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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  cr 10141  0cc0 10142  1c1 10143   + caddc 10145   · cmul 10147  *cxr 10279   < clt 10280  cle 10281  cmin 10472   / cdiv 10890  cn 11226  2c2 11276  4c4 11278  +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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