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Theorem uniioombllem5 23575
 Description: Lemma for uniioombl 23577. (Contributed by Mario Carneiro, 25-Aug-2014.)
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
uniioombllem5 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 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 uniioombllem5
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 inss1 3981 . . . . 5 (𝐸𝐴) ⊆ 𝐸
21a1i 11 . . . 4 (𝜑 → (𝐸𝐴) ⊆ 𝐸)
3 uniioombl.s . . . . 5 (𝜑𝐸 ran ((,) ∘ 𝐺))
4 uniioombl.g . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
54uniiccdif 23566 . . . . . . 7 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
65simpld 482 . . . . . 6 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
7 ovolficcss 23457 . . . . . . 7 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
84, 7syl 17 . . . . . 6 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
96, 8sstrd 3762 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
103, 9sstrd 3762 . . . 4 (𝜑𝐸 ⊆ ℝ)
11 uniioombl.e . . . 4 (𝜑 → (vol*‘𝐸) ∈ ℝ)
12 ovolsscl 23474 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
132, 10, 11, 12syl3anc 1476 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
14 difssd 3889 . . . 4 (𝜑 → (𝐸𝐴) ⊆ 𝐸)
15 ovolsscl 23474 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
1614, 10, 11, 15syl3anc 1476 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
1713, 16readdcld 10275 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ∈ ℝ)
18 inss1 3981 . . . . . 6 (𝐾𝐴) ⊆ 𝐾
1918a1i 11 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
20 uniioombl.k . . . . . . . 8 𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))
21 imassrn 5617 . . . . . . . . 9 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
2221unissi 4598 . . . . . . . 8 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
2320, 22eqsstri 3784 . . . . . . 7 𝐾 ran ((,) ∘ 𝐺)
2423a1i 11 . . . . . 6 (𝜑𝐾 ran ((,) ∘ 𝐺))
2524, 9sstrd 3762 . . . . 5 (𝜑𝐾 ⊆ ℝ)
26 uniioombl.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27 uniioombl.2 . . . . . . . 8 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
28 uniioombl.3 . . . . . . . 8 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
29 uniioombl.a . . . . . . . 8 𝐴 = ran ((,) ∘ 𝐹)
30 uniioombl.c . . . . . . . 8 (𝜑𝐶 ∈ ℝ+)
31 uniioombl.t . . . . . . . 8 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
32 uniioombl.v . . . . . . . 8 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
3326, 27, 28, 29, 11, 30, 4, 3, 31, 32uniioombllem1 23569 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
34 ssid 3773 . . . . . . . 8 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
3531ovollb 23467 . . . . . . . 8 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
364, 34, 35sylancl 574 . . . . . . 7 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
37 ovollecl 23471 . . . . . . 7 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
389, 33, 36, 37syl3anc 1476 . . . . . 6 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
39 ovolsscl 23474 . . . . . 6 ((𝐾 ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘𝐾) ∈ ℝ)
4024, 9, 38, 39syl3anc 1476 . . . . 5 (𝜑 → (vol*‘𝐾) ∈ ℝ)
41 ovolsscl 23474 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4219, 25, 40, 41syl3anc 1476 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
43 difssd 3889 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
44 ovolsscl 23474 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4543, 25, 40, 44syl3anc 1476 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
4642, 45readdcld 10275 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ∈ ℝ)
4730rpred 12075 . . . 4 (𝜑𝐶 ∈ ℝ)
4847, 47readdcld 10275 . . 3 (𝜑 → (𝐶 + 𝐶) ∈ ℝ)
4946, 48readdcld 10275 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ∈ ℝ)
50 4re 11303 . . . 4 4 ∈ ℝ
51 remulcl 10227 . . . 4 ((4 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (4 · 𝐶) ∈ ℝ)
5250, 47, 51sylancr 575 . . 3 (𝜑 → (4 · 𝐶) ∈ ℝ)
5311, 52readdcld 10275 . 2 (𝜑 → ((vol*‘𝐸) + (4 · 𝐶)) ∈ ℝ)
54 uniioombl.m . . . 4 (𝜑𝑀 ∈ ℕ)
55 uniioombl.m2 . . . 4 (𝜑 → (abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
5626, 27, 28, 29, 11, 30, 4, 3, 31, 32, 54, 55, 20uniioombllem3 23573 . . 3 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
5717, 49, 56ltled 10391 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
5811, 48readdcld 10275 . . . 4 (𝜑 → ((vol*‘𝐸) + (𝐶 + 𝐶)) ∈ ℝ)
5940, 47readdcld 10275 . . . . 5 (𝜑 → ((vol*‘𝐾) + 𝐶) ∈ ℝ)
60 inss1 3981 . . . . . . . . . 10 (𝐾𝐿) ⊆ 𝐾
6160a1i 11 . . . . . . . . 9 (𝜑 → (𝐾𝐿) ⊆ 𝐾)
62 ovolsscl 23474 . . . . . . . . 9 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
6361, 25, 40, 62syl3anc 1476 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℝ)
6463, 47readdcld 10275 . . . . . . 7 (𝜑 → ((vol*‘(𝐾𝐿)) + 𝐶) ∈ ℝ)
65 difssd 3889 . . . . . . . 8 (𝜑 → (𝐾𝐿) ⊆ 𝐾)
66 ovolsscl 23474 . . . . . . . 8 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
6765, 25, 40, 66syl3anc 1476 . . . . . . 7 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℝ)
68 uniioombl.n . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
69 uniioombl.n2 . . . . . . . 8 (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
70 uniioombl.l . . . . . . . 8 𝐿 = (((,) ∘ 𝐹) “ (1...𝑁))
7126, 27, 28, 29, 11, 30, 4, 3, 31, 32, 54, 55, 20, 68, 69, 70uniioombllem4 23574 . . . . . . 7 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
72 imassrn 5617 . . . . . . . . . . 11 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
7372unissi 4598 . . . . . . . . . 10 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
7473, 70, 293sstr4i 3793 . . . . . . . . 9 𝐿𝐴
75 sscon 3895 . . . . . . . . 9 (𝐿𝐴 → (𝐾𝐴) ⊆ (𝐾𝐿))
7674, 75mp1i 13 . . . . . . . 8 (𝜑 → (𝐾𝐴) ⊆ (𝐾𝐿))
7765, 25sstrd 3762 . . . . . . . 8 (𝜑 → (𝐾𝐿) ⊆ ℝ)
78 ovolss 23473 . . . . . . . 8 (((𝐾𝐴) ⊆ (𝐾𝐿) ∧ (𝐾𝐿) ⊆ ℝ) → (vol*‘(𝐾𝐴)) ≤ (vol*‘(𝐾𝐿)))
7976, 77, 78syl2anc 573 . . . . . . 7 (𝜑 → (vol*‘(𝐾𝐴)) ≤ (vol*‘(𝐾𝐿)))
8042, 45, 64, 67, 71, 79le2addd 10852 . . . . . 6 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))))
8163recnd 10274 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℂ)
8247recnd 10274 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
8367recnd 10274 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℂ)
8481, 82, 83add32d 10469 . . . . . . 7 (𝜑 → (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))) = (((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))) + 𝐶))
85 ioof 12477 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
86 inss2 3982 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
87 rexpssxrxp 10290 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
8886, 87sstri 3761 . . . . . . . . . . . . . 14 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
89 fss 6197 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
9026, 88, 89sylancl 574 . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
91 fco 6199 . . . . . . . . . . . . 13 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9285, 90, 91sylancr 575 . . . . . . . . . . . 12 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
93 ffun 6187 . . . . . . . . . . . 12 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐹))
94 funiunfv 6652 . . . . . . . . . . . 12 (Fun ((,) ∘ 𝐹) → 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐹) “ (1...𝑁)))
9592, 93, 943syl 18 . . . . . . . . . . 11 (𝜑 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐹) “ (1...𝑁)))
9695, 70syl6eqr 2823 . . . . . . . . . 10 (𝜑 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = 𝐿)
97 fzfid 12980 . . . . . . . . . . 11 (𝜑 → (1...𝑁) ∈ Fin)
98 elfznn 12577 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
99 fvco3 6419 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
10026, 98, 99syl2an 583 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
101 ffvelrn 6502 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
10226, 98, 101syl2an 583 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
10386, 102sseldi 3750 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) ∈ (ℝ × ℝ))
104 1st2nd2 7358 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) ∈ (ℝ × ℝ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
105103, 104syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
106105fveq2d 6337 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
107 df-ov 6799 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
108106, 107syl6eqr 2823 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
109100, 108eqtrd 2805 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
110 ioombl 23553 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol
111109, 110syl6eqel 2858 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
112111ralrimiva 3115 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
113 finiunmbl 23532 . . . . . . . . . . 11 (((1...𝑁) ∈ Fin ∧ ∀𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol) → 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
11497, 112, 113syl2anc 573 . . . . . . . . . 10 (𝜑 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
11596, 114eqeltrrd 2851 . . . . . . . . 9 (𝜑𝐿 ∈ dom vol)
116 mblsplit 23520 . . . . . . . . 9 ((𝐿 ∈ dom vol ∧ 𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘𝐾) = ((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))))
117115, 25, 40, 116syl3anc 1476 . . . . . . . 8 (𝜑 → (vol*‘𝐾) = ((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))))
118117oveq1d 6811 . . . . . . 7 (𝜑 → ((vol*‘𝐾) + 𝐶) = (((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))) + 𝐶))
11984, 118eqtr4d 2808 . . . . . 6 (𝜑 → (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))) = ((vol*‘𝐾) + 𝐶))
12080, 119breqtrd 4813 . . . . 5 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ ((vol*‘𝐾) + 𝐶))
12111, 47readdcld 10275 . . . . . . 7 (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ)
12231ovollb 23467 . . . . . . . . 9 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐾 ran ((,) ∘ 𝐺)) → (vol*‘𝐾) ≤ sup(ran 𝑇, ℝ*, < ))
1234, 23, 122sylancl 574 . . . . . . . 8 (𝜑 → (vol*‘𝐾) ≤ sup(ran 𝑇, ℝ*, < ))
12440, 33, 121, 123, 32letrd 10400 . . . . . . 7 (𝜑 → (vol*‘𝐾) ≤ ((vol*‘𝐸) + 𝐶))
12540, 121, 47, 124leadd1dd 10847 . . . . . 6 (𝜑 → ((vol*‘𝐾) + 𝐶) ≤ (((vol*‘𝐸) + 𝐶) + 𝐶))
12611recnd 10274 . . . . . . 7 (𝜑 → (vol*‘𝐸) ∈ ℂ)
127126, 82, 82addassd 10268 . . . . . 6 (𝜑 → (((vol*‘𝐸) + 𝐶) + 𝐶) = ((vol*‘𝐸) + (𝐶 + 𝐶)))
128125, 127breqtrd 4813 . . . . 5 (𝜑 → ((vol*‘𝐾) + 𝐶) ≤ ((vol*‘𝐸) + (𝐶 + 𝐶)))
12946, 59, 58, 120, 128letrd 10400 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ ((vol*‘𝐸) + (𝐶 + 𝐶)))
13046, 58, 48, 129leadd1dd 10847 . . 3 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ≤ (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)))
13148recnd 10274 . . . . 5 (𝜑 → (𝐶 + 𝐶) ∈ ℂ)
132126, 131, 131addassd 10268 . . . 4 (𝜑 → (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol*‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
133 2t2e4 11384 . . . . . . 7 (2 · 2) = 4
134133oveq1i 6806 . . . . . 6 ((2 · 2) · 𝐶) = (4 · 𝐶)
135 2cnd 11299 . . . . . . . 8 (𝜑 → 2 ∈ ℂ)
136135, 135, 82mulassd 10269 . . . . . . 7 (𝜑 → ((2 · 2) · 𝐶) = (2 · (2 · 𝐶)))
137822timesd 11482 . . . . . . . 8 (𝜑 → (2 · 𝐶) = (𝐶 + 𝐶))
138137oveq2d 6812 . . . . . . 7 (𝜑 → (2 · (2 · 𝐶)) = (2 · (𝐶 + 𝐶)))
1391312timesd 11482 . . . . . . 7 (𝜑 → (2 · (𝐶 + 𝐶)) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
140136, 138, 1393eqtrd 2809 . . . . . 6 (𝜑 → ((2 · 2) · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
141134, 140syl5eqr 2819 . . . . 5 (𝜑 → (4 · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
142141oveq2d 6812 . . . 4 (𝜑 → ((vol*‘𝐸) + (4 · 𝐶)) = ((vol*‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
143132, 142eqtr4d 2808 . . 3 (𝜑 → (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol*‘𝐸) + (4 · 𝐶)))
144130, 143breqtrd 4813 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
14517, 49, 53, 57, 144letrd 10400 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ∖ cdif 3720   ∩ cin 3722   ⊆ wss 3723  𝒫 cpw 4298  ⟨cop 4323  ∪ cuni 4575  ∪ ciun 4655  Disj wdisj 4755   class class class wbr 4787   × cxp 5248  dom cdm 5250  ran crn 5251   “ cima 5253   ∘ ccom 5254  Fun wfun 6024  ⟶wf 6026  ‘cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  Fincfn 8113  supcsup 8506  ℝcr 10141  0cc0 10142  1c1 10143   + caddc 10145   · cmul 10147  ℝ*cxr 10279   < clt 10280   ≤ cle 10281   − cmin 10472   / cdiv 10890  ℕcn 11226  2c2 11276  4c4 11278  ℝ+crp 12035  (,)cioo 12380  [,]cicc 12383  ...cfz 12533  seqcseq 13008  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 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-inf2 8706  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220 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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-disj 4756  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-of 7048  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fi 8477  df-sup 8508  df-inf 8509  df-oi 8575  df-card 8969  df-acn 8972  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-n0 11500  df-z 11585  df-uz 11894  df-q 11997  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:  uniioombllem6  23576
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