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Theorem uniioombllem4 24856
Description: Lemma for uniioombl 24859. (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 4175 . . 3 (𝐾𝐴) ⊆ 𝐾
2 uniioombl.k . . . . 5 𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))
3 imassrn 6010 . . . . . 6 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
43unissi 4861 . . . . 5 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
52, 4eqsstri 3966 . . . 4 𝐾 ran ((,) ∘ 𝐺)
6 uniioombl.g . . . . . . 7 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
76uniiccdif 24848 . . . . . 6 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
87simpld 495 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
9 ovolficcss 24739 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
106, 9syl 17 . . . . 5 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
118, 10sstrd 3942 . . . 4 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
125, 11sstrid 3943 . . 3 (𝜑𝐾 ⊆ ℝ)
13 uniioombl.1 . . . . . 6 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
14 uniioombl.2 . . . . . 6 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
15 uniioombl.3 . . . . . 6 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
16 uniioombl.a . . . . . 6 𝐴 = ran ((,) ∘ 𝐹)
17 uniioombl.e . . . . . 6 (𝜑 → (vol*‘𝐸) ∈ ℝ)
18 uniioombl.c . . . . . 6 (𝜑𝐶 ∈ ℝ+)
19 uniioombl.s . . . . . 6 (𝜑𝐸 ran ((,) ∘ 𝐺))
20 uniioombl.t . . . . . 6 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
21 uniioombl.v . . . . . 6 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
2213, 14, 15, 16, 17, 18, 6, 19, 20, 21uniioombllem1 24851 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
23 ssid 3954 . . . . . 6 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
2420ovollb 24749 . . . . . 6 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
256, 23, 24sylancl 586 . . . . 5 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
26 ovollecl 24753 . . . . 5 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
2711, 22, 25, 26syl3anc 1370 . . . 4 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
28 ovolsscl 24756 . . . 4 ((𝐾 ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘𝐾) ∈ ℝ)
295, 11, 27, 28mp3an2i 1465 . . 3 (𝜑 → (vol*‘𝐾) ∈ ℝ)
30 ovolsscl 24756 . . 3 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
311, 12, 29, 30mp3an2i 1465 . 2 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
32 inss1 4175 . . . 4 (𝐾𝐿) ⊆ 𝐾
33 ovolsscl 24756 . . . 4 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
3432, 12, 29, 33mp3an2i 1465 . . 3 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℝ)
35 ssun2 4120 . . . . . 6 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
36 nnuz 12722 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
37 uniioombl.n . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℕ)
3837peano2nnd 12091 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 + 1) ∈ ℕ)
3938, 36eleqtrdi 2847 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 + 1) ∈ (ℤ‘1))
40 uzsplit 13429 . . . . . . . . . . . . . . 15 ((𝑁 + 1) ∈ (ℤ‘1) → (ℤ‘1) = ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
4139, 40syl 17 . . . . . . . . . . . . . 14 (𝜑 → (ℤ‘1) = ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
4236, 41eqtrid 2788 . . . . . . . . . . . . 13 (𝜑 → ℕ = ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
4337nncnd 12090 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℂ)
44 ax-1cn 11030 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
45 pncan 11328 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
4643, 44, 45sylancl 586 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
4746oveq2d 7353 . . . . . . . . . . . . . 14 (𝜑 → (1...((𝑁 + 1) − 1)) = (1...𝑁))
4847uneq1d 4109 . . . . . . . . . . . . 13 (𝜑 → ((1...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))) = ((1...𝑁) ∪ (ℤ‘(𝑁 + 1))))
4942, 48eqtrd 2776 . . . . . . . . . . . 12 (𝜑 → ℕ = ((1...𝑁) ∪ (ℤ‘(𝑁 + 1))))
5049iuneq1d 4968 . . . . . . . . . . 11 (𝜑 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)) = 𝑖 ∈ ((1...𝑁) ∪ (ℤ‘(𝑁 + 1)))((,)‘(𝐹𝑖)))
51 iunxun 5041 . . . . . . . . . . 11 𝑖 ∈ ((1...𝑁) ∪ (ℤ‘(𝑁 + 1)))((,)‘(𝐹𝑖)) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∪ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
5250, 51eqtrdi 2792 . . . . . . . . . 10 (𝜑 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∪ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
53 ioof 13280 . . . . . . . . . . . . . 14 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
54 inss2 4176 . . . . . . . . . . . . . . . 16 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
55 rexpssxrxp 11121 . . . . . . . . . . . . . . . 16 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5654, 55sstri 3941 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
57 fss 6668 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
5813, 56, 57sylancl 586 . . . . . . . . . . . . . 14 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
59 fco 6675 . . . . . . . . . . . . . 14 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
6053, 58, 59sylancr 587 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
61 ffn 6651 . . . . . . . . . . . . 13 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
62 fniunfv 7176 . . . . . . . . . . . . 13 (((,) ∘ 𝐹) Fn ℕ → 𝑖 ∈ ℕ (((,) ∘ 𝐹)‘𝑖) = ran ((,) ∘ 𝐹))
6360, 61, 623syl 18 . . . . . . . . . . . 12 (𝜑 𝑖 ∈ ℕ (((,) ∘ 𝐹)‘𝑖) = ran ((,) ∘ 𝐹))
64 fvco3 6923 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑖 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑖) = ((,)‘(𝐹𝑖)))
6513, 64sylan 580 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑖) = ((,)‘(𝐹𝑖)))
6665iuneq2dv 4965 . . . . . . . . . . . 12 (𝜑 𝑖 ∈ ℕ (((,) ∘ 𝐹)‘𝑖) = 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
6763, 66eqtr3d 2778 . . . . . . . . . . 11 (𝜑 ran ((,) ∘ 𝐹) = 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
6816, 67eqtrid 2788 . . . . . . . . . 10 (𝜑𝐴 = 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
69 uniioombl.l . . . . . . . . . . . 12 𝐿 = (((,) ∘ 𝐹) “ (1...𝑁))
70 ffun 6654 . . . . . . . . . . . . . 14 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐹))
71 funiunfv 7177 . . . . . . . . . . . . . 14 (Fun ((,) ∘ 𝐹) → 𝑖 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑖) = (((,) ∘ 𝐹) “ (1...𝑁)))
7260, 70, 713syl 18 . . . . . . . . . . . . 13 (𝜑 𝑖 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑖) = (((,) ∘ 𝐹) “ (1...𝑁)))
73 elfznn 13386 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...𝑁) → 𝑖 ∈ ℕ)
7413, 73, 64syl2an 596 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑖) = ((,)‘(𝐹𝑖)))
7574iuneq2dv 4965 . . . . . . . . . . . . 13 (𝜑 𝑖 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑖) = 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
7672, 75eqtr3d 2778 . . . . . . . . . . . 12 (𝜑 (((,) ∘ 𝐹) “ (1...𝑁)) = 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
7769, 76eqtrid 2788 . . . . . . . . . . 11 (𝜑𝐿 = 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
7877uneq1d 4109 . . . . . . . . . 10 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∪ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
7952, 68, 783eqtr4d 2786 . . . . . . . . 9 (𝜑𝐴 = (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
8079ineq2d 4159 . . . . . . . 8 (𝜑 → (𝐾𝐴) = (𝐾 ∩ (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))))
81 indi 4220 . . . . . . . 8 (𝐾 ∩ (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))) = ((𝐾𝐿) ∪ (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
8280, 81eqtrdi 2792 . . . . . . 7 (𝜑 → (𝐾𝐴) = ((𝐾𝐿) ∪ (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))))
83 fss 6668 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
846, 56, 83sylancl 586 . . . . . . . . . . . . . 14 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
85 fco 6675 . . . . . . . . . . . . . 14 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
8653, 84, 85sylancr 587 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
87 ffun 6654 . . . . . . . . . . . . 13 (((,) ∘ 𝐺):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐺))
88 funiunfv 7177 . . . . . . . . . . . . 13 (Fun ((,) ∘ 𝐺) → 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = (((,) ∘ 𝐺) “ (1...𝑀)))
8986, 87, 883syl 18 . . . . . . . . . . . 12 (𝜑 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = (((,) ∘ 𝐺) “ (1...𝑀)))
90 elfznn 13386 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
91 fvco3 6923 . . . . . . . . . . . . . 14 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑗) = ((,)‘(𝐺𝑗)))
926, 90, 91syl2an 596 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (((,) ∘ 𝐺)‘𝑗) = ((,)‘(𝐺𝑗)))
9392iuneq2dv 4965 . . . . . . . . . . . 12 (𝜑 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
9489, 93eqtr3d 2778 . . . . . . . . . . 11 (𝜑 (((,) ∘ 𝐺) “ (1...𝑀)) = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
952, 94eqtrid 2788 . . . . . . . . . 10 (𝜑𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
9695ineq2d 4159 . . . . . . . . 9 (𝜑 → ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝐾) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗))))
97 incom 4148 . . . . . . . . 9 (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝐾)
98 iunin2 5018 . . . . . . . . . . . . 13 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
99 incom 4148 . . . . . . . . . . . . . . 15 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖)))
10099a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (ℤ‘(𝑁 + 1)) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))))
101100iuneq2i 4962 . . . . . . . . . . . . 13 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖)))
102 incom 4148 . . . . . . . . . . . . 13 ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
10398, 101, 1023eqtr4ri 2775 . . . . . . . . . . . 12 ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))
104103a1i 11 . . . . . . . . . . 11 (𝑗 ∈ (1...𝑀) → ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
105104iuneq2i 4962 . . . . . . . . . 10 𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))
106 iunin2 5018 . . . . . . . . . 10 𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
107105, 106eqtr3i 2766 . . . . . . . . 9 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = ( 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)) ∩ 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
10896, 97, 1073eqtr4g 2801 . . . . . . . 8 (𝜑 → (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
109108uneq2d 4110 . . . . . . 7 (𝜑 → ((𝐾𝐿) ∪ (𝐾 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))) = ((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
11082, 109eqtrd 2776 . . . . . 6 (𝜑 → (𝐾𝐴) = ((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
11135, 110sseqtrrid 3985 . . . . 5 (𝜑 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ (𝐾𝐴))
112111, 1sstrdi 3944 . . . 4 (𝜑 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
113 ovolsscl 24756 . . . 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)
114112, 12, 29, 113syl3anc 1370 . . 3 (𝜑 → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
11534, 114readdcld 11105 . 2 (𝜑 → ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ∈ ℝ)
11618rpred 12873 . . 3 (𝜑𝐶 ∈ ℝ)
11734, 116readdcld 11105 . 2 (𝜑 → ((vol*‘(𝐾𝐿)) + 𝐶) ∈ ℝ)
118110fveq2d 6829 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) = (vol*‘((𝐾𝐿) ∪ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
11932, 12sstrid 3943 . . . 4 (𝜑 → (𝐾𝐿) ⊆ ℝ)
120112, 12sstrd 3942 . . . 4 (𝜑 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ)
121 ovolun 24769 . . . 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))))))
122119, 34, 120, 114, 121syl22anc 836 . . 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))))))
123118, 122eqbrtrd 5114 . 2 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
124 fzfid 13794 . . . . 5 (𝜑 → (1...𝑀) ∈ Fin)
125 iunss 4992 . . . . . . . 8 ( 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾 ↔ ∀𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
126112, 125sylib 217 . . . . . . 7 (𝜑 → ∀𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
127126r19.21bi 3230 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾)
12812adantr 481 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → 𝐾 ⊆ ℝ)
12929adantr 481 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘𝐾) ∈ ℝ)
130 ovolsscl 24756 . . . . . 6 (( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
131127, 128, 129, 130syl3anc 1370 . . . . 5 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
132124, 131fsumrecl 15545 . . . 4 (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
133127, 128sstrd 3942 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ)
134133, 131jca 512 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → ( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ ∧ (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
135134ralrimiva 3139 . . . . 5 (𝜑 → ∀𝑗 ∈ (1...𝑀)( 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ℝ ∧ (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
136 ovolfiniun 24771 . . . . 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)))))
137124, 135, 136syl2anc 584 . . . 4 (𝜑 → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
138 uniioombl.m . . . . . . . 8 (𝜑𝑀 ∈ ℕ)
139116, 138nndivred 12128 . . . . . . 7 (𝜑 → (𝐶 / 𝑀) ∈ ℝ)
140139adantr 481 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐶 / 𝑀) ∈ ℝ)
14177ineq2d 4159 . . . . . . . . . . . . 13 (𝜑 → (((,)‘(𝐺𝑗)) ∩ 𝐿) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖))))
142141adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖))))
14399a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑁) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))))
144143iuneq2i 4962 . . . . . . . . . . . . 13 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (1...𝑁)(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖)))
145 iunin2 5018 . . . . . . . . . . . . 13 𝑖 ∈ (1...𝑁)(((,)‘(𝐺𝑗)) ∩ ((,)‘(𝐹𝑖))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
146144, 145eqtri 2764 . . . . . . . . . . . 12 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)))
147142, 146eqtr4di 2794 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) = 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
148 fzfid 13794 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (1...𝑁) ∈ Fin)
149 ffvelcdm 7015 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑖 ∈ ℕ) → (𝐹𝑖) ∈ ( ≤ ∩ (ℝ × ℝ)))
15013, 73, 149syl2an 596 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (1...𝑁)) → (𝐹𝑖) ∈ ( ≤ ∩ (ℝ × ℝ)))
151150elin2d 4146 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (1...𝑁)) → (𝐹𝑖) ∈ (ℝ × ℝ))
152 1st2nd2 7938 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑖) ∈ (ℝ × ℝ) → (𝐹𝑖) = ⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩)
153151, 152syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑁)) → (𝐹𝑖) = ⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩)
154153fveq2d 6829 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) = ((,)‘⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩))
155 df-ov 7340 . . . . . . . . . . . . . . . . 17 ((1st ‘(𝐹𝑖))(,)(2nd ‘(𝐹𝑖))) = ((,)‘⟨(1st ‘(𝐹𝑖)), (2nd ‘(𝐹𝑖))⟩)
156154, 155eqtr4di 2794 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) = ((1st ‘(𝐹𝑖))(,)(2nd ‘(𝐹𝑖))))
157 ioombl 24835 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑖))(,)(2nd ‘(𝐹𝑖))) ∈ dom vol
158156, 157eqeltrdi 2845 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) ∈ dom vol)
159158adantlr 712 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((,)‘(𝐹𝑖)) ∈ dom vol)
160 ffvelcdm 7015 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
1616, 90, 160syl2an 596 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
162161elin2d 4146 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ (ℝ × ℝ))
163 1st2nd2 7938 . . . . . . . . . . . . . . . . . . 19 ((𝐺𝑗) ∈ (ℝ × ℝ) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
164162, 163syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
165164fveq2d 6829 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
166 df-ov 7340 . . . . . . . . . . . . . . . . 17 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
167165, 166eqtr4di 2794 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
168 ioombl 24835 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ∈ dom vol
169167, 168eqeltrdi 2845 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ∈ dom vol)
170169adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((,)‘(𝐺𝑗)) ∈ dom vol)
171 inmbl 24812 . . . . . . . . . . . . . 14 ((((,)‘(𝐹𝑖)) ∈ dom vol ∧ ((,)‘(𝐺𝑗)) ∈ dom vol) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
172159, 170, 171syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
173172ralrimiva 3139 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ∀𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
174 finiunmbl 24814 . . . . . . . . . . . 12 (((1...𝑁) ∈ Fin ∧ ∀𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol) → 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
175148, 173, 174syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol)
176147, 175eqeltrd 2837 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) ∈ dom vol)
177 inss2 4176 . . . . . . . . . . 11 (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ 𝐴
17813uniiccdif 24848 . . . . . . . . . . . . . . 15 (𝜑 → ( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ (vol*‘( ran ([,] ∘ 𝐹) ∖ ran ((,) ∘ 𝐹))) = 0))
179178simpld 495 . . . . . . . . . . . . . 14 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
180 ovolficcss 24739 . . . . . . . . . . . . . . 15 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
18113, 180syl 17 . . . . . . . . . . . . . 14 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
182179, 181sstrd 3942 . . . . . . . . . . . . 13 (𝜑 ran ((,) ∘ 𝐹) ⊆ ℝ)
18316, 182eqsstrid 3980 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ ℝ)
184183adantr 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → 𝐴 ⊆ ℝ)
185177, 184sstrid 3943 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ℝ)
186 inss1 4175 . . . . . . . . . . 11 (((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ((,)‘(𝐺𝑗))
187 ioossre 13241 . . . . . . . . . . . 12 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
188167, 187eqsstrdi 3986 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
189167fveq2d 6829 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺𝑗))) = (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))))
190 ovolfcl 24736 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
1916, 90, 190syl2an 596 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
192 ovolioo 24838 . . . . . . . . . . . . . 14 (((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))) → (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
193191, 192syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗)))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
194189, 193eqtrd 2776 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺𝑗))) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
195191simp2d 1142 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
196191simp1d 1141 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
197195, 196resubcld 11504 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
198194, 197eqeltrd 2837 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
199 ovolsscl 24756 . . . . . . . . . . 11 (((((,)‘(𝐺𝑗)) ∩ 𝐴) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ∈ ℝ)
200186, 188, 198, 199mp3an2i 1465 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ∈ ℝ)
201 mblsplit 24802 . . . . . . . . . 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)))))
202176, 185, 200, 201syl3anc 1370 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = ((vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿))) + (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿)))))
203 imassrn 6010 . . . . . . . . . . . . . . 15 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
204203unissi 4861 . . . . . . . . . . . . . 14 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
205204, 69, 163sstr4i 3975 . . . . . . . . . . . . 13 𝐿𝐴
206 sslin 4181 . . . . . . . . . . . . 13 (𝐿𝐴 → (((,)‘(𝐺𝑗)) ∩ 𝐿) ⊆ (((,)‘(𝐺𝑗)) ∩ 𝐴))
207205, 206mp1i 13 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ 𝐿) ⊆ (((,)‘(𝐺𝑗)) ∩ 𝐴))
208 sseqin2 4162 . . . . . . . . . . . 12 ((((,)‘(𝐺𝑗)) ∩ 𝐿) ⊆ (((,)‘(𝐺𝑗)) ∩ 𝐴) ↔ ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿)) = (((,)‘(𝐺𝑗)) ∩ 𝐿))
209207, 208sylib 217 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿)) = (((,)‘(𝐺𝑗)) ∩ 𝐿))
210209fveq2d 6829 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿))) = (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)))
211 indifdir 4231 . . . . . . . . . . . . 13 ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = ((𝐴 ∩ ((,)‘(𝐺𝑗))) ∖ (𝐿 ∩ ((,)‘(𝐺𝑗))))
212 incom 4148 . . . . . . . . . . . . . 14 (𝐴 ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝐴)
213 incom 4148 . . . . . . . . . . . . . 14 (𝐿 ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝐿)
214212, 213difeq12i 4067 . . . . . . . . . . . . 13 ((𝐴 ∩ ((,)‘(𝐺𝑗))) ∖ (𝐿 ∩ ((,)‘(𝐺𝑗)))) = ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿))
215211, 214eqtri 2764 . . . . . . . . . . . 12 ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿))
21679eqcomd 2742 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝐴)
21777ineq1d 4158 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
218 2fveq3 6830 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑖 → ((,)‘(𝐹𝑥)) = ((,)‘(𝐹𝑖)))
219218cbvdisjv 5068 . . . . . . . . . . . . . . . . . . . 20 (Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)) ↔ Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
22014, 219sylib 217 . . . . . . . . . . . . . . . . . . 19 (𝜑Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
221 fz1ssnn 13388 . . . . . . . . . . . . . . . . . . . 20 (1...𝑁) ⊆ ℕ
222221a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...𝑁) ⊆ ℕ)
223 uzss 12706 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 + 1) ∈ (ℤ‘1) → (ℤ‘(𝑁 + 1)) ⊆ (ℤ‘1))
22439, 223syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (ℤ‘(𝑁 + 1)) ⊆ (ℤ‘1))
225224, 36sseqtrrdi 3983 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℤ‘(𝑁 + 1)) ⊆ ℕ)
22647ineq1d 4158 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((1...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ((1...𝑁) ∩ (ℤ‘(𝑁 + 1))))
227 uzdisj 13430 . . . . . . . . . . . . . . . . . . . 20 ((1...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ∅
228226, 227eqtr3di 2791 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅)
229 disjiun 5079 . . . . . . . . . . . . . . . . . . 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))) = ∅)
230220, 222, 225, 228, 229syl13anc 1371 . . . . . . . . . . . . . . . . . 18 (𝜑 → ( 𝑖 ∈ (1...𝑁)((,)‘(𝐹𝑖)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅)
231217, 230eqtrd 2776 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅)
232 uneqdifeq 4437 . . . . . . . . . . . . . . . . 17 ((𝐿𝐴 ∧ (𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = ∅) → ((𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝐴 ↔ (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
233205, 231, 232sylancr 587 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐿 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))) = 𝐴 ↔ (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
234216, 233mpbid 231 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
235234adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐴𝐿) = 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
236235ineq2d 4159 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺𝑗)) ∩ (𝐴𝐿)) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖))))
237 incom 4148 . . . . . . . . . . . . 13 ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ (𝐴𝐿))
238101, 98eqtri 2764 . . . . . . . . . . . . 13 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) = (((,)‘(𝐺𝑗)) ∩ 𝑖 ∈ (ℤ‘(𝑁 + 1))((,)‘(𝐹𝑖)))
239236, 237, 2383eqtr4g 2801 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → ((𝐴𝐿) ∩ ((,)‘(𝐺𝑗))) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
240215, 239eqtr3id 2790 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿)) = 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
241240fveq2d 6829 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿))) = (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
242210, 241oveq12d 7355 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∩ (((,)‘(𝐺𝑗)) ∩ 𝐿))) + (vol*‘((((,)‘(𝐺𝑗)) ∩ 𝐴) ∖ (((,)‘(𝐺𝑗)) ∩ 𝐿)))) = ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
243202, 242eqtrd 2776 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) = ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))))
244200, 140resubcld 11504 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ∈ ℝ)
245 inss2 4176 . . . . . . . . . . . . 13 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗))
246188adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
247198adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ)
248 ovolsscl 24756 . . . . . . . . . . . . 13 (((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐺𝑗)) ∧ ((,)‘(𝐺𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺𝑗))) ∈ ℝ) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
249245, 246, 247, 248mp3an2i 1465 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
250148, 249fsumrecl 15545 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
251 uniioombl.n2 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
252251r19.21bi 3230 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
253250, 200, 140absdifltd 15244 . . . . . . . . . . . . 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)))))
254252, 253mpbid 231 . . . . . . . . . . . 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))))
255254simpld 495 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) < Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
256244, 250, 255ltled 11224 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ≤ Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
257147fveq2d 6829 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) = (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
258 mblvol 24800 . . . . . . . . . . . . . . . . 17 ((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol → (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
259172, 258syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
260259, 249eqeltrd 2837 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ)
261172, 260jca 512 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (1...𝑀)) ∧ 𝑖 ∈ (1...𝑁)) → ((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol ∧ (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
262261ralrimiva 3139 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → ∀𝑖 ∈ (1...𝑁)((((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol ∧ (vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ∈ ℝ))
263 inss1 4175 . . . . . . . . . . . . . . . 16 (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐹𝑖))
264263rgenw 3065 . . . . . . . . . . . . . . 15 𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐹𝑖))
265220adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (1...𝑀)) → Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)))
266 disjss2 5060 . . . . . . . . . . . . . . 15 (∀𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ⊆ ((,)‘(𝐹𝑖)) → (Disj 𝑖 ∈ ℕ ((,)‘(𝐹𝑖)) → Disj 𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
267264, 265, 266mpsyl 68 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (1...𝑀)) → Disj 𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
268 disjss1 5063 . . . . . . . . . . . . . 14 ((1...𝑁) ⊆ ℕ → (Disj 𝑖 ∈ ℕ (((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) → Disj 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
269221, 267, 268mpsyl 68 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → Disj 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))
270 volfiniun 24817 . . . . . . . . . . . . 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)))))
271148, 262, 269, 270syl3anc 1370 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (vol‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
272 mblvol 24800 . . . . . . . . . . . . 13 ( 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))) ∈ dom vol → (vol‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
273175, 272syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (vol‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
274259sumeq2dv 15514 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → Σ𝑖 ∈ (1...𝑁)(vol‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
275271, 273, 2743eqtr3d 2784 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘ 𝑖 ∈ (1...𝑁)(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) = Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
276257, 275eqtrd 2776 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) = Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))))
277256, 276breqtrrd 5120 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ≤ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)))
278276, 250eqeltrd 2837 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) ∈ ℝ)
279200, 140, 278lesubaddd 11673 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → (((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) − (𝐶 / 𝑀)) ≤ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) ↔ (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ≤ ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (𝐶 / 𝑀))))
280277, 279mpbid 231 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)) ≤ ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (𝐶 / 𝑀)))
281243, 280eqbrtrrd 5116 . . . . . . 7 ((𝜑𝑗 ∈ (1...𝑀)) → ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ≤ ((vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐿)) + (𝐶 / 𝑀)))
282131, 140, 278leadd2d 11671 . . . . . . 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))))
283281, 282mpbird 256 . . . . . 6 ((𝜑𝑗 ∈ (1...𝑀)) → (vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ (𝐶 / 𝑀))
284124, 131, 140, 283fsumle 15610 . . . . 5 (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀))
285139recnd 11104 . . . . . . 7 (𝜑 → (𝐶 / 𝑀) ∈ ℂ)
286 fsumconst 15601 . . . . . . 7 (((1...𝑀) ∈ Fin ∧ (𝐶 / 𝑀) ∈ ℂ) → Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀) = ((♯‘(1...𝑀)) · (𝐶 / 𝑀)))
287124, 285, 286syl2anc 584 . . . . . 6 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀) = ((♯‘(1...𝑀)) · (𝐶 / 𝑀)))
288 nnnn0 12341 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
289 hashfz1 14161 . . . . . . . 8 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
290138, 288, 2893syl 18 . . . . . . 7 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
291290oveq1d 7352 . . . . . 6 (𝜑 → ((♯‘(1...𝑀)) · (𝐶 / 𝑀)) = (𝑀 · (𝐶 / 𝑀)))
292116recnd 11104 . . . . . . 7 (𝜑𝐶 ∈ ℂ)
293138nncnd 12090 . . . . . . 7 (𝜑𝑀 ∈ ℂ)
294138nnne0d 12124 . . . . . . 7 (𝜑𝑀 ≠ 0)
295292, 293, 294divcan2d 11854 . . . . . 6 (𝜑 → (𝑀 · (𝐶 / 𝑀)) = 𝐶)
296287, 291, 2953eqtrd 2780 . . . . 5 (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝐶 / 𝑀) = 𝐶)
297284, 296breqtrd 5118 . . . 4 (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘ 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ 𝐶)
298114, 132, 116, 137, 297letrd 11233 . . 3 (𝜑 → (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) ≤ 𝐶)
299114, 116, 34, 298leadd2dd 11691 . 2 (𝜑 → ((vol*‘(𝐾𝐿)) + (vol*‘ 𝑗 ∈ (1...𝑀) 𝑖 ∈ (ℤ‘(𝑁 + 1))(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗))))) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
30031, 115, 117, 123, 299letrd 11233 1 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3061  cdif 3895  cun 3896  cin 3897  wss 3898  c0 4269  𝒫 cpw 4547  cop 4579   cuni 4852   ciun 4941  Disj wdisj 5057   class class class wbr 5092   × cxp 5618  dom cdm 5620  ran crn 5621  cima 5623  ccom 5624  Fun wfun 6473   Fn wfn 6474  wf 6475  cfv 6479  (class class class)co 7337  1st c1st 7897  2nd c2nd 7898  Fincfn 8804  supcsup 9297  cc 10970  cr 10971  0cc0 10972  1c1 10973   + caddc 10975   · cmul 10977  *cxr 11109   < clt 11110  cle 11111  cmin 11306   / cdiv 11733  cn 12074  0cn0 12334  cuz 12683  +crp 12831  (,)cioo 13180  [,]cicc 13183  ...cfz 13340  seqcseq 13822  chash 14145  abscabs 15044  Σcsu 15496  vol*covol 24732  volcvol 24733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-inf2 9498  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-disj 5058  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-of 7595  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-2o 8368  df-er 8569  df-map 8688  df-pm 8689  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-fi 9268  df-sup 9299  df-inf 9300  df-oi 9367  df-dju 9758  df-card 9796  df-acn 9799  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-n0 12335  df-z 12421  df-uz 12684  df-q 12790  df-rp 12832  df-xneg 12949  df-xadd 12950  df-xmul 12951  df-ioo 13184  df-ico 13186  df-icc 13187  df-fz 13341  df-fzo 13484  df-fl 13613  df-seq 13823  df-exp 13884  df-hash 14146  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-abs 15046  df-clim 15296  df-rlim 15297  df-sum 15497  df-rest 17230  df-topgen 17251  df-psmet 20695  df-xmet 20696  df-met 20697  df-bl 20698  df-mopn 20699  df-top 22149  df-topon 22166  df-bases 22202  df-cmp 22644  df-ovol 24734  df-vol 24735
This theorem is referenced by:  uniioombllem5  24857
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