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Theorem uniioombllem5 24857
Description: Lemma for uniioombl 24859. (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 4175 . . . 4 (𝐸𝐴) ⊆ 𝐸
2 uniioombl.s . . . . 5 (𝜑𝐸 ran ((,) ∘ 𝐺))
3 uniioombl.g . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
43uniiccdif 24848 . . . . . . 7 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
54simpld 495 . . . . . 6 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
6 ovolficcss 24739 . . . . . . 7 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
73, 6syl 17 . . . . . 6 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
85, 7sstrd 3942 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
92, 8sstrd 3942 . . . 4 (𝜑𝐸 ⊆ ℝ)
10 uniioombl.e . . . 4 (𝜑 → (vol*‘𝐸) ∈ ℝ)
11 ovolsscl 24756 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
121, 9, 10, 11mp3an2i 1465 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
13 difssd 4079 . . . 4 (𝜑 → (𝐸𝐴) ⊆ 𝐸)
14 ovolsscl 24756 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
1513, 9, 10, 14syl3anc 1370 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
1612, 15readdcld 11105 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ∈ ℝ)
17 inss1 4175 . . . . 5 (𝐾𝐴) ⊆ 𝐾
18 uniioombl.k . . . . . . 7 𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))
19 imassrn 6010 . . . . . . . 8 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
2019unissi 4861 . . . . . . 7 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
2118, 20eqsstri 3966 . . . . . 6 𝐾 ran ((,) ∘ 𝐺)
2221, 8sstrid 3943 . . . . 5 (𝜑𝐾 ⊆ ℝ)
23 uniioombl.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24 uniioombl.2 . . . . . . . 8 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
25 uniioombl.3 . . . . . . . 8 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
26 uniioombl.a . . . . . . . 8 𝐴 = ran ((,) ∘ 𝐹)
27 uniioombl.c . . . . . . . 8 (𝜑𝐶 ∈ ℝ+)
28 uniioombl.t . . . . . . . 8 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
29 uniioombl.v . . . . . . . 8 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
3023, 24, 25, 26, 10, 27, 3, 2, 28, 29uniioombllem1 24851 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
31 ssid 3954 . . . . . . . 8 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
3228ovollb 24749 . . . . . . . 8 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
333, 31, 32sylancl 586 . . . . . . 7 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
34 ovollecl 24753 . . . . . . 7 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
358, 30, 33, 34syl3anc 1370 . . . . . 6 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
36 ovolsscl 24756 . . . . . 6 ((𝐾 ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘𝐾) ∈ ℝ)
3721, 8, 35, 36mp3an2i 1465 . . . . 5 (𝜑 → (vol*‘𝐾) ∈ ℝ)
38 ovolsscl 24756 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
3917, 22, 37, 38mp3an2i 1465 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
40 difssd 4079 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
41 ovolsscl 24756 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4240, 22, 37, 41syl3anc 1370 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
4339, 42readdcld 11105 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ∈ ℝ)
4427rpred 12873 . . . 4 (𝜑𝐶 ∈ ℝ)
4544, 44readdcld 11105 . . 3 (𝜑 → (𝐶 + 𝐶) ∈ ℝ)
4643, 45readdcld 11105 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ∈ ℝ)
47 4re 12158 . . . 4 4 ∈ ℝ
48 remulcl 11057 . . . 4 ((4 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (4 · 𝐶) ∈ ℝ)
4947, 44, 48sylancr 587 . . 3 (𝜑 → (4 · 𝐶) ∈ ℝ)
5010, 49readdcld 11105 . 2 (𝜑 → ((vol*‘𝐸) + (4 · 𝐶)) ∈ ℝ)
51 uniioombl.m . . . 4 (𝜑𝑀 ∈ ℕ)
52 uniioombl.m2 . . . 4 (𝜑 → (abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
5323, 24, 25, 26, 10, 27, 3, 2, 28, 29, 51, 52, 18uniioombllem3 24855 . . 3 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
5416, 46, 53ltled 11224 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
5510, 45readdcld 11105 . . . 4 (𝜑 → ((vol*‘𝐸) + (𝐶 + 𝐶)) ∈ ℝ)
5637, 44readdcld 11105 . . . . 5 (𝜑 → ((vol*‘𝐾) + 𝐶) ∈ ℝ)
57 inss1 4175 . . . . . . . . 9 (𝐾𝐿) ⊆ 𝐾
58 ovolsscl 24756 . . . . . . . . 9 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
5957, 22, 37, 58mp3an2i 1465 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℝ)
6059, 44readdcld 11105 . . . . . . 7 (𝜑 → ((vol*‘(𝐾𝐿)) + 𝐶) ∈ ℝ)
61 difssd 4079 . . . . . . . 8 (𝜑 → (𝐾𝐿) ⊆ 𝐾)
62 ovolsscl 24756 . . . . . . . 8 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
6361, 22, 37, 62syl3anc 1370 . . . . . . 7 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℝ)
64 uniioombl.n . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
65 uniioombl.n2 . . . . . . . 8 (𝜑 → ∀𝑗 ∈ (1...𝑀)(abs‘(Σ𝑖 ∈ (1...𝑁)(vol*‘(((,)‘(𝐹𝑖)) ∩ ((,)‘(𝐺𝑗)))) − (vol*‘(((,)‘(𝐺𝑗)) ∩ 𝐴)))) < (𝐶 / 𝑀))
66 uniioombl.l . . . . . . . 8 𝐿 = (((,) ∘ 𝐹) “ (1...𝑁))
6723, 24, 25, 26, 10, 27, 3, 2, 28, 29, 51, 52, 18, 64, 65, 66uniioombllem4 24856 . . . . . . 7 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
68 imassrn 6010 . . . . . . . . . . 11 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
6968unissi 4861 . . . . . . . . . 10 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
7069, 66, 263sstr4i 3975 . . . . . . . . 9 𝐿𝐴
71 sscon 4085 . . . . . . . . 9 (𝐿𝐴 → (𝐾𝐴) ⊆ (𝐾𝐿))
7270, 71mp1i 13 . . . . . . . 8 (𝜑 → (𝐾𝐴) ⊆ (𝐾𝐿))
7361, 22sstrd 3942 . . . . . . . 8 (𝜑 → (𝐾𝐿) ⊆ ℝ)
74 ovolss 24755 . . . . . . . 8 (((𝐾𝐴) ⊆ (𝐾𝐿) ∧ (𝐾𝐿) ⊆ ℝ) → (vol*‘(𝐾𝐴)) ≤ (vol*‘(𝐾𝐿)))
7572, 73, 74syl2anc 584 . . . . . . 7 (𝜑 → (vol*‘(𝐾𝐴)) ≤ (vol*‘(𝐾𝐿)))
7639, 42, 60, 63, 67, 75le2addd 11695 . . . . . 6 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))))
7759recnd 11104 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℂ)
7844recnd 11104 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
7963recnd 11104 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℂ)
8077, 78, 79add32d 11303 . . . . . . 7 (𝜑 → (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))) = (((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))) + 𝐶))
81 ioof 13280 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
82 inss2 4176 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
83 rexpssxrxp 11121 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
8482, 83sstri 3941 . . . . . . . . . . . . . 14 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
85 fss 6668 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
8623, 84, 85sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
87 fco 6675 . . . . . . . . . . . . 13 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
8881, 86, 87sylancr 587 . . . . . . . . . . . 12 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
89 ffun 6654 . . . . . . . . . . . 12 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐹))
90 funiunfv 7177 . . . . . . . . . . . 12 (Fun ((,) ∘ 𝐹) → 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐹) “ (1...𝑁)))
9188, 89, 903syl 18 . . . . . . . . . . 11 (𝜑 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐹) “ (1...𝑁)))
9291, 66eqtr4di 2794 . . . . . . . . . 10 (𝜑 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = 𝐿)
93 fzfid 13794 . . . . . . . . . . 11 (𝜑 → (1...𝑁) ∈ Fin)
94 elfznn 13386 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
95 fvco3 6923 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
9623, 94, 95syl2an 596 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
97 ffvelcdm 7015 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
9823, 94, 97syl2an 596 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
9998elin2d 4146 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) ∈ (ℝ × ℝ))
100 1st2nd2 7938 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) ∈ (ℝ × ℝ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
10199, 100syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
102101fveq2d 6829 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
103 df-ov 7340 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
104102, 103eqtr4di 2794 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
10596, 104eqtrd 2776 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
106 ioombl 24835 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol
107105, 106eqeltrdi 2845 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
108107ralrimiva 3139 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
109 finiunmbl 24814 . . . . . . . . . . 11 (((1...𝑁) ∈ Fin ∧ ∀𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol) → 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
11093, 108, 109syl2anc 584 . . . . . . . . . 10 (𝜑 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
11192, 110eqeltrrd 2838 . . . . . . . . 9 (𝜑𝐿 ∈ dom vol)
112 mblsplit 24802 . . . . . . . . 9 ((𝐿 ∈ dom vol ∧ 𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘𝐾) = ((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))))
113111, 22, 37, 112syl3anc 1370 . . . . . . . 8 (𝜑 → (vol*‘𝐾) = ((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))))
114113oveq1d 7352 . . . . . . 7 (𝜑 → ((vol*‘𝐾) + 𝐶) = (((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))) + 𝐶))
11580, 114eqtr4d 2779 . . . . . 6 (𝜑 → (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))) = ((vol*‘𝐾) + 𝐶))
11676, 115breqtrd 5118 . . . . 5 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ ((vol*‘𝐾) + 𝐶))
11710, 44readdcld 11105 . . . . . . 7 (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ)
11828ovollb 24749 . . . . . . . . 9 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐾 ran ((,) ∘ 𝐺)) → (vol*‘𝐾) ≤ sup(ran 𝑇, ℝ*, < ))
1193, 21, 118sylancl 586 . . . . . . . 8 (𝜑 → (vol*‘𝐾) ≤ sup(ran 𝑇, ℝ*, < ))
12037, 30, 117, 119, 29letrd 11233 . . . . . . 7 (𝜑 → (vol*‘𝐾) ≤ ((vol*‘𝐸) + 𝐶))
12137, 117, 44, 120leadd1dd 11690 . . . . . 6 (𝜑 → ((vol*‘𝐾) + 𝐶) ≤ (((vol*‘𝐸) + 𝐶) + 𝐶))
12210recnd 11104 . . . . . . 7 (𝜑 → (vol*‘𝐸) ∈ ℂ)
123122, 78, 78addassd 11098 . . . . . 6 (𝜑 → (((vol*‘𝐸) + 𝐶) + 𝐶) = ((vol*‘𝐸) + (𝐶 + 𝐶)))
124121, 123breqtrd 5118 . . . . 5 (𝜑 → ((vol*‘𝐾) + 𝐶) ≤ ((vol*‘𝐸) + (𝐶 + 𝐶)))
12543, 56, 55, 116, 124letrd 11233 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ ((vol*‘𝐸) + (𝐶 + 𝐶)))
12643, 55, 45, 125leadd1dd 11690 . . 3 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ≤ (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)))
12745recnd 11104 . . . . 5 (𝜑 → (𝐶 + 𝐶) ∈ ℂ)
128122, 127, 127addassd 11098 . . . 4 (𝜑 → (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol*‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
129 2t2e4 12238 . . . . . . 7 (2 · 2) = 4
130129oveq1i 7347 . . . . . 6 ((2 · 2) · 𝐶) = (4 · 𝐶)
131 2cnd 12152 . . . . . . . 8 (𝜑 → 2 ∈ ℂ)
132131, 131, 78mulassd 11099 . . . . . . 7 (𝜑 → ((2 · 2) · 𝐶) = (2 · (2 · 𝐶)))
133782timesd 12317 . . . . . . . 8 (𝜑 → (2 · 𝐶) = (𝐶 + 𝐶))
134133oveq2d 7353 . . . . . . 7 (𝜑 → (2 · (2 · 𝐶)) = (2 · (𝐶 + 𝐶)))
1351272timesd 12317 . . . . . . 7 (𝜑 → (2 · (𝐶 + 𝐶)) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
136132, 134, 1353eqtrd 2780 . . . . . 6 (𝜑 → ((2 · 2) · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
137130, 136eqtr3id 2790 . . . . 5 (𝜑 → (4 · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
138137oveq2d 7353 . . . 4 (𝜑 → ((vol*‘𝐸) + (4 · 𝐶)) = ((vol*‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
139128, 138eqtr4d 2779 . . 3 (𝜑 → (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol*‘𝐸) + (4 · 𝐶)))
140126, 139breqtrd 5118 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
14116, 46, 50, 54, 140letrd 11233 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  wral 3061  cdif 3895  cin 3897  wss 3898  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  wf 6475  cfv 6479  (class class class)co 7337  1st c1st 7897  2nd c2nd 7898  Fincfn 8804  supcsup 9297  cr 10971  0cc0 10972  1c1 10973   + caddc 10975   · cmul 10977  *cxr 11109   < clt 11110  cle 11111  cmin 11306   / cdiv 11733  cn 12074  2c2 12129  4c4 12131  +crp 12831  (,)cioo 13180  [,]cicc 13183  ...cfz 13340  seqcseq 13822  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-4 12139  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:  uniioombllem6  24858
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