 96syl2an 495 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
9893, 97sseldi 3742 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) ∈ (ℝ × ℝ))
99 1st2nd2 7372 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑗) ∈ (ℝ × ℝ) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
10098, 99syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
101100fveq2d 6356 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
102 df-ov 6816 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
103101, 102syl6eqr 2812 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
104 ioossre 12428 . . . . . . . . . . . . . . 15 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
105103, 104syl6eqss 3796 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
106105ad2antrr 764 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
107103fveq2d 6356 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) = (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))))
108 ovolfcl 23435 . . . . . . . . . . . . . . . . . 18 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
10994, 95, 108syl2an 495 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
110 ovolioo 23536 . . . . . . . . . . . . . . . . 17 (((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))) → (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
111109, 110syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
112107, 111eqtrd 2794 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
113109simp2d 1138 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
114109simp1d 1137 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
115113, 114resubcld 10650 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
116112, 115eqeltrd 2839 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
117116ad2antrr 764 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
118 ovolsscl 23454 . . . . . . . . . . . . 13 (((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
11992, 106, 117, 118syl3anc 1477 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
120119recnd 10260 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℂ)
12188, 90, 120fsumser 14660 . . . . . . . . . 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))
122121eqcomd 2766 . . . . . . . . 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)))))
123 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑧 = 𝑘 → (𝐹𝑧) = (𝐹𝑘))
124123fveq2d 6356 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((,)‘(𝐹𝑧)) = ((,)‘(𝐹𝑘)))
125124ineq1d 3956 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑘)) ∩ ((,)‘(𝐺𝑗))))
126125cbvmptv 4902 . . . . . . . . . . . 12 (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (𝑘 ∈ ℕ ↦ (((,)‘(𝐹𝑘)) ∩ ((,)‘(𝐺𝑗))))
127 eqeq1 2764 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑧 = ∅ ↔ 𝑥 = ∅))
128 infeq1 8547 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → inf(𝑧, ℝ*, < ) = inf(𝑥, ℝ*, < ))
129 supeq1 8516 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → sup(𝑧, ℝ*, < ) = sup(𝑥, ℝ*, < ))
130128, 129opeq12d 4561 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → ⟨inf(𝑧, ℝ*, < ), sup(𝑧, ℝ*, < )⟩ = ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩)
131127, 130ifbieq2d 4255 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → if(𝑧 = ∅, ⟨0, 0⟩, ⟨inf(𝑧, ℝ*, < ), sup(𝑧, ℝ*, < )⟩) = if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))
132131cbvmptv 4902 . . . . . . . . . . . 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*, < )⟩))
13316, 17, 18, 19, 20, 3, 7, 21, 5, 22, 126, 132uniioombllem2 23551 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
13495, 133sylan2 492 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑚)) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
135134adantlr 753 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
1361, 65, 69, 122, 135climi2 14441 . . . . . . . 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))
137 1z 11599 . . . . . . . . 9 1 ∈ ℤ
1381rexuz3 14287 . . . . . . . . 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)))
139137, 138ax-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))
140136, 139sylib 208 . . . . . . 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))
141140ralrimiva 3104 . . . . . 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))
142 fzfi 12965 . . . . . . 7 (1...𝑚) ∈ Fin
143 rexfiuz 14286 . . . . . . 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)))
144142, 143ax-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))
145141, 144sylibr 224 . . . . 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))
1461rexuz3 14287 . . . . . 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)))
147137, 146ax-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))
148145, 147sylibr 224 . . . 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))
1491r19.2uz 14290 . . . 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))
150148, 149syl 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))
15116adantr 472 . . . . 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)))
15217adantr 472 . . . . 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)))
15320adantr 472 . . . . 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)
1543adantr 472 . . . . 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+)
1557adantr 472 . . . . 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)))
15621adantr 472 . . . . 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))
15722adantr 472 . . . . 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))
158 simprll 821 . . . . 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)
159 simprlr 822 . . . . 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)
160 eqid 2760 . . . . 5 (((,) ∘ 𝐺) “ (1...𝑚)) = (((,) ∘ 𝐺) “ (1...𝑚))
161 simprrl 823 . . . . 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)
162 simprrr 824 . . . . . 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))
163 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑖 = 𝑧 → (𝐹𝑖) = (𝐹𝑧))
164163fveq2d 6356 . . . . . . . . . . . . . 14 (𝑖 = 𝑧 → ((,)‘(𝐹𝑖)) = ((,)‘(𝐹𝑧)))
165164ineq1d 3956 . . . . . . . . . . . . 13 (𝑖 = 𝑧 → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
166165fveq2d 6356 . . . . . . . . . . . 12 (𝑖 = 𝑧 → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))
167166cbvsumv 14625 . . . . . . . . . . 11 Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
168 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (𝐺𝑗) = (𝐺𝑘))
169168fveq2d 6356 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → ((,)‘(𝐺𝑗)) = ((,)‘(𝐺𝑘)))
170169ineq2d 3957 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘))))
171170fveq2d 6356 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
172171sumeq2sdv 14634 . . . . . . . . . . 11 (𝑗 = 𝑘 → Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
173167, 172syl5eq 2806 . . . . . . . . . 10 (𝑗 = 𝑘 → Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
174169ineq1d 3956 . . . . . . . . . . 11 (𝑗 = 𝑘 → (((,)‘(𝐺𝑗)) ∩ 𝐴) = (((,)‘(𝐺𝑘)) ∩ 𝐴))
175174fveq2d 6356 . . . . . . . . . 10 (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))
176173, 175oveq12d 6831 . . . . . . . . 9 (𝑗 = 𝑘 → (Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴))) = (Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴))))
177176fveq2d 6356 . . . . . . . 8 (𝑗 = 𝑘 → (abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) = (abs‘(Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))))
178177breq1d 4814 . . . . . . 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)))
179178cbvralv 3310 . . . . . 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))
180162, 179sylib 208 . . . . 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))
181 eqid 2760 . . . . 5 (((,) ∘ 𝐹) “ (1...𝑛)) = (((,) ∘ 𝐹) “ (1...𝑛))
182151, 152, 18, 19, 153, 154, 155, 156, 5, 157, 158, 159, 160, 161, 180, 181uniioombllem5 23555 . . . 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)))
183182anassrs 683 . . 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)))
184150, 183rexlimddv 3173 . 2 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
18564, 184rexlimddv 3173 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  cin 3714  wss 3715  c0 4058  ifcif 4230  cop 4327   cuni 4588  Disj wdisj 4772   class class class wbr 4804  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267  cima 5269  ccom 5270   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  Fincfn 8121  supcsup 8511  infcinf 8512  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   · cmul 10133  +∞cpnf 10263  *cxr 10265   < clt 10266  cle 10267  cmin 10458   / cdiv 10876  cn 11212  4c4 11264  cz 11569  cuz 11879  +crp 12025  (,)cioo 12368  [,)cico 12370  ...cfz 12519  seqcseq 12995  abscabs 14173  cli 14414  Σcsu 14615  vol*covol 23431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fi 8482  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-acn 8958  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-n0 11485  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ioo 12372  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-rlim 14419  df-sum 14616  df-rest 16285  df-topgen 16306  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-top 20901  df-topon 20918  df-bases 20952  df-cmp 21392  df-ovol 23433  df-vol 23434
This theorem is referenced by:  uniioombl  23557
  Copyright terms: Public domain W3C validator