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Theorem uniioombllem4 23400
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))
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 3866 . . . 4 (𝐾𝐴) ⊆ 𝐾
21a1i 11 . . 3 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
3 uniioombl.k . . . . . 6 𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))
4 imassrn 5512 . . . . . . 7 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
54unissi 4493 . . . . . 6 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
63, 5eqsstri 3668 . . . . 5 𝐾 ran ((,) ∘ 𝐺)
76a1i 11 . . . 4 (𝜑𝐾 ran ((,) ∘ 𝐺))
8 uniioombl.g . . . . . . 7 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
98uniiccdif 23392 . . . . . 6 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
109simpld 474 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
11 ovolficcss 23284 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
128, 11syl 17 . . . . 5 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
1310, 12sstrd 3646 . . . 4 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
147, 13sstrd 3646 . . 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 23395 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
25 ssid 3657 . . . . . 6 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
2622ovollb 23293 . . . . . 6 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
278, 25, 26sylancl 695 . . . . 5 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
28 ovollecl 23297 . . . . 5 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
2913, 24, 27, 28syl3anc 1366 . . . 4 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
30 ovolsscl 23300 . . . 4 ((𝐾 ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘𝐾) ∈ ℝ)
317, 13, 29, 30syl3anc 1366 . . 3 (𝜑 → (vol*‘𝐾) ∈ ℝ)
32 ovolsscl 23300 . . 3 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
332, 14, 31, 32syl3anc 1366 . 2 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
34 inss1 3866 . . . . 5 (𝐾𝐿) ⊆ 𝐾
3534a1i 11 . . . 4 (𝜑 → (𝐾𝐿) ⊆ 𝐾)
36 ovolsscl 23300 . . . 4 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
3735, 14, 31, 36syl3anc 1366 . . 3 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℝ)
38 ssun2 3810 . . . . . 6 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
39 nnuz 11761 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
40 uniioombl.n . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℕ)
4140peano2nnd 11075 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 + 1) ∈ ℕ)
4241, 39syl6eleq 2740 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 + 1) ∈ (ℤ‘1))
43 uzsplit 12450 . . . . . . . . . . . . . . 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 2697 . . . . . . . . . . . . 13 (𝜑 → ℕ = ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
4640nncnd 11074 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℂ)
47 ax-1cn 10032 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
48 pncan 10325 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
4946, 47, 48sylancl 695 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
5049oveq2d 6706 . . . . . . . . . . . . . 14 (𝜑 → (1...((𝑁 + 1) − 1)) = (1...𝑁))
5150uneq1d 3799 . . . . . . . . . . . . 13 (𝜑 → ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))) = ((1...𝑁) ∪ (ℤ‘(𝑁 + 1))))
5245, 51eqtrd 2685 . . . . . . . . . . . 12 (𝜑 → ℕ = ((1...𝑁) ∪ (ℤ‘(𝑁 + 1))))
5352iuneq1d 4577 . . . . . . . . . . 11 (𝜑 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)) = 𝑖 ∈ ((1...𝑁) ∪ (ℤ‘(𝑁 + 1)))((,)‘(𝐹𝑖)))
54 iunxun 4637 . . . . . . . . . . 11 𝑖 ∈ ((1...𝑁) ∪ (ℤ‘(𝑁 + 1)))((,)‘(𝐹𝑖)) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∪ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
5553, 54syl6eq 2701 . . . . . . . . . 10 (𝜑 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∪ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
56 ioof 12309 . . . . . . . . . . . . . 14 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
57 inss2 3867 . . . . . . . . . . . . . . . 16 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
58 rexpssxrxp 10122 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5957, 58sstri 3645 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
60 fss 6094 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
6115, 59, 60sylancl 695 . . . . . . . . . . . . . 14 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
62 fco 6096 . . . . . . . . . . . . . 14 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
6356, 61, 62sylancr 696 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
64 ffn 6083 . . . . . . . . . . . . 13 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
65 fniunfv 6545 . . . . . . . . . . . . 13 (((,) ∘ 𝐹) Fn ℕ → 𝑖 ∈ ℕ (((,) ∘ 𝐹)‘𝑖) = ran ((,) ∘ 𝐹))
6663, 64, 653syl 18 . . . . . . . . . . . 12 (𝜑 𝑖 ∈ ℕ (((,) ∘ 𝐹)‘𝑖) = ran ((,) ∘ 𝐹))
67 fvco3 6314 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑖 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑖) = ((,)‘(𝐹𝑖)))
6815, 67sylan 487 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑖) = ((,)‘(𝐹𝑖)))
6968iuneq2dv 4574 . . . . . . . . . . . 12 (𝜑 𝑖 ∈ ℕ (((,) ∘ 𝐹)‘𝑖) = 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
7066, 69eqtr3d 2687 . . . . . . . . . . 11 (𝜑 ran ((,) ∘ 𝐹) = 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
7118, 70syl5eq 2697 . . . . . . . . . 10 (𝜑𝐴 = 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
72 uniioombl.l . . . . . . . . . . . 12 𝐿 = (((,) ∘ 𝐹) “ (1...𝑁))
73 ffun 6086 . . . . . . . . . . . . . 14 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐹))
74 funiunfv 6546 . . . . . . . . . . . . . 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 12408 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...𝑁) → 𝑖 ∈ ℕ)
7715, 76, 67syl2an 493 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑖) = ((,)‘(𝐹𝑖)))
7877iuneq2dv 4574 . . . . . . . . . . . . 13 (𝜑 𝑖 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑖) = 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
7975, 78eqtr3d 2687 . . . . . . . . . . . 12 (𝜑 (((,) ∘ 𝐹) “ (1...𝑁)) = 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
8072, 79syl5eq 2697 . . . . . . . . . . 11 (𝜑𝐿 = 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
8180uneq1d 3799 . . . . . . . . . 10 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∪ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
8255, 71, 813eqtr4d 2695 . . . . . . . . 9 (𝜑𝐴 = (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
8382ineq2d 3847 . . . . . . . 8 (𝜑 → (𝐾𝐴) = (𝐾 ∩ (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))))
84 indi 3906 . . . . . . . 8 (𝐾 ∩ (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))) = ((𝐾𝐿) ∪ (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
8583, 84syl6eq 2701 . . . . . . 7 (𝜑 → (𝐾𝐴) = ((𝐾𝐿) ∪ (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))))
86 fss 6094 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
878, 59, 86sylancl 695 . . . . . . . . . . . . . 14 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
88 fco 6096 . . . . . . . . . . . . . 14 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
8956, 87, 88sylancr 696 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
90 ffun 6086 . . . . . . . . . . . . 13 (((,) ∘ 𝐺):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐺))
91 funiunfv 6546 . . . . . . . . . . . . 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 12408 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
94 fvco3 6314 . . . . . . . . . . . . . 14 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑗) = ((,)‘(𝐺𝑗)))
958, 93, 94syl2an 493 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (((,) ∘ 𝐺)‘𝑗) = ((,)‘(𝐺𝑗)))
9695iuneq2dv 4574 . . . . . . . . . . . 12 (𝜑 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
9792, 96eqtr3d 2687 . . . . . . . . . . 11 (𝜑 (((,) ∘ 𝐺) “ (1...𝑀)) = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
983, 97syl5eq 2697 . . . . . . . . . 10 (𝜑𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
9998ineq2d 3847 . . . . . . . . 9 (𝜑 → ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝐾) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗))))
100 incom 3838 . . . . . . . . 9 (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝐾)
101 iunin2 4616 . . . . . . . . . . . . 13 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
102 incom 3838 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖)))
103102a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (ℤ‘(𝑁 + 1)) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))))
104103iuneq2i 4571 . . . . . . . . . . . . 13 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖)))
105 incom 3838 . . . . . . . . . . . . 13 ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
106101, 104, 1053eqtr4ri 2684 . . . . . . . . . . . 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 4571 . . . . . . . . . 10 𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))
109 iunin2 4616 . . . . . . . . . 10 𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
110108, 109eqtr3i 2675 . . . . . . . . 9 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
11199, 100, 1103eqtr4g 2710 . . . . . . . 8 (𝜑 → (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
112111uneq2d 3800 . . . . . . 7 (𝜑 → ((𝐾𝐿) ∪ (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))) = ((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
11385, 112eqtrd 2685 . . . . . 6 (𝜑 → (𝐾𝐴) = ((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
11438, 113syl5sseqr 3687 . . . . 5 (𝜑 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ (𝐾𝐴))
115114, 1syl6ss 3648 . . . 4 (𝜑 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
116 ovolsscl 23300 . . . 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 1366 . . 3 (𝜑 → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
11837, 117readdcld 10107 . 2 (𝜑 → ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ∈ ℝ)
11920rpred 11910 . . 3 (𝜑𝐶 ∈ ℝ)
12037, 119readdcld 10107 . 2 (𝜑 → ((vol*‘(𝐾𝐿)) + 𝐶) ∈ ℝ)
121113fveq2d 6233 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) = (vol*‘((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
12235, 14sstrd 3646 . . . 4 (𝜑 → (𝐾𝐿) ⊆ ℝ)
123115, 14sstrd 3646 . . . 4 (𝜑 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ)
124 ovolun 23313 . . . 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 1367 . . 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 4707 . 2 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
127 fzfid 12812 . . . . 5 (𝜑 → (1...𝑀) ∈ Fin)
128 iunss 4593 . . . . . . . 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 2961 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
13114adantr 480 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → 𝐾 ⊆ ℝ)
13231adantr 480 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘𝐾) ∈ ℝ)
133 ovolsscl 23300 . . . . . 6 (( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
134130, 131, 132, 133syl3anc 1366 . . . . 5 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
135127, 134fsumrecl 14509 . . . 4 (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
136130, 131sstrd 3646 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ)
137136, 134jca 553 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → ( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ ∧ (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
138137ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ ∧ (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
139 ovolfiniun 23315 . . . . 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 694 . . . 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 11107 . . . . . . 7 (𝜑 → (𝐶 / 𝑀) ∈ ℝ)
143142adantr 480 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐶 / 𝑀) ∈ ℝ)
14480ineq2d 3847 . . . . . . . . . . . . 13 (𝜑 → (((,)‘(𝐺𝑗)) ∩ 𝐿) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖))))
145144adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖))))
146102a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑁) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))))
147146iuneq2i 4571 . . . . . . . . . . . . 13 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (1...𝑁)(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖)))
148 iunin2 4616 . . . . . . . . . . . . 13 𝑖 ∈ (1...𝑁)(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
149147, 148eqtri 2673 . . . . . . . . . . . 12 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
150145, 149syl6eqr 2703 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) = 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
151 fzfid 12812 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (1...𝑁) ∈ Fin)
152 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑖 ∈ ℕ) → (𝐹𝑖) ∈ ( ≤ ∩ (ℝ × ℝ)))
15315, 76, 152syl2an 493 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (1...𝑁)) → (𝐹𝑖) ∈ ( ≤ ∩ (ℝ × ℝ)))
15457, 153sseldi 3634 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (1...𝑁)) → (𝐹𝑖) ∈ (ℝ × ℝ))
155 1st2nd2 7249 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑖) ∈ (ℝ × ℝ) → (𝐹𝑖) = ⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩)
156154, 155syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑁)) → (𝐹𝑖) = ⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩)
157156fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) = ((,)‘⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩))
158 df-ov 6693 . . . . . . . . . . . . . . . . 17 ((1st ‘(𝐹𝑖))(,)(2nd ‘(𝐹𝑖))) = ((,)‘⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩)
159157, 158syl6eqr 2703 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) = ((1st ‘(𝐹𝑖))(,)(2nd ‘(𝐹𝑖))))
160 ioombl 23379 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑖))(,)(2nd ‘(𝐹𝑖))) ∈ dom vol
161159, 160syl6eqel 2738 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) ∈ dom vol)
162161adantlr 751 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) ∈ dom vol)
163 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
1648, 93, 163syl2an 493 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
16557, 164sseldi 3634 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ (ℝ × ℝ))
166 1st2nd2 7249 . . . . . . . . . . . . . . . . . . 19 ((𝐺𝑗) ∈ (ℝ × ℝ) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
167165, 166syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
168167fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
169 df-ov 6693 . . . . . . . . . . . . . . . . 17 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
170168, 169syl6eqr 2703 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
171 ioombl 23379 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ∈ dom vol
172170, 171syl6eqel 2738 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ∈ dom vol)
173172adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((,)‘(𝐺𝑗)) ∈ dom vol)
174 inmbl 23356 . . . . . . . . . . . . . 14 ((((,)‘(𝐹𝑖)) ∈ dom vol ∧ ((,)‘(𝐺𝑗)) ∈ dom vol) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
175162, 173, 174syl2anc 694 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
176175ralrimiva 2995 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ∀𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
177 finiunmbl 23358 . . . . . . . . . . . 12 (((1...𝑁) ∈ Fin ∧ ∀𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol) → 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
178151, 176, 177syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
179150, 178eqeltrd 2730 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) ∈ dom vol)
180 inss2 3867 . . . . . . . . . . 11 (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ 𝐴
18115uniiccdif 23392 . . . . . . . . . . . . . . 15 (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))
182181simpld 474 . . . . . . . . . . . . . 14 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
183 ovolficcss 23284 . . . . . . . . . . . . . . 15 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
18415, 183syl 17 . . . . . . . . . . . . . 14 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
185182, 184sstrd 3646 . . . . . . . . . . . . 13 (𝜑 ran ((,) ∘ 𝐹) ⊆ ℝ)
18618, 185syl5eqss 3682 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ ℝ)
187186adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → 𝐴 ⊆ ℝ)
188180, 187syl5ss 3647 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ℝ)
189 inss1 3866 . . . . . . . . . . . 12 (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ((,)‘(𝐺𝑗))
190189a1i 11 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ((,)‘(𝐺𝑗)))
191 ioossre 12273 . . . . . . . . . . . 12 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
192170, 191syl6eqss 3688 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
193170fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺𝑗))) = (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))))
194 ovolfcl 23281 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
1958, 93, 194syl2an 493 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
196 ovolioo 23382 . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺𝑗))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
199195simp2d 1094 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
200195simp1d 1093 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
201199, 200resubcld 10496 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
202198, 201eqeltrd 2730 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
203 ovolsscl 23300 . . . . . . . . . . 11 (((((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ∈ ℝ)
204190, 192, 202, 203syl3anc 1366 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ∈ ℝ)
205 mblsplit 23346 . . . . . . . . . 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 1366 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = ((vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿))) + (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿)))))
207 imassrn 5512 . . . . . . . . . . . . . . 15 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
208207unissi 4493 . . . . . . . . . . . . . 14 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
209208, 72, 183sstr4i 3677 . . . . . . . . . . . . 13 𝐿𝐴
210 sslin 3872 . . . . . . . . . . . . 13 (𝐿𝐴 → (((,)‘(𝐺𝑗)) ∩ 𝐿) ⊆ (((,)‘(𝐺𝑗)) ∩ 𝐴))
211209, 210mp1i 13 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) ⊆ (((,)‘(𝐺𝑗)) ∩ 𝐴))
212 sseqin2 3850 . . . . . . . . . . . 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 6233 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿))) = (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)))
215 indifdir 3916 . . . . . . . . . . . . 13 ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = ((𝐴 ∩ ((,)‘(𝐺𝑗))) ∖ (𝐿 ∩ ((,)‘(𝐺𝑗))))
216 incom 3838 . . . . . . . . . . . . . 14 (𝐴 ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝐴)
217 incom 3838 . . . . . . . . . . . . . 14 (𝐿 ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝐿)
218216, 217difeq12i 3759 . . . . . . . . . . . . 13 ((𝐴 ∩ ((,)‘(𝐺𝑗))) ∖ (𝐿 ∩ ((,)‘(𝐺𝑗)))) = ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿))
219215, 218eqtri 2673 . . . . . . . . . . . 12 ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿))
22082eqcomd 2657 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝐴)
22180ineq1d 3846 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
222 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑖 → (𝐹𝑥) = (𝐹𝑖))
223222fveq2d 6233 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑖 → ((,)‘(𝐹𝑥)) = ((,)‘(𝐹𝑖)))
224223cbvdisjv 4663 . . . . . . . . . . . . . . . . . . . 20 (Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)) ↔ Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
22516, 224sylib 208 . . . . . . . . . . . . . . . . . . 19 (𝜑Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
22676ssriv 3640 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ⊆ ℕ
227226a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...𝑁) ⊆ ℕ)
228 uzss 11746 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 + 1) ∈ (ℤ‘1) → (ℤ‘(𝑁 + 1)) ⊆ (ℤ‘1))
22942, 228syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (ℤ‘(𝑁 + 1)) ⊆ (ℤ‘1))
230229, 39syl6sseqr 3685 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℤ‘(𝑁 + 1)) ⊆ ℕ)
231 uzdisj 12451 . . . . . . . . . . . . . . . . . . . 20 ((1...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ∅
23250ineq1d 3846 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((1...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ((1...𝑁) ∩ (ℤ‘(𝑁 + 1))))
233231, 232syl5reqr 2700 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅)
234 disjiun 4672 . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . 18 (𝜑 → ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅)
236221, 235eqtrd 2685 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅)
237 uneqdifeq 4090 . . . . . . . . . . . . . . . . 17 ((𝐿𝐴 ∧ (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅) → ((𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝐴 ↔ (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
238209, 236, 237sylancr 696 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
241240ineq2d 3847 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ (𝐴𝐿)) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
242 incom 3838 . . . . . . . . . . . . 13 ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ (𝐴𝐿))
243104, 101eqtri 2673 . . . . . . . . . . . . 13 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
244241, 242, 2433eqtr4g 2710 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
245219, 244syl5eqr 2699 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿)) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
246245fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿))) = (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
247214, 246oveq12d 6708 . . . . . . . . 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 2685 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
249204, 143resubcld 10496 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ∈ ℝ)
250 inss2 3867 . . . . . . . . . . . . . 14 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗))
251250a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)))
252192adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
253202adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
254 ovolsscl 23300 . . . . . . . . . . . . 13 (((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
255251, 252, 253, 254syl3anc 1366 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
256151, 255fsumrecl 14509 . . . . . . . . . . 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 2961 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
259256, 204, 143absdifltd 14216 . . . . . . . . . . . . 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 474 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) < Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
262249, 256, 261ltled 10223 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ≤ Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
263150fveq2d 6233 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) = (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
264 mblvol 23344 . . . . . . . . . . . . . . . . 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 2730 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
267175, 266jca 553 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol ∧ (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
268267ralrimiva 2995 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ∀𝑖 ∈ (1...𝑁)((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol ∧ (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
269 inss1 3866 . . . . . . . . . . . . . . . 16 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐹𝑖))
270269rgenw 2953 . . . . . . . . . . . . . . 15 𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐹𝑖))
271225adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
272 disjss2 4655 . . . . . . . . . . . . . . 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 4658 . . . . . . . . . . . . . 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 23361 . . . . . . . . . . . . 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 1366 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (vol‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
278 mblvol 23344 . . . . . . . . . . . . 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 14477 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → Σ𝑖 ∈ (1...𝑁)(vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
281277, 279, 2803eqtr3d 2693 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
282263, 281eqtrd 2685 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) = Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
283262, 282breqtrrd 4713 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ≤ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)))
284282, 256eqeltrd 2730 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) ∈ ℝ)
285204, 143, 284lesubaddd 10662 . . . . . . . . 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 4709 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ≤ ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (𝐶 / 𝑀)))
288134, 143, 284leadd2d 10660 . . . . . . 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 14575 . . . . 5 (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀))
291142recnd 10106 . . . . . . 7 (𝜑 → (𝐶 / 𝑀) ∈ ℂ)
292 fsumconst 14566 . . . . . . 7 (((1...𝑀) ∈ Fin ∧ (𝐶 / 𝑀) ∈ ℂ) → Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀) = ((#‘(1...𝑀)) · (𝐶 / 𝑀)))
293127, 291, 292syl2anc 694 . . . . . 6 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀) = ((#‘(1...𝑀)) · (𝐶 / 𝑀)))
294 nnnn0 11337 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
295 hashfz1 13174 . . . . . . . 8 (𝑀 ∈ ℕ0 → (#‘(1...𝑀)) = 𝑀)
296141, 294, 2953syl 18 . . . . . . 7 (𝜑 → (#‘(1...𝑀)) = 𝑀)
297296oveq1d 6705 . . . . . 6 (𝜑 → ((#‘(1...𝑀)) · (𝐶 / 𝑀)) = (𝑀 · (𝐶 / 𝑀)))
298119recnd 10106 . . . . . . 7 (𝜑𝐶 ∈ ℂ)
299141nncnd 11074 . . . . . . 7 (𝜑𝑀 ∈ ℂ)
300141nnne0d 11103 . . . . . . 7 (𝜑𝑀 ≠ 0)
301298, 299, 300divcan2d 10841 . . . . . 6 (𝜑 → (𝑀 · (𝐶 / 𝑀)) = 𝐶)
302293, 297, 3013eqtrd 2689 . . . . 5 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀) = 𝐶)
303290, 302breqtrd 4711 . . . 4 (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ 𝐶)
304117, 135, 119, 140, 303letrd 10232 . . 3 (𝜑 → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ 𝐶)
305117, 119, 37, 304leadd2dd 10680 . 2 (𝜑 → ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
30633, 118, 120, 126, 305letrd 10232 1 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  cop 4216   cuni 4468   ciun 4552  Disj wdisj 4652   class class class wbr 4685   × cxp 5141  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Fincfn 7997  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  0cn0 11330  cuz 11725  +crp 11870  (,)cioo 12213  [,]cicc 12216  ...cfz 12364  seqcseq 12841  #chash 13157  abscabs 14018  Σcsu 14460  vol*covol 23277  volcvol 23278
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-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:  uniioombllem5  23401
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