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Theorem uniioombllem5 25622
Description: Lemma for uniioombl 25624. (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 4237 . . . 4 (𝐸𝐴) ⊆ 𝐸
2 uniioombl.s . . . . 5 (𝜑𝐸 ran ((,) ∘ 𝐺))
3 uniioombl.g . . . . . . . 8 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
43uniiccdif 25613 . . . . . . 7 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
54simpld 494 . . . . . 6 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
6 ovolficcss 25504 . . . . . . 7 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
73, 6syl 17 . . . . . 6 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
85, 7sstrd 3994 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
92, 8sstrd 3994 . . . 4 (𝜑𝐸 ⊆ ℝ)
10 uniioombl.e . . . 4 (𝜑 → (vol*‘𝐸) ∈ ℝ)
11 ovolsscl 25521 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
121, 9, 10, 11mp3an2i 1468 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
13 difssd 4137 . . . 4 (𝜑 → (𝐸𝐴) ⊆ 𝐸)
14 ovolsscl 25521 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
1513, 9, 10, 14syl3anc 1373 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
1612, 15readdcld 11290 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ∈ ℝ)
17 inss1 4237 . . . . 5 (𝐾𝐴) ⊆ 𝐾
18 uniioombl.k . . . . . . 7 𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))
19 imassrn 6089 . . . . . . . 8 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
2019unissi 4916 . . . . . . 7 (((,) ∘ 𝐺) “ (1...𝑀)) ⊆ ran ((,) ∘ 𝐺)
2118, 20eqsstri 4030 . . . . . 6 𝐾 ran ((,) ∘ 𝐺)
2221, 8sstrid 3995 . . . . 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 25616 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
31 ssid 4006 . . . . . . . 8 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
3228ovollb 25514 . . . . . . . 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 25518 . . . . . . 7 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
358, 30, 33, 34syl3anc 1373 . . . . . 6 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
36 ovolsscl 25521 . . . . . 6 ((𝐾 ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘𝐾) ∈ ℝ)
3721, 8, 35, 36mp3an2i 1468 . . . . 5 (𝜑 → (vol*‘𝐾) ∈ ℝ)
38 ovolsscl 25521 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
3917, 22, 37, 38mp3an2i 1468 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
40 difssd 4137 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
41 ovolsscl 25521 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4240, 22, 37, 41syl3anc 1373 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
4339, 42readdcld 11290 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ∈ ℝ)
4427rpred 13077 . . . 4 (𝜑𝐶 ∈ ℝ)
4544, 44readdcld 11290 . . 3 (𝜑 → (𝐶 + 𝐶) ∈ ℝ)
4643, 45readdcld 11290 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ∈ ℝ)
47 4re 12350 . . . 4 4 ∈ ℝ
48 remulcl 11240 . . . 4 ((4 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (4 · 𝐶) ∈ ℝ)
4947, 44, 48sylancr 587 . . 3 (𝜑 → (4 · 𝐶) ∈ ℝ)
5010, 49readdcld 11290 . 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 25620 . . 3 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
5416, 46, 53ltled 11409 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
5510, 45readdcld 11290 . . . 4 (𝜑 → ((vol*‘𝐸) + (𝐶 + 𝐶)) ∈ ℝ)
5637, 44readdcld 11290 . . . . 5 (𝜑 → ((vol*‘𝐾) + 𝐶) ∈ ℝ)
57 inss1 4237 . . . . . . . . 9 (𝐾𝐿) ⊆ 𝐾
58 ovolsscl 25521 . . . . . . . . 9 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
5957, 22, 37, 58mp3an2i 1468 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℝ)
6059, 44readdcld 11290 . . . . . . 7 (𝜑 → ((vol*‘(𝐾𝐿)) + 𝐶) ∈ ℝ)
61 difssd 4137 . . . . . . . 8 (𝜑 → (𝐾𝐿) ⊆ 𝐾)
62 ovolsscl 25521 . . . . . . . 8 (((𝐾𝐿) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐿)) ∈ ℝ)
6361, 22, 37, 62syl3anc 1373 . . . . . . 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 25621 . . . . . . 7 (𝜑 → (vol*‘(𝐾𝐴)) ≤ ((vol*‘(𝐾𝐿)) + 𝐶))
68 imassrn 6089 . . . . . . . . . . 11 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
6968unissi 4916 . . . . . . . . . 10 (((,) ∘ 𝐹) “ (1...𝑁)) ⊆ ran ((,) ∘ 𝐹)
7069, 66, 263sstr4i 4035 . . . . . . . . 9 𝐿𝐴
71 sscon 4143 . . . . . . . . 9 (𝐿𝐴 → (𝐾𝐴) ⊆ (𝐾𝐿))
7270, 71mp1i 13 . . . . . . . 8 (𝜑 → (𝐾𝐴) ⊆ (𝐾𝐿))
7361, 22sstrd 3994 . . . . . . . 8 (𝜑 → (𝐾𝐿) ⊆ ℝ)
74 ovolss 25520 . . . . . . . 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 11882 . . . . . 6 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))))
7759recnd 11289 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℂ)
7844recnd 11289 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
7963recnd 11289 . . . . . . . 8 (𝜑 → (vol*‘(𝐾𝐿)) ∈ ℂ)
8077, 78, 79add32d 11489 . . . . . . 7 (𝜑 → (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))) = (((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))) + 𝐶))
81 ioof 13487 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
82 inss2 4238 . . . . . . . . . . . . . . 15 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
83 rexpssxrxp 11306 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
8482, 83sstri 3993 . . . . . . . . . . . . . 14 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
85 fss 6752 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
8623, 84, 85sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
87 fco 6760 . . . . . . . . . . . . 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 6739 . . . . . . . . . . . 12 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐹))
90 funiunfv 7268 . . . . . . . . . . . 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 2795 . . . . . . . . . 10 (𝜑 𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) = 𝐿)
93 fzfid 14014 . . . . . . . . . . 11 (𝜑 → (1...𝑁) ∈ Fin)
94 elfznn 13593 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
95 fvco3 7008 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
9623, 94, 95syl2an 596 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
97 ffvelcdm 7101 . . . . . . . . . . . . . . . . . . 19 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
9823, 94, 97syl2an 596 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
9998elin2d 4205 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) ∈ (ℝ × ℝ))
100 1st2nd2 8053 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) ∈ (ℝ × ℝ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
10199, 100syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑁)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
102101fveq2d 6910 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
103 df-ov 7434 . . . . . . . . . . . . . . 15 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
104102, 103eqtr4di 2795 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → ((,)‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
10596, 104eqtrd 2777 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
106 ioombl 25600 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol
107105, 106eqeltrdi 2849 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
108107ralrimiva 3146 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ (1...𝑁)(((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
109 finiunmbl 25579 . . . . . . . . . . 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 2842 . . . . . . . . 9 (𝜑𝐿 ∈ dom vol)
112 mblsplit 25567 . . . . . . . . 9 ((𝐿 ∈ dom vol ∧ 𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘𝐾) = ((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))))
113111, 22, 37, 112syl3anc 1373 . . . . . . . 8 (𝜑 → (vol*‘𝐾) = ((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))))
114113oveq1d 7446 . . . . . . 7 (𝜑 → ((vol*‘𝐾) + 𝐶) = (((vol*‘(𝐾𝐿)) + (vol*‘(𝐾𝐿))) + 𝐶))
11580, 114eqtr4d 2780 . . . . . 6 (𝜑 → (((vol*‘(𝐾𝐿)) + 𝐶) + (vol*‘(𝐾𝐿))) = ((vol*‘𝐾) + 𝐶))
11676, 115breqtrd 5169 . . . . 5 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ ((vol*‘𝐾) + 𝐶))
11710, 44readdcld 11290 . . . . . . 7 (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ)
11828ovollb 25514 . . . . . . . . 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 11418 . . . . . . 7 (𝜑 → (vol*‘𝐾) ≤ ((vol*‘𝐸) + 𝐶))
12137, 117, 44, 120leadd1dd 11877 . . . . . 6 (𝜑 → ((vol*‘𝐾) + 𝐶) ≤ (((vol*‘𝐸) + 𝐶) + 𝐶))
12210recnd 11289 . . . . . . 7 (𝜑 → (vol*‘𝐸) ∈ ℂ)
123122, 78, 78addassd 11283 . . . . . 6 (𝜑 → (((vol*‘𝐸) + 𝐶) + 𝐶) = ((vol*‘𝐸) + (𝐶 + 𝐶)))
124121, 123breqtrd 5169 . . . . 5 (𝜑 → ((vol*‘𝐾) + 𝐶) ≤ ((vol*‘𝐸) + (𝐶 + 𝐶)))
12543, 56, 55, 116, 124letrd 11418 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) ≤ ((vol*‘𝐸) + (𝐶 + 𝐶)))
12643, 55, 45, 125leadd1dd 11877 . . 3 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ≤ (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)))
12745recnd 11289 . . . . 5 (𝜑 → (𝐶 + 𝐶) ∈ ℂ)
128122, 127, 127addassd 11283 . . . 4 (𝜑 → (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol*‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
129 2t2e4 12430 . . . . . . 7 (2 · 2) = 4
130129oveq1i 7441 . . . . . 6 ((2 · 2) · 𝐶) = (4 · 𝐶)
131 2cnd 12344 . . . . . . . 8 (𝜑 → 2 ∈ ℂ)
132131, 131, 78mulassd 11284 . . . . . . 7 (𝜑 → ((2 · 2) · 𝐶) = (2 · (2 · 𝐶)))
133782timesd 12509 . . . . . . . 8 (𝜑 → (2 · 𝐶) = (𝐶 + 𝐶))
134133oveq2d 7447 . . . . . . 7 (𝜑 → (2 · (2 · 𝐶)) = (2 · (𝐶 + 𝐶)))
1351272timesd 12509 . . . . . . 7 (𝜑 → (2 · (𝐶 + 𝐶)) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
136132, 134, 1353eqtrd 2781 . . . . . 6 (𝜑 → ((2 · 2) · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
137130, 136eqtr3id 2791 . . . . 5 (𝜑 → (4 · 𝐶) = ((𝐶 + 𝐶) + (𝐶 + 𝐶)))
138137oveq2d 7447 . . . 4 (𝜑 → ((vol*‘𝐸) + (4 · 𝐶)) = ((vol*‘𝐸) + ((𝐶 + 𝐶) + (𝐶 + 𝐶))))
139128, 138eqtr4d 2780 . . 3 (𝜑 → (((vol*‘𝐸) + (𝐶 + 𝐶)) + (𝐶 + 𝐶)) = ((vol*‘𝐸) + (4 · 𝐶)))
140126, 139breqtrd 5169 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
14116, 46, 50, 54, 140letrd 11418 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) ≤ ((vol*‘𝐸) + (4 · 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cin 3950  wss 3951  𝒫 cpw 4600  cop 4632   cuni 4907   ciun 4991  Disj wdisj 5110   class class class wbr 5143   × cxp 5683  dom cdm 5685  ran crn 5686  cima 5688  ccom 5689  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Fincfn 8985  supcsup 9480  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  *cxr 11294   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  2c2 12321  4c4 12323  +crp 13034  (,)cioo 13387  [,]cicc 13390  ...cfz 13547  seqcseq 14042  abscabs 15273  Σcsu 15722  vol*covol 25497  volcvol 25498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500
This theorem is referenced by:  uniioombllem6  25623
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