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Theorem uniioombllem5 23161
 Description: Lemma for uniioombl 23163. (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 3795 . . . . 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 23152 . . . . . . 7 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
65simpld 474 . . . . . 6 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
7 ovolficcss 23045 . . . . . . 7 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
84, 7syl 17 . . . . . 6 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
96, 8sstrd 3578 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
103, 9sstrd 3578 . . . 4 (𝜑𝐸 ⊆ ℝ)
11 uniioombl.e . . . 4 (𝜑 → (vol*‘𝐸) ∈ ℝ)
12 ovolsscl 23061 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
132, 10, 11, 12syl3anc 1318 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
14 difssd 3700 . . . 4 (𝜑 → (𝐸𝐴) ⊆ 𝐸)
15 ovolsscl 23061 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
1614, 10, 11, 15syl3anc 1318 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
1713, 16readdcld 9948 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ∈ ℝ)
18 inss1 3795 . . . . . 6 (𝐾𝐴) ⊆ 𝐾
1918a1i 11 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
20 uniioombl.k . . . . . . . 8 𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))
21 imassrn 5396 . . . . . . . . 9 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
2221unissi 4397 . . . . . . . 8 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
2320, 22eqsstri 3598 . . . . . . 7 𝐾 ran ((,) ∘ 𝐺)
2423a1i 11 . . . . . 6 (𝜑𝐾 ran ((,) ∘ 𝐺))
2524, 9sstrd 3578 . . . . 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 23155 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
34 ssid 3587 . . . . . . . 8 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
3531ovollb 23054 . . . . . . . 8 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
364, 34, 35sylancl 693 . . . . . . 7 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
37 ovollecl 23058 . . . . . . 7 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
389, 33, 36, 37syl3anc 1318 . . . . . 6 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
39 ovolsscl 23061 . . . . . 6 ((𝐾 ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘𝐾) ∈ ℝ)
4024, 9, 38, 39syl3anc 1318 . . . . 5 (𝜑 → (vol*‘𝐾) ∈ ℝ)
41 ovolsscl 23061 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4219, 25, 40, 41syl3anc 1318 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
43 difssd 3700 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
44 ovolsscl 23061 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4543, 25, 40, 44syl3anc 1318 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
4642, 45readdcld 9948 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ∈ ℝ)
4730rpred 11748 . . . 4 (𝜑𝐶 ∈ ℝ)
4847, 47readdcld 9948 . . 3 (𝜑 → (𝐶 + 𝐶) ∈ ℝ)
4946, 48readdcld 9948 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ∈ ℝ)
50 4re 10974 . . . 4 4 ∈ ℝ
51 remulcl 9900 . . . 4 ((4 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (4 · 𝐶) ∈ ℝ)
5250, 47, 51sylancr 694 . . 3 (𝜑 → (4 · 𝐶) ∈ ℝ)
5311, 52readdcld 9948 . 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 23159 . . 3 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
5717, 49, 56ltled 10064 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
5811, 48readdcld 9948 . . . 4 (𝜑 → ((vol*‘𝐸) + (𝐶 + 𝐶)) ∈ ℝ)
5940, 47readdcld 9948 . . . . 5 (𝜑 → ((vol*‘𝐾) + 𝐶) ∈ ℝ)
60 inss1 3795 . . . . . . . . . 10 (𝐾𝐿) ⊆ 𝐾
6160a1i 11 . . . . . . . . 9 (𝜑 → (𝐾𝐿) ⊆ 𝐾)
62 ovolsscl 23061 . . . . . . . . 9 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
6361, 25, 40, 62syl3anc 1318 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℝ)
6463, 47readdcld 9948 . . . . . . 7 (𝜑 → ((vol*‘(𝐾𝐿)) + 𝐶) ∈ ℝ)
65 difssd 3700 . . . . . . . 8 (𝜑 → (𝐾𝐿) ⊆ 𝐾)
66 ovolsscl 23061 . . . . . . . 8 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
6765, 25, 40, 66syl3anc 1318 . . . . . . 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 23160 . . . . . . 7 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
72 imassrn 5396 . . . . . . . . . . 11 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
7372unissi 4397 . . . . . . . . . 10 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
7473, 70, 293sstr4i 3607 . . . . . . . . 9 𝐿𝐴
75 sscon 3706 . . . . . . . . 9 (𝐿𝐴 → (𝐾𝐴) ⊆ (𝐾𝐿))
7674, 75mp1i 13 . . . . . . . 8 (𝜑 → (𝐾𝐴) ⊆ (𝐾𝐿))
7765, 25sstrd 3578 . . . . . . . 8 (𝜑 → (𝐾𝐿) ⊆ ℝ)
78 ovolss 23060 . . . . . . . 8 (((𝐾𝐴) ⊆ (𝐾𝐿) ∧ (𝐾𝐿) ⊆ ℝ) → (vol*‘(𝐾𝐴)) ≤ (vol*‘(𝐾𝐿)))
7976, 77, 78syl2anc 691 . . . . . . 7 (𝜑 → (vol*‘(𝐾𝐴)) ≤ (vol*‘(𝐾𝐿)))
8042, 45, 64, 67, 71, 79le2addd 10525 . . . . . 6 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))))
8163recnd 9947 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℂ)
8247recnd 9947 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
8367recnd 9947 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℂ)
8481, 82, 83add32d 10142 . . . . . . 7 (𝜑 → (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))) = (((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))) + 𝐶))
85 ioof 12142 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
86 inss2 3796 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
87 rexpssxrxp 9963 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
8886, 87sstri 3577 . . . . . . . . . . . . . 14 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
89 fss 5969 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
9026, 88, 89sylancl 693 . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
91 fco 5971 . . . . . . . . . . . . 13 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9285, 90, 91sylancr 694 . . . . . . . . . . . 12 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
93 ffun 5961 . . . . . . . . . . . 12 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐹))
94 funiunfv 6410 . . . . . . . . . . . 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 2662 . . . . . . . . . 10 (𝜑 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = 𝐿)
97 fzfid 12634 . . . . . . . . . . 11 (𝜑 → (1...𝑁) ∈ Fin)
98 elfznn 12241 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
99 fvco3 6185 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
10026, 98, 99syl2an 493 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
101 ffvelrn 6265 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
10226, 98, 101syl2an 493 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
10386, 102sseldi 3566 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) ∈ (ℝ × ℝ))
104 1st2nd2 7096 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) ∈ (ℝ × ℝ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
105103, 104syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
106105fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
107 df-ov 6552 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
108106, 107syl6eqr 2662 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
109100, 108eqtrd 2644 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
110 ioombl 23140 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol
111109, 110syl6eqel 2696 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
112111ralrimiva 2949 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
113 finiunmbl 23119 . . . . . . . . . . 11 (((1...𝑁) ∈ Fin ∧ ∀𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol) → 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
11497, 112, 113syl2anc 691 . . . . . . . . . 10 (𝜑 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
11596, 114eqeltrrd 2689 . . . . . . . . 9 (𝜑𝐿 ∈ dom vol)
116 mblsplit 23107 . . . . . . . . 9 ((𝐿 ∈ dom vol ∧ 𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘𝐾) = ((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))))
117115, 25, 40, 116syl3anc 1318 . . . . . . . 8 (𝜑 → (vol*‘𝐾) = ((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))))
118117oveq1d 6564 . . . . . . 7 (𝜑 → ((vol*‘𝐾) + 𝐶) = (((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))) + 𝐶))
11984, 118eqtr4d 2647 . . . . . 6 (𝜑 → (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))) = ((vol*‘𝐾) + 𝐶))
12080, 119breqtrd 4609 . . . . 5 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ ((vol*‘𝐾) + 𝐶))
12111, 47readdcld 9948 . . . . . . 7 (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ)
12231ovollb 23054 . . . . . . . . 9 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐾 ran ((,) ∘ 𝐺)) → (vol*‘𝐾) ≤ sup(ran 𝑇, ℝ*, < ))
1234, 23, 122sylancl 693 . . . . . . . 8 (𝜑 → (vol*‘𝐾) ≤ sup(ran 𝑇, ℝ*, < ))
12440, 33, 121, 123, 32letrd 10073 . . . . . . 7 (𝜑 → (vol*‘𝐾) ≤ ((vol*‘𝐸) + 𝐶))
12540, 121, 47, 124leadd1dd 10520 . . . . . 6 (𝜑 → ((vol*‘𝐾) + 𝐶) ≤ (((vol*‘𝐸) + 𝐶) + 𝐶))
12611recnd 9947 . . . . . . 7 (𝜑 → (vol*‘𝐸) ∈ ℂ)
127126, 82, 82addassd 9941 . . . . . 6 (𝜑 → (((vol*‘𝐸) + 𝐶) + 𝐶) = ((vol*‘𝐸) + (𝐶 + 𝐶)))
128125, 127breqtrd 4609 . . . . 5 (𝜑 → ((vol*‘𝐾) + 𝐶) ≤ ((vol*‘𝐸) + (𝐶 + 𝐶)))
12946, 59, 58, 120, 128letrd 10073 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ ((vol*‘𝐸) + (𝐶 + 𝐶)))
13046, 58, 48, 129leadd1dd 10520 . . 3 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ≤ (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)))
13148recnd 9947 . . . . 5 (𝜑 → (𝐶 + 𝐶) ∈ ℂ)
132126, 131, 131addassd 9941 . . . 4 (𝜑 → (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol*‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
133 2t2e4 11054 . . . . . . 7 (2 · 2) = 4
134133oveq1i 6559 . . . . . 6 ((2 · 2) · 𝐶) = (4 · 𝐶)
135 2cnd 10970 . . . . . . . 8 (𝜑 → 2 ∈ ℂ)
136135, 135, 82mulassd 9942 . . . . . . 7 (𝜑 → ((2 · 2) · 𝐶) = (2 · (2 · 𝐶)))
137822timesd 11152 . . . . . . . 8 (𝜑 → (2 · 𝐶) = (𝐶 + 𝐶))
138137oveq2d 6565 . . . . . . 7 (𝜑 → (2 · (2 · 𝐶)) = (2 · (𝐶 + 𝐶)))
1391312timesd 11152 . . . . . . 7 (𝜑 → (2 · (𝐶 + 𝐶)) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
140136, 138, 1393eqtrd 2648 . . . . . 6 (𝜑 → ((2 · 2) · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
141134, 140syl5eqr 2658 . . . . 5 (𝜑 → (4 · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
142141oveq2d 6565 . . . 4 (𝜑 → ((vol*‘𝐸) + (4 · 𝐶)) = ((vol*‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
143132, 142eqtr4d 2647 . . 3 (𝜑 → (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol*‘𝐸) + (4 · 𝐶)))
144130, 143breqtrd 4609 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
14517, 49, 53, 57, 144letrd 10073 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583   × cxp 5036  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  supcsup 8229  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  2c2 10947  4c4 10949  ℝ+crp 11708  (,)cioo 12046  [,]cicc 12049  ...cfz 12197  seqcseq 12663  abscabs 13822  Σcsu 14264  vol*covol 23038  volcvol 23039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041 This theorem is referenced by:  uniioombllem6  23162
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