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Theorem uniioombllem4 23574
Description: Lemma for uniioombl 23577. (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))
uniioombl.m (𝜑𝑀 ∈ ℕ)
uniioombl.m2 (𝜑 → (abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
uniioombl.k 𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))
uniioombl.n (𝜑𝑁 ∈ ℕ)
uniioombl.n2 (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
uniioombl.l 𝐿 = (((,) ∘ 𝐹) “ (1...𝑁))
Assertion
Ref Expression
uniioombllem4 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
Distinct variable groups:   𝑖,𝑗,𝑥,𝐹   𝑖,𝐺,𝑗,𝑥   𝑗,𝐾,𝑥   𝐴,𝑗,𝑥   𝐶,𝑖,𝑗,𝑥   𝑖,𝑀,𝑗,𝑥   𝑖,𝑁,𝑗   𝜑,𝑖,𝑗,𝑥   𝑇,𝑖,𝑗,𝑥
Allowed substitution hints:   𝐴(𝑖)   𝑆(𝑥,𝑖,𝑗)   𝐸(𝑥,𝑖,𝑗)   𝐾(𝑖)   𝐿(𝑥,𝑖,𝑗)   𝑁(𝑥)

Proof of Theorem uniioombllem4
StepHypRef Expression
1 inss1 3981 . . . 4 (𝐾𝐴) ⊆ 𝐾
21a1i 11 . . 3 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
3 uniioombl.k . . . . . 6 𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))
4 imassrn 5618 . . . . . . 7 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
54unissi 4597 . . . . . 6 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
63, 5eqsstri 3784 . . . . 5 𝐾 ran ((,) ∘ 𝐺)
76a1i 11 . . . 4 (𝜑𝐾 ran ((,) ∘ 𝐺))
8 uniioombl.g . . . . . . 7 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
98uniiccdif 23566 . . . . . 6 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
109simpld 482 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
11 ovolficcss 23457 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
128, 11syl 17 . . . . 5 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
1310, 12sstrd 3762 . . . 4 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
147, 13sstrd 3762 . . 3 (𝜑𝐾 ⊆ ℝ)
15 uniioombl.1 . . . . . 6 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
16 uniioombl.2 . . . . . 6 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
17 uniioombl.3 . . . . . 6 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
18 uniioombl.a . . . . . 6 𝐴 = ran ((,) ∘ 𝐹)
19 uniioombl.e . . . . . 6 (𝜑 → (vol*‘𝐸) ∈ ℝ)
20 uniioombl.c . . . . . 6 (𝜑𝐶 ∈ ℝ+)
21 uniioombl.s . . . . . 6 (𝜑𝐸 ran ((,) ∘ 𝐺))
22 uniioombl.t . . . . . 6 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
23 uniioombl.v . . . . . 6 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
2415, 16, 17, 18, 19, 20, 8, 21, 22, 23uniioombllem1 23569 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
25 ssid 3773 . . . . . 6 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
2622ovollb 23467 . . . . . 6 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
278, 25, 26sylancl 574 . . . . 5 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
28 ovollecl 23471 . . . . 5 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
2913, 24, 27, 28syl3anc 1476 . . . 4 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
30 ovolsscl 23474 . . . 4 ((𝐾 ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘𝐾) ∈ ℝ)
317, 13, 29, 30syl3anc 1476 . . 3 (𝜑 → (vol*‘𝐾) ∈ ℝ)
32 ovolsscl 23474 . . 3 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
332, 14, 31, 32syl3anc 1476 . 2 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
34 inss1 3981 . . . . 5 (𝐾𝐿) ⊆ 𝐾
3534a1i 11 . . . 4 (𝜑 → (𝐾𝐿) ⊆ 𝐾)
36 ovolsscl 23474 . . . 4 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
3735, 14, 31, 36syl3anc 1476 . . 3 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℝ)
38 ssun2 3928 . . . . . 6 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
39 nnuz 11925 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
40 uniioombl.n . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℕ)
4140peano2nnd 11239 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 + 1) ∈ ℕ)
4241, 39syl6eleq 2860 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 + 1) ∈ (ℤ‘1))
43 uzsplit 12619 . . . . . . . . . . . . . . 15 ((𝑁 + 1) ∈ (ℤ‘1) → (ℤ‘1) = ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
4442, 43syl 17 . . . . . . . . . . . . . 14 (𝜑 → (ℤ‘1) = ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
4539, 44syl5eq 2817 . . . . . . . . . . . . 13 (𝜑 → ℕ = ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
4640nncnd 11238 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℂ)
47 ax-1cn 10196 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
48 pncan 10489 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
4946, 47, 48sylancl 574 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
5049oveq2d 6809 . . . . . . . . . . . . . 14 (𝜑 → (1...((𝑁 + 1) − 1)) = (1...𝑁))
5150uneq1d 3917 . . . . . . . . . . . . 13 (𝜑 → ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))) = ((1...𝑁) ∪ (ℤ‘(𝑁 + 1))))
5245, 51eqtrd 2805 . . . . . . . . . . . 12 (𝜑 → ℕ = ((1...𝑁) ∪ (ℤ‘(𝑁 + 1))))
5352iuneq1d 4679 . . . . . . . . . . 11 (𝜑 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)) = 𝑖 ∈ ((1...𝑁) ∪ (ℤ‘(𝑁 + 1)))((,)‘(𝐹𝑖)))
54 iunxun 4739 . . . . . . . . . . 11 𝑖 ∈ ((1...𝑁) ∪ (ℤ‘(𝑁 + 1)))((,)‘(𝐹𝑖)) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∪ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
5553, 54syl6eq 2821 . . . . . . . . . 10 (𝜑 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∪ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
56 ioof 12477 . . . . . . . . . . . . . 14 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
57 inss2 3982 . . . . . . . . . . . . . . . 16 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
58 rexpssxrxp 10286 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5957, 58sstri 3761 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
60 fss 6196 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
6115, 59, 60sylancl 574 . . . . . . . . . . . . . 14 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
62 fco 6198 . . . . . . . . . . . . . 14 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
6356, 61, 62sylancr 575 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
64 ffn 6185 . . . . . . . . . . . . 13 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
65 fniunfv 6648 . . . . . . . . . . . . 13 (((,) ∘ 𝐹) Fn ℕ → 𝑖 ∈ ℕ (((,) ∘ 𝐹)‘𝑖) = ran ((,) ∘ 𝐹))
6663, 64, 653syl 18 . . . . . . . . . . . 12 (𝜑 𝑖 ∈ ℕ (((,) ∘ 𝐹)‘𝑖) = ran ((,) ∘ 𝐹))
67 fvco3 6417 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑖 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑖) = ((,)‘(𝐹𝑖)))
6815, 67sylan 569 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑖) = ((,)‘(𝐹𝑖)))
6968iuneq2dv 4676 . . . . . . . . . . . 12 (𝜑 𝑖 ∈ ℕ (((,) ∘ 𝐹)‘𝑖) = 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
7066, 69eqtr3d 2807 . . . . . . . . . . 11 (𝜑 ran ((,) ∘ 𝐹) = 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
7118, 70syl5eq 2817 . . . . . . . . . 10 (𝜑𝐴 = 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
72 uniioombl.l . . . . . . . . . . . 12 𝐿 = (((,) ∘ 𝐹) “ (1...𝑁))
73 ffun 6188 . . . . . . . . . . . . . 14 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐹))
74 funiunfv 6649 . . . . . . . . . . . . . 14 (Fun ((,) ∘ 𝐹) → 𝑖 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑖) = (((,) ∘ 𝐹) “ (1...𝑁)))
7563, 73, 743syl 18 . . . . . . . . . . . . 13 (𝜑 𝑖 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑖) = (((,) ∘ 𝐹) “ (1...𝑁)))
76 elfznn 12577 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...𝑁) → 𝑖 ∈ ℕ)
7715, 76, 67syl2an 583 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑖) = ((,)‘(𝐹𝑖)))
7877iuneq2dv 4676 . . . . . . . . . . . . 13 (𝜑 𝑖 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑖) = 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
7975, 78eqtr3d 2807 . . . . . . . . . . . 12 (𝜑 (((,) ∘ 𝐹) “ (1...𝑁)) = 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
8072, 79syl5eq 2817 . . . . . . . . . . 11 (𝜑𝐿 = 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
8180uneq1d 3917 . . . . . . . . . 10 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∪ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
8255, 71, 813eqtr4d 2815 . . . . . . . . 9 (𝜑𝐴 = (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
8382ineq2d 3965 . . . . . . . 8 (𝜑 → (𝐾𝐴) = (𝐾 ∩ (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))))
84 indi 4022 . . . . . . . 8 (𝐾 ∩ (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))) = ((𝐾𝐿) ∪ (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
8583, 84syl6eq 2821 . . . . . . 7 (𝜑 → (𝐾𝐴) = ((𝐾𝐿) ∪ (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))))
86 fss 6196 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
878, 59, 86sylancl 574 . . . . . . . . . . . . . 14 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
88 fco 6198 . . . . . . . . . . . . . 14 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
8956, 87, 88sylancr 575 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
90 ffun 6188 . . . . . . . . . . . . 13 (((,) ∘ 𝐺):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐺))
91 funiunfv 6649 . . . . . . . . . . . . 13 (Fun ((,) ∘ 𝐺) → 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = (((,) ∘ 𝐺) “ (1...𝑀)))
9289, 90, 913syl 18 . . . . . . . . . . . 12 (𝜑 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = (((,) ∘ 𝐺) “ (1...𝑀)))
93 elfznn 12577 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
94 fvco3 6417 . . . . . . . . . . . . . 14 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑗) = ((,)‘(𝐺𝑗)))
958, 93, 94syl2an 583 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (((,) ∘ 𝐺)‘𝑗) = ((,)‘(𝐺𝑗)))
9695iuneq2dv 4676 . . . . . . . . . . . 12 (𝜑 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
9792, 96eqtr3d 2807 . . . . . . . . . . 11 (𝜑 (((,) ∘ 𝐺) “ (1...𝑀)) = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
983, 97syl5eq 2817 . . . . . . . . . 10 (𝜑𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
9998ineq2d 3965 . . . . . . . . 9 (𝜑 → ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝐾) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗))))
100 incom 3956 . . . . . . . . 9 (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝐾)
101 iunin2 4718 . . . . . . . . . . . . 13 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
102 incom 3956 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖)))
103102a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (ℤ‘(𝑁 + 1)) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))))
104103iuneq2i 4673 . . . . . . . . . . . . 13 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖)))
105 incom 3956 . . . . . . . . . . . . 13 ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
106101, 104, 1053eqtr4ri 2804 . . . . . . . . . . . 12 ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))
107106a1i 11 . . . . . . . . . . 11 (𝑗 ∈ (1...𝑀) → ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
108107iuneq2i 4673 . . . . . . . . . 10 𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))
109 iunin2 4718 . . . . . . . . . 10 𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
110108, 109eqtr3i 2795 . . . . . . . . 9 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
11199, 100, 1103eqtr4g 2830 . . . . . . . 8 (𝜑 → (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
112111uneq2d 3918 . . . . . . 7 (𝜑 → ((𝐾𝐿) ∪ (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))) = ((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
11385, 112eqtrd 2805 . . . . . 6 (𝜑 → (𝐾𝐴) = ((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
11438, 113syl5sseqr 3803 . . . . 5 (𝜑 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ (𝐾𝐴))
115114, 1syl6ss 3764 . . . 4 (𝜑 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
116 ovolsscl 23474 . . . 4 (( 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
117115, 14, 31, 116syl3anc 1476 . . 3 (𝜑 → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
11837, 117readdcld 10271 . 2 (𝜑 → ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ∈ ℝ)
11920rpred 12075 . . 3 (𝜑𝐶 ∈ ℝ)
12037, 119readdcld 10271 . 2 (𝜑 → ((vol*‘(𝐾𝐿)) + 𝐶) ∈ ℝ)
121113fveq2d 6336 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) = (vol*‘((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
12235, 14sstrd 3762 . . . 4 (𝜑 → (𝐾𝐿) ⊆ ℝ)
123115, 14sstrd 3762 . . . 4 (𝜑 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ)
124 ovolun 23487 . . . 4 ((((𝐾𝐿) ⊆ ℝ ∧ (vol*‘(𝐾𝐿)) ∈ ℝ) ∧ ( 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ ∧ (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)) → (vol*‘((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ≤ ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
125122, 37, 123, 117, 124syl22anc 1477 . . 3 (𝜑 → (vol*‘((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ≤ ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
126121, 125eqbrtrd 4808 . 2 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
127 fzfid 12980 . . . . 5 (𝜑 → (1...𝑀) ∈ Fin)
128 iunss 4695 . . . . . . . 8 ( 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾 ↔ ∀𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
129115, 128sylib 208 . . . . . . 7 (𝜑 → ∀𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
130129r19.21bi 3081 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
13114adantr 466 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → 𝐾 ⊆ ℝ)
13231adantr 466 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘𝐾) ∈ ℝ)
133 ovolsscl 23474 . . . . . 6 (( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
134130, 131, 132, 133syl3anc 1476 . . . . 5 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
135127, 134fsumrecl 14673 . . . 4 (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
136130, 131sstrd 3762 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ)
137136, 134jca 501 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → ( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ ∧ (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
138137ralrimiva 3115 . . . . 5 (𝜑 → ∀𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ ∧ (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
139 ovolfiniun 23489 . . . . 5 (((1...𝑀) ∈ Fin ∧ ∀𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ ∧ (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)) → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
140127, 138, 139syl2anc 573 . . . 4 (𝜑 → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
141 uniioombl.m . . . . . . . 8 (𝜑𝑀 ∈ ℕ)
142119, 141nndivred 11271 . . . . . . 7 (𝜑 → (𝐶 / 𝑀) ∈ ℝ)
143142adantr 466 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐶 / 𝑀) ∈ ℝ)
14480ineq2d 3965 . . . . . . . . . . . . 13 (𝜑 → (((,)‘(𝐺𝑗)) ∩ 𝐿) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖))))
145144adantr 466 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖))))
146102a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑁) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))))
147146iuneq2i 4673 . . . . . . . . . . . . 13 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (1...𝑁)(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖)))
148 iunin2 4718 . . . . . . . . . . . . 13 𝑖 ∈ (1...𝑁)(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
149147, 148eqtri 2793 . . . . . . . . . . . 12 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
150145, 149syl6eqr 2823 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) = 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
151 fzfid 12980 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (1...𝑁) ∈ Fin)
152 ffvelrn 6500 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑖 ∈ ℕ) → (𝐹𝑖) ∈ ( ≤ ∩ (ℝ × ℝ)))
15315, 76, 152syl2an 583 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (1...𝑁)) → (𝐹𝑖) ∈ ( ≤ ∩ (ℝ × ℝ)))
15457, 153sseldi 3750 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (1...𝑁)) → (𝐹𝑖) ∈ (ℝ × ℝ))
155 1st2nd2 7354 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑖) ∈ (ℝ × ℝ) → (𝐹𝑖) = ⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩)
156154, 155syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑁)) → (𝐹𝑖) = ⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩)
157156fveq2d 6336 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) = ((,)‘⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩))
158 df-ov 6796 . . . . . . . . . . . . . . . . 17 ((1st ‘(𝐹𝑖))(,)(2nd ‘(𝐹𝑖))) = ((,)‘⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩)
159157, 158syl6eqr 2823 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) = ((1st ‘(𝐹𝑖))(,)(2nd ‘(𝐹𝑖))))
160 ioombl 23553 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑖))(,)(2nd ‘(𝐹𝑖))) ∈ dom vol
161159, 160syl6eqel 2858 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) ∈ dom vol)
162161adantlr 694 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) ∈ dom vol)
163 ffvelrn 6500 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
1648, 93, 163syl2an 583 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
16557, 164sseldi 3750 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ (ℝ × ℝ))
166 1st2nd2 7354 . . . . . . . . . . . . . . . . . . 19 ((𝐺𝑗) ∈ (ℝ × ℝ) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
167165, 166syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
168167fveq2d 6336 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
169 df-ov 6796 . . . . . . . . . . . . . . . . 17 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
170168, 169syl6eqr 2823 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
171 ioombl 23553 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ∈ dom vol
172170, 171syl6eqel 2858 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ∈ dom vol)
173172adantr 466 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((,)‘(𝐺𝑗)) ∈ dom vol)
174 inmbl 23530 . . . . . . . . . . . . . 14 ((((,)‘(𝐹𝑖)) ∈ dom vol ∧ ((,)‘(𝐺𝑗)) ∈ dom vol) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
175162, 173, 174syl2anc 573 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
176175ralrimiva 3115 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ∀𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
177 finiunmbl 23532 . . . . . . . . . . . 12 (((1...𝑁) ∈ Fin ∧ ∀𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol) → 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
178151, 176, 177syl2anc 573 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
179150, 178eqeltrd 2850 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) ∈ dom vol)
180 inss2 3982 . . . . . . . . . . 11 (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ 𝐴
18115uniiccdif 23566 . . . . . . . . . . . . . . 15 (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))
182181simpld 482 . . . . . . . . . . . . . 14 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
183 ovolficcss 23457 . . . . . . . . . . . . . . 15 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
18415, 183syl 17 . . . . . . . . . . . . . 14 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
185182, 184sstrd 3762 . . . . . . . . . . . . 13 (𝜑 ran ((,) ∘ 𝐹) ⊆ ℝ)
18618, 185syl5eqss 3798 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ ℝ)
187186adantr 466 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → 𝐴 ⊆ ℝ)
188180, 187syl5ss 3763 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ℝ)
189 inss1 3981 . . . . . . . . . . . 12 (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ((,)‘(𝐺𝑗))
190189a1i 11 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ((,)‘(𝐺𝑗)))
191 ioossre 12440 . . . . . . . . . . . 12 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
192170, 191syl6eqss 3804 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
193170fveq2d 6336 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺𝑗))) = (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))))
194 ovolfcl 23454 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
1958, 93, 194syl2an 583 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
196 ovolioo 23556 . . . . . . . . . . . . . 14 (((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))) → (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
197195, 196syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
198193, 197eqtrd 2805 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺𝑗))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
199195simp2d 1137 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
200195simp1d 1136 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
201199, 200resubcld 10660 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
202198, 201eqeltrd 2850 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
203 ovolsscl 23474 . . . . . . . . . . 11 (((((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ∈ ℝ)
204190, 192, 202, 203syl3anc 1476 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ∈ ℝ)
205 mblsplit 23520 . . . . . . . . . 10 (((((,)‘(𝐺𝑗)) ∩ 𝐿) ∈ dom vol ∧ (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ℝ ∧ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ∈ ℝ) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = ((vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿))) + (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿)))))
206179, 188, 204, 205syl3anc 1476 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = ((vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿))) + (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿)))))
207 imassrn 5618 . . . . . . . . . . . . . . 15 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
208207unissi 4597 . . . . . . . . . . . . . 14 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
209208, 72, 183sstr4i 3793 . . . . . . . . . . . . 13 𝐿𝐴
210 sslin 3987 . . . . . . . . . . . . 13 (𝐿𝐴 → (((,)‘(𝐺𝑗)) ∩ 𝐿) ⊆ (((,)‘(𝐺𝑗)) ∩ 𝐴))
211209, 210mp1i 13 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) ⊆ (((,)‘(𝐺𝑗)) ∩ 𝐴))
212 sseqin2 3968 . . . . . . . . . . . 12 ((((,)‘(𝐺𝑗)) ∩ 𝐿) ⊆ (((,)‘(𝐺𝑗)) ∩ 𝐴) ↔ ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿)) = (((,)‘(𝐺𝑗)) ∩ 𝐿))
213211, 212sylib 208 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿)) = (((,)‘(𝐺𝑗)) ∩ 𝐿))
214213fveq2d 6336 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿))) = (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)))
215 indifdir 4032 . . . . . . . . . . . . 13 ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = ((𝐴 ∩ ((,)‘(𝐺𝑗))) ∖ (𝐿 ∩ ((,)‘(𝐺𝑗))))
216 incom 3956 . . . . . . . . . . . . . 14 (𝐴 ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝐴)
217 incom 3956 . . . . . . . . . . . . . 14 (𝐿 ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝐿)
218216, 217difeq12i 3877 . . . . . . . . . . . . 13 ((𝐴 ∩ ((,)‘(𝐺𝑗))) ∖ (𝐿 ∩ ((,)‘(𝐺𝑗)))) = ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿))
219215, 218eqtri 2793 . . . . . . . . . . . 12 ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿))
22082eqcomd 2777 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝐴)
22180ineq1d 3964 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
222 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑖 → (𝐹𝑥) = (𝐹𝑖))
223222fveq2d 6336 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑖 → ((,)‘(𝐹𝑥)) = ((,)‘(𝐹𝑖)))
224223cbvdisjv 4765 . . . . . . . . . . . . . . . . . . . 20 (Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)) ↔ Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
22516, 224sylib 208 . . . . . . . . . . . . . . . . . . 19 (𝜑Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
22676ssriv 3756 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ⊆ ℕ
227226a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...𝑁) ⊆ ℕ)
228 uzss 11909 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 + 1) ∈ (ℤ‘1) → (ℤ‘(𝑁 + 1)) ⊆ (ℤ‘1))
22942, 228syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (ℤ‘(𝑁 + 1)) ⊆ (ℤ‘1))
230229, 39syl6sseqr 3801 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℤ‘(𝑁 + 1)) ⊆ ℕ)
231 uzdisj 12620 . . . . . . . . . . . . . . . . . . . 20 ((1...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ∅
23250ineq1d 3964 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((1...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ((1...𝑁) ∩ (ℤ‘(𝑁 + 1))))
233231, 232syl5reqr 2820 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅)
234 disjiun 4774 . . . . . . . . . . . . . . . . . . 19 ((Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)) ∧ ((1...𝑁) ⊆ ℕ ∧ (ℤ‘(𝑁 + 1)) ⊆ ℕ ∧ ((1...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅)) → ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅)
235225, 227, 230, 233, 234syl13anc 1478 . . . . . . . . . . . . . . . . . 18 (𝜑 → ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅)
236221, 235eqtrd 2805 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅)
237 uneqdifeq 4199 . . . . . . . . . . . . . . . . 17 ((𝐿𝐴 ∧ (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅) → ((𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝐴 ↔ (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
238209, 236, 237sylancr 575 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝐴 ↔ (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
239220, 238mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
240239adantr 466 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
241240ineq2d 3965 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ (𝐴𝐿)) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
242 incom 3956 . . . . . . . . . . . . 13 ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ (𝐴𝐿))
243104, 101eqtri 2793 . . . . . . . . . . . . 13 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
244241, 242, 2433eqtr4g 2830 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
245219, 244syl5eqr 2819 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿)) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
246245fveq2d 6336 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿))) = (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
247214, 246oveq12d 6811 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿))) + (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿)))) = ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
248206, 247eqtrd 2805 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
249204, 143resubcld 10660 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ∈ ℝ)
250 inss2 3982 . . . . . . . . . . . . . 14 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗))
251250a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)))
252192adantr 466 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
253202adantr 466 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
254 ovolsscl 23474 . . . . . . . . . . . . 13 (((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
255251, 252, 253, 254syl3anc 1476 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
256151, 255fsumrecl 14673 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
257 uniioombl.n2 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
258257r19.21bi 3081 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
259256, 204, 143absdifltd 14380 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ((abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀) ↔ (((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) < Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∧ Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) < ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) + (𝐶 / 𝑀)))))
260258, 259mpbid 222 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) < Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∧ Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) < ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) + (𝐶 / 𝑀))))
261260simpld 482 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) < Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
262249, 256, 261ltled 10387 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ≤ Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
263150fveq2d 6336 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) = (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
264 mblvol 23518 . . . . . . . . . . . . . . . . 17 ((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol → (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
265175, 264syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
266265, 255eqeltrd 2850 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
267175, 266jca 501 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol ∧ (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
268267ralrimiva 3115 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ∀𝑖 ∈ (1...𝑁)((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol ∧ (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
269 inss1 3981 . . . . . . . . . . . . . . . 16 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐹𝑖))
270269rgenw 3073 . . . . . . . . . . . . . . 15 𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐹𝑖))
271225adantr 466 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
272 disjss2 4757 . . . . . . . . . . . . . . 15 (∀𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐹𝑖)) → (Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)) → Disj 𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
273270, 271, 272mpsyl 68 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → Disj 𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
274 disjss1 4760 . . . . . . . . . . . . . 14 ((1...𝑁) ⊆ ℕ → (Disj 𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) → Disj 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
275226, 273, 274mpsyl 68 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → Disj 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
276 volfiniun 23535 . . . . . . . . . . . . 13 (((1...𝑁) ∈ Fin ∧ ∀𝑖 ∈ (1...𝑁)((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol ∧ (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ) ∧ Disj 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) → (vol‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
277151, 268, 275, 276syl3anc 1476 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (vol‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
278 mblvol 23518 . . . . . . . . . . . . 13 ( 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol → (vol‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
279178, 278syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (vol‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
280265sumeq2dv 14641 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → Σ𝑖 ∈ (1...𝑁)(vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
281277, 279, 2803eqtr3d 2813 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
282263, 281eqtrd 2805 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) = Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
283262, 282breqtrrd 4814 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ≤ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)))
284282, 256eqeltrd 2850 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) ∈ ℝ)
285204, 143, 284lesubaddd 10826 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → (((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ≤ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) ↔ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ≤ ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (𝐶 / 𝑀))))
286283, 285mpbid 222 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ≤ ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (𝐶 / 𝑀)))
287248, 286eqbrtrrd 4810 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ≤ ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (𝐶 / 𝑀)))
288134, 143, 284leadd2d 10824 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ (𝐶 / 𝑀) ↔ ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ≤ ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (𝐶 / 𝑀))))
289287, 288mpbird 247 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ (𝐶 / 𝑀))
290127, 134, 143, 289fsumle 14738 . . . . 5 (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀))
291142recnd 10270 . . . . . . 7 (𝜑 → (𝐶 / 𝑀) ∈ ℂ)
292 fsumconst 14729 . . . . . . 7 (((1...𝑀) ∈ Fin ∧ (𝐶 / 𝑀) ∈ ℂ) → Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀) = ((♯‘(1...𝑀)) · (𝐶 / 𝑀)))
293127, 291, 292syl2anc 573 . . . . . 6 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀) = ((♯‘(1...𝑀)) · (𝐶 / 𝑀)))
294 nnnn0 11501 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
295 hashfz1 13338 . . . . . . . 8 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
296141, 294, 2953syl 18 . . . . . . 7 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
297296oveq1d 6808 . . . . . 6 (𝜑 → ((♯‘(1...𝑀)) · (𝐶 / 𝑀)) = (𝑀 · (𝐶 / 𝑀)))
298119recnd 10270 . . . . . . 7 (𝜑𝐶 ∈ ℂ)
299141nncnd 11238 . . . . . . 7 (𝜑𝑀 ∈ ℂ)
300141nnne0d 11267 . . . . . . 7 (𝜑𝑀 ≠ 0)
301298, 299, 300divcan2d 11005 . . . . . 6 (𝜑 → (𝑀 · (𝐶 / 𝑀)) = 𝐶)
302293, 297, 3013eqtrd 2809 . . . . 5 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀) = 𝐶)
303290, 302breqtrd 4812 . . . 4 (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ 𝐶)
304117, 135, 119, 140, 303letrd 10396 . . 3 (𝜑 → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ 𝐶)
305117, 119, 37, 304leadd2dd 10844 . 2 (𝜑 → ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
30633, 118, 120, 126, 305letrd 10396 1 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  cop 4322   cuni 4574   ciun 4654  Disj wdisj 4754   class class class wbr 4786   × cxp 5247  dom cdm 5249  ran crn 5250  cima 5252  ccom 5253  Fun wfun 6025   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  Fincfn 8109  supcsup 8502  cc 10136  cr 10137  0cc0 10138  1c1 10139   + caddc 10141   · cmul 10143  *cxr 10275   < clt 10276  cle 10277  cmin 10468   / cdiv 10886  cn 11222  0cn0 11494  cuz 11888  +crp 12035  (,)cioo 12380  [,]cicc 12383  ...cfz 12533  seqcseq 13008  chash 13321  abscabs 14182  Σcsu 14624  vol*covol 23450  volcvol 23451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-disj 4755  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-acn 8968  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-rlim 14428  df-sum 14625  df-rest 16291  df-topgen 16312  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-top 20919  df-topon 20936  df-bases 20971  df-cmp 21411  df-ovol 23452  df-vol 23453
This theorem is referenced by:  uniioombllem5  23575
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