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Theorem uniioombllem6 23402
Description: Lemma for uniioombl 23403. (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 11761 . . . 4 ℕ = (ℤ‘1)
2 1zzd 11446 . . . 4 (𝜑 → 1 ∈ ℤ)
3 uniioombl.c . . . 4 (𝜑𝐶 ∈ ℝ+)
4 eqidd 2652 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) = (𝑇𝑚))
5 uniioombl.t . . . . . 6 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
6 eqidd 2652 . . . . . 6 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) = (((abs ∘ − ) ∘ 𝐺)‘𝑎))
7 uniioombl.g . . . . . . . . . 10 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
8 eqid 2651 . . . . . . . . . . 11 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
98ovolfsf 23286 . . . . . . . . . 10 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
107, 9syl 17 . . . . . . . . 9 (𝜑 → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞))
1110ffvelrnda 6399 . . . . . . . 8 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ (0[,)+∞))
12 elrege0 12316 . . . . . . . 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 474 . . . . . 6 ((𝜑𝑎 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑎) ∈ ℝ)
1513simprd 478 . . . . . 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 23395 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
248, 5ovolsf 23287 . . . . . . . . . . . . 13 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
257, 24syl 17 . . . . . . . . . . . 12 (𝜑𝑇:ℕ⟶(0[,)+∞))
26 frn 6091 . . . . . . . . . . . 12 (𝑇:ℕ⟶(0[,)+∞) → ran 𝑇 ⊆ (0[,)+∞))
2725, 26syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
28 icossxr 12296 . . . . . . . . . . 11 (0[,)+∞) ⊆ ℝ*
2927, 28syl6ss 3648 . . . . . . . . . 10 (𝜑 → ran 𝑇 ⊆ ℝ*)
30 supxrub 12192 . . . . . . . . . 10 ((ran 𝑇 ⊆ ℝ*𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
3129, 30sylan 487 . . . . . . . . 9 ((𝜑𝑥 ∈ ran 𝑇) → 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
3231ralrimiva 2995 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < ))
33 ffn 6083 . . . . . . . . . 10 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
3425, 33syl 17 . . . . . . . . 9 (𝜑𝑇 Fn ℕ)
35 breq1 4688 . . . . . . . . . 10 (𝑥 = (𝑇𝑚) → (𝑥 ≤ sup(ran 𝑇, ℝ*, < ) ↔ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
3635ralrn 6402 . . . . . . . . 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 4689 . . . . . . . . 9 (𝑥 = sup(ran 𝑇, ℝ*, < ) → ((𝑇𝑚) ≤ 𝑥 ↔ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
4039ralbidv 3015 . . . . . . . 8 (𝑥 = sup(ran 𝑇, ℝ*, < ) → (∀𝑚 ∈ ℕ (𝑇𝑚) ≤ 𝑥 ↔ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )))
4140rspcev 3340 . . . . . . 7 ((sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ sup(ran 𝑇, ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ 𝑥)
4223, 38, 41syl2anc 694 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ ℕ (𝑇𝑚) ≤ 𝑥)
431, 5, 2, 6, 14, 15, 42isumsup2 14622 . . . . 5 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ, < ))
44 rge0ssre 12318 . . . . . . 7 (0[,)+∞) ⊆ ℝ
4527, 44syl6ss 3648 . . . . . 6 (𝜑 → ran 𝑇 ⊆ ℝ)
46 1nn 11069 . . . . . . . . 9 1 ∈ ℕ
47 fdm 6089 . . . . . . . . . 10 (𝑇:ℕ⟶(0[,)+∞) → dom 𝑇 = ℕ)
4825, 47syl 17 . . . . . . . . 9 (𝜑 → dom 𝑇 = ℕ)
4946, 48syl5eleqr 2737 . . . . . . . 8 (𝜑 → 1 ∈ dom 𝑇)
50 ne0i 3954 . . . . . . . 8 (1 ∈ dom 𝑇 → dom 𝑇 ≠ ∅)
5149, 50syl 17 . . . . . . 7 (𝜑 → dom 𝑇 ≠ ∅)
52 dm0rn0 5374 . . . . . . . 8 (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
5352necon3bii 2875 . . . . . . 7 (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
5451, 53sylib 208 . . . . . 6 (𝜑 → ran 𝑇 ≠ ∅)
55 breq2 4689 . . . . . . . . 9 (𝑦 = sup(ran 𝑇, ℝ*, < ) → (𝑥𝑦𝑥 ≤ sup(ran 𝑇, ℝ*, < )))
5655ralbidv 3015 . . . . . . . 8 (𝑦 = sup(ran 𝑇, ℝ*, < ) → (∀𝑥 ∈ ran 𝑇 𝑥𝑦 ↔ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < )))
5756rspcev 3340 . . . . . . 7 ((sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ ∀𝑥 ∈ ran 𝑇 𝑥 ≤ sup(ran 𝑇, ℝ*, < )) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦)
5823, 32, 57syl2anc 694 . . . . . 6 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦)
59 supxrre 12195 . . . . . 6 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ ran 𝑇 𝑥𝑦) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
6045, 54, 58, 59syl3anc 1366 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
6143, 60breqtrrd 4713 . . . 4 (𝜑𝑇 ⇝ sup(ran 𝑇, ℝ*, < ))
621, 2, 3, 4, 61climi2 14286 . . 3 (𝜑 → ∃𝑗 ∈ ℕ ∀𝑚 ∈ (ℤ𝑗)(abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
631r19.2uz 14135 . . 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 11446 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 1 ∈ ℤ)
663ad2antrr 762 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝐶 ∈ ℝ+)
67 simplrl 817 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℕ)
6867nnrpd 11908 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → 𝑚 ∈ ℝ+)
6966, 68rpdivcld 11927 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐶 / 𝑚) ∈ ℝ+)
70 fvex 6239 . . . . . . . . . . . . . . . 16 ((,)‘(𝐹𝑧)) ∈ V
7170inex1 4832 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V
7271rgenw 2953 . . . . . . . . . . . . . 14 𝑧 ∈ ℕ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) ∈ V
73 eqid 2651 . . . . . . . . . . . . . . 15 (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
7473fnmpt 6058 . . . . . . . . . . . . . 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 12408 . . . . . . . . . . . . 13 (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℕ)
77 fvco2 6312 . . . . . . . . . . . . 13 (((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) Fn ℕ ∧ 𝑖 ∈ ℕ) → ((vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))‘𝑖) = (vol*‘((𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))‘𝑖)))
7875, 76, 77syl2an 493 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℕ)
80 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑖 → (𝐹𝑧) = (𝐹𝑖))
8180fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑖 → ((,)‘(𝐹𝑧)) = ((,)‘(𝐹𝑖)))
8281ineq1d 3846 . . . . . . . . . . . . . . 15 (𝑧 = 𝑖 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
83 fvex 6239 . . . . . . . . . . . . . . . 16 ((,)‘(𝐹𝑖)) ∈ V
8483inex1 4832 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ V
8582, 73, 84fvmpt 6321 . . . . . . . . . . . . . 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 6233 . . . . . . . . . . . 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 2685 . . . . . . . . . . 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 476 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9089, 1syl6eleq 2740 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
91 inss2 3867 . . . . . . . . . . . . . 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 3867 . . . . . . . . . . . . . . . . . . 19 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
947adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
95 elfznn 12408 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
96 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
9794, 95, 96syl2an 493 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
9893, 97sseldi 3634 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (𝐺𝑗) ∈ (ℝ × ℝ))
99 1st2nd2 7249 . . . . . . . . . . . . . . . . . 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 6233 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
102 df-ov 6693 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
103101, 102syl6eqr 2703 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
104 ioossre 12273 . . . . . . . . . . . . . . 15 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
105103, 104syl6eqss 3688 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
106105ad2antrr 762 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
107103fveq2d 6233 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) = (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))))
108 ovolfcl 23281 . . . . . . . . . . . . . . . . . 18 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
10994, 95, 108syl2an 493 . . . . . . . . . . . . . . . . 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 23382 . . . . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
113109simp2d 1094 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
114109simp1d 1093 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
115113, 114resubcld 10496 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
116112, 115eqeltrd 2730 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
117116ad2antrr 762 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
118 ovolsscl 23300 . . . . . . . . . . . . 13 (((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
11992, 106, 117, 118syl3anc 1366 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) ∧ 𝑗 ∈ (1...𝑚)) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (1...𝑛)) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
120119recnd 10106 . . . . . . . . . . 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 14505 . . . . . . . . . 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 2657 . . . . . . . . 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 6229 . . . . . . . . . . . . . . 15 (𝑧 = 𝑘 → (𝐹𝑧) = (𝐹𝑘))
124123fveq2d 6233 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((,)‘(𝐹𝑧)) = ((,)‘(𝐹𝑘)))
125124ineq1d 3846 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑘)) ∩ ((,)‘(𝐺𝑗))))
126125cbvmptv 4783 . . . . . . . . . . . 12 (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (𝑘 ∈ ℕ ↦ (((,)‘(𝐹𝑘)) ∩ ((,)‘(𝐺𝑗))))
127 eqeq1 2655 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑧 = ∅ ↔ 𝑥 = ∅))
128 infeq1 8423 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → inf(𝑧, ℝ*, < ) = inf(𝑥, ℝ*, < ))
129 supeq1 8392 . . . . . . . . . . . . . . 15 (𝑧 = 𝑥 → sup(𝑧, ℝ*, < ) = sup(𝑥, ℝ*, < ))
130128, 129opeq12d 4441 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → ⟨inf(𝑧, ℝ*, < ), sup(𝑧, ℝ*, < )⟩ = ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩)
131127, 130ifbieq2d 4144 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → if(𝑧 = ∅, ⟨0, 0⟩, ⟨inf(𝑧, ℝ*, < ), sup(𝑧, ℝ*, < )⟩) = if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩))
132131cbvmptv 4783 . . . . . . . . . . . 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 23397 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
13495, 133sylan2 490 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑚)) → seq1( + , (vol* ∘ (𝑧 ∈ ℕ ↦ (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))) ⇝ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))
135134adantlr 751 . . . . . . . . 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 14286 . . . . . . . 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 11445 . . . . . . . . 9 1 ∈ ℤ
1381rexuz3 14132 . . . . . . . . 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 2995 . . . . . 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 12811 . . . . . . 7 (1...𝑚) ∈ Fin
143 rexfiuz 14131 . . . . . . 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 14132 . . . . . 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 14135 . . . 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 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)))
15217adantr 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)))
15320adantr 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)
1543adantr 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+)
1557adantr 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)))
15621adantr 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))
15722adantr 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))
158 simprll 819 . . . . 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 820 . . . . 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 2651 . . . . 5 (((,) ∘ 𝐺) “ (1...𝑚)) = (((,) ∘ 𝐺) “ (1...𝑚))
161 simprrl 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)))) → 𝑛 ∈ ℕ)
162 simprrr 822 . . . . . 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 6229 . . . . . . . . . . . . . . 15 (𝑖 = 𝑧 → (𝐹𝑖) = (𝐹𝑧))
164163fveq2d 6233 . . . . . . . . . . . . . 14 (𝑖 = 𝑧 → ((,)‘(𝐹𝑖)) = ((,)‘(𝐹𝑧)))
165164ineq1d 3846 . . . . . . . . . . . . 13 (𝑖 = 𝑧 → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
166165fveq2d 6233 . . . . . . . . . . . 12 (𝑖 = 𝑧 → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))))
167166cbvsumv 14470 . . . . . . . . . . 11 Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))))
168 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (𝐺𝑗) = (𝐺𝑘))
169168fveq2d 6233 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → ((,)‘(𝐺𝑗)) = ((,)‘(𝐺𝑘)))
170169ineq2d 3847 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘))))
171170fveq2d 6233 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
172171sumeq2sdv 14479 . . . . . . . . . . 11 (𝑗 = 𝑘 → Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
173167, 172syl5eq 2697 . . . . . . . . . 10 (𝑗 = 𝑘 → Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))))
174169ineq1d 3846 . . . . . . . . . . 11 (𝑗 = 𝑘 → (((,)‘(𝐺𝑗)) ∩ 𝐴) = (((,)‘(𝐺𝑘)) ∩ 𝐴))
175174fveq2d 6233 . . . . . . . . . 10 (𝑗 = 𝑘 → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))
176173, 175oveq12d 6708 . . . . . . . . 9 (𝑗 = 𝑘 → (Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴))) = (Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴))))
177176fveq2d 6233 . . . . . . . 8 (𝑗 = 𝑘 → (abs‘(Σ𝑖 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) = (abs‘(Σ𝑧 ∈ (1...𝑛)(vol*‘(((,)‘(𝐹𝑧)) ∩ ((,)‘(𝐺𝑘)))) − (vol*‘(((,)‘(𝐺𝑘)) ∩ 𝐴)))))
178177breq1d 4695 . . . . . . 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 3201 . . . . . 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 2651 . . . . 5 (((,) ∘ 𝐹) “ (1...𝑛)) = (((,) ∘ 𝐹) “ (1...𝑛))
182151, 152, 18, 19, 153, 154, 155, 156, 5, 157, 158, 159, 160, 161, 180, 181uniioombllem5 23401 . . . 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 681 . . 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 3064 . 2 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (abs‘((𝑇𝑚) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)) → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
18564, 184rexlimddv 3064 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 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  ifcif 4119  cop 4216   cuni 4468  Disj wdisj 4652   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Fincfn 7997  supcsup 8387  infcinf 8388  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  4c4 11110  cz 11415  cuz 11725  +crp 11870  (,)cioo 12213  [,)cico 12215  ...cfz 12364  seqcseq 12841  abscabs 14018  cli 14259  Σcsu 14460  vol*covol 23277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280
This theorem is referenced by:  uniioombl  23403
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