MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniioombllem3 Structured version   Visualization version   GIF version

Theorem uniioombllem3 24185
Description: Lemma for uniioombl 24189. (Contributed by Mario Carneiro, 26-Mar-2015.)
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))
Assertion
Ref Expression
uniioombllem3 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝐾   𝑥,𝐴   𝑥,𝐶   𝑥,𝑀   𝜑,𝑥   𝑥,𝑇
Allowed substitution hints:   𝑆(𝑥)   𝐸(𝑥)

Proof of Theorem uniioombllem3
Dummy variables 𝑗 𝑘 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 4204 . . . . 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 24178 . . . . . . 7 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
65simpld 497 . . . . . 6 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
7 ovolficcss 24069 . . . . . . 7 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
84, 7syl 17 . . . . . 6 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
96, 8sstrd 3976 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
103, 9sstrd 3976 . . . 4 (𝜑𝐸 ⊆ ℝ)
11 uniioombl.e . . . 4 (𝜑 → (vol*‘𝐸) ∈ ℝ)
12 ovolsscl 24086 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
132, 10, 11, 12syl3anc 1367 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
14 difssd 4108 . . . 4 (𝜑 → (𝐸𝐴) ⊆ 𝐸)
15 ovolsscl 24086 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
1614, 10, 11, 15syl3anc 1367 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
17 inss1 4204 . . . . . 6 (𝐾𝐴) ⊆ 𝐾
1817a1i 11 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
19 uniioombl.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
20 uniioombl.2 . . . . . . . 8 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
21 uniioombl.3 . . . . . . . 8 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
22 uniioombl.a . . . . . . . 8 𝐴 = ran ((,) ∘ 𝐹)
23 uniioombl.c . . . . . . . 8 (𝜑𝐶 ∈ ℝ+)
24 uniioombl.t . . . . . . . 8 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
25 uniioombl.v . . . . . . . 8 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
26 uniioombl.m . . . . . . . 8 (𝜑𝑀 ∈ ℕ)
27 uniioombl.m2 . . . . . . . 8 (𝜑 → (abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
28 uniioombl.k . . . . . . . 8 𝐾 = (((,) ∘ 𝐺) “ (1...𝑀))
2919, 20, 21, 22, 11, 23, 4, 3, 24, 25, 26, 27, 28uniioombllem3a 24184 . . . . . . 7 (𝜑 → (𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ∧ (vol*‘𝐾) ∈ ℝ))
3029simpld 497 . . . . . 6 (𝜑𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
31 inss2 4205 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
32 elfznn 12935 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
33 ffvelrn 6848 . . . . . . . . . . . . . 14 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
344, 32, 33syl2an 597 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
3531, 34sseldi 3964 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ (ℝ × ℝ))
36 1st2nd2 7727 . . . . . . . . . . . 12 ((𝐺𝑗) ∈ (ℝ × ℝ) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
3735, 36syl 17 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
3837fveq2d 6673 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
39 df-ov 7158 . . . . . . . . . 10 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
4038, 39syl6eqr 2874 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
41 ioossre 12797 . . . . . . . . 9 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
4240, 41eqsstrdi 4020 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
4342ralrimiva 3182 . . . . . . 7 (𝜑 → ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
44 iunss 4968 . . . . . . 7 ( 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ ↔ ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
4543, 44sylibr 236 . . . . . 6 (𝜑 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
4630, 45eqsstrd 4004 . . . . 5 (𝜑𝐾 ⊆ ℝ)
4729simprd 498 . . . . 5 (𝜑 → (vol*‘𝐾) ∈ ℝ)
48 ovolsscl 24086 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4918, 46, 47, 48syl3anc 1367 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
5023rpred 12430 . . . 4 (𝜑𝐶 ∈ ℝ)
5149, 50readdcld 10669 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + 𝐶) ∈ ℝ)
52 difssd 4108 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
53 ovolsscl 24086 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
5452, 46, 47, 53syl3anc 1367 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
5554, 50readdcld 10669 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + 𝐶) ∈ ℝ)
56 ssun2 4148 . . . . . . 7 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
57 ioof 12834 . . . . . . . . . . . . . . 15 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
58 rexpssxrxp 10685 . . . . . . . . . . . . . . . . 17 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5931, 58sstri 3975 . . . . . . . . . . . . . . . 16 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
60 fss 6526 . . . . . . . . . . . . . . . 16 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
614, 59, 60sylancl 588 . . . . . . . . . . . . . . 15 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
62 fco 6530 . . . . . . . . . . . . . . 15 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
6357, 61, 62sylancr 589 . . . . . . . . . . . . . 14 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
6463ffnd 6514 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
65 fnima 6477 . . . . . . . . . . . . 13 (((,) ∘ 𝐺) Fn ℕ → (((,) ∘ 𝐺) “ ℕ) = ran ((,) ∘ 𝐺))
6664, 65syl 17 . . . . . . . . . . . 12 (𝜑 → (((,) ∘ 𝐺) “ ℕ) = ran ((,) ∘ 𝐺))
67 nnuz 12280 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6826peano2nnd 11654 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 + 1) ∈ ℕ)
6968, 67eleqtrdi 2923 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 + 1) ∈ (ℤ‘1))
70 uzsplit 12978 . . . . . . . . . . . . . . . 16 ((𝑀 + 1) ∈ (ℤ‘1) → (ℤ‘1) = ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))))
7169, 70syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (ℤ‘1) = ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))))
7267, 71syl5eq 2868 . . . . . . . . . . . . . 14 (𝜑 → ℕ = ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))))
7326nncnd 11653 . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ ℂ)
74 ax-1cn 10594 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
75 pncan 10891 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
7673, 74, 75sylancl 588 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
7776oveq2d 7171 . . . . . . . . . . . . . . 15 (𝜑 → (1...((𝑀 + 1) − 1)) = (1...𝑀))
7877uneq1d 4137 . . . . . . . . . . . . . 14 (𝜑 → ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))) = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
7972, 78eqtrd 2856 . . . . . . . . . . . . 13 (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
8079imaeq2d 5928 . . . . . . . . . . . 12 (𝜑 → (((,) ∘ 𝐺) “ ℕ) = (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
8166, 80eqtr3d 2858 . . . . . . . . . . 11 (𝜑 → ran ((,) ∘ 𝐺) = (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
82 imaundi 6007 . . . . . . . . . . 11 (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8381, 82syl6eq 2872 . . . . . . . . . 10 (𝜑 → ran ((,) ∘ 𝐺) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8483unieqd 4851 . . . . . . . . 9 (𝜑 ran ((,) ∘ 𝐺) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
85 uniun 4860 . . . . . . . . 9 ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8684, 85syl6eq 2872 . . . . . . . 8 (𝜑 ran ((,) ∘ 𝐺) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8728uneq1i 4134 . . . . . . . 8 (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8886, 87syl6eqr 2874 . . . . . . 7 (𝜑 ran ((,) ∘ 𝐺) = (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8956, 88sseqtrrid 4019 . . . . . 6 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ 𝐺))
9019, 20, 21, 22, 11, 23, 4, 3, 24, 25uniioombllem1 24181 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
91 ssid 3988 . . . . . . . 8 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
9224ovollb 24079 . . . . . . . 8 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
934, 91, 92sylancl 588 . . . . . . 7 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
94 ovollecl 24083 . . . . . . 7 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
959, 90, 93, 94syl3anc 1367 . . . . . 6 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
96 ovolsscl 24086 . . . . . 6 (( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)
9789, 9, 95, 96syl3anc 1367 . . . . 5 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)
9849, 97readdcld 10669 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
99 unss1 4154 . . . . . . . 8 ((𝐾𝐴) ⊆ 𝐾 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
10017, 99ax-mp 5 . . . . . . 7 ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
101100, 88sseqtrrid 4019 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺))
102 ovolsscl 24086 . . . . . 6 ((((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
103101, 9, 95, 102syl3anc 1367 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
1043, 88sseqtrd 4006 . . . . . . . 8 (𝜑𝐸 ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
105104ssrind 4211 . . . . . . 7 (𝜑 → (𝐸𝐴) ⊆ ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴))
106 indir 4251 . . . . . . . 8 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴) = ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴))
107 inss1 4204 . . . . . . . . 9 ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))
108 unss2 4156 . . . . . . . . 9 (( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) → ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
109107, 108ax-mp 5 . . . . . . . 8 ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
110106, 109eqsstri 4000 . . . . . . 7 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
111105, 110sstrdi 3978 . . . . . 6 (𝜑 → (𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
112101, 9sstrd 3976 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ)
113 ovolss 24085 . . . . . 6 (((𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∧ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ) → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
114111, 112, 113syl2anc 586 . . . . 5 (𝜑 → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
11518, 46sstrd 3976 . . . . . 6 (𝜑 → (𝐾𝐴) ⊆ ℝ)
11689, 9sstrd 3976 . . . . . 6 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ℝ)
117 ovolun 24099 . . . . . 6 ((((𝐾𝐴) ⊆ ℝ ∧ (vol*‘(𝐾𝐴)) ∈ ℝ) ∧ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ℝ ∧ (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)) → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
118115, 49, 116, 97, 117syl22anc 836 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
11913, 103, 98, 114, 118letrd 10796 . . . 4 (𝜑 → (vol*‘(𝐸𝐴)) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
120 rge0ssre 12843 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
121 eqid 2821 . . . . . . . . . . 11 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
122121, 24ovolsf 24072 . . . . . . . . . 10 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
1234, 122syl 17 . . . . . . . . 9 (𝜑𝑇:ℕ⟶(0[,)+∞))
124123, 26ffvelrnd 6851 . . . . . . . 8 (𝜑 → (𝑇𝑀) ∈ (0[,)+∞))
125120, 124sseldi 3964 . . . . . . 7 (𝜑 → (𝑇𝑀) ∈ ℝ)
12690, 125resubcld 11067 . . . . . 6 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ∈ ℝ)
12797rexrd 10690 . . . . . . 7 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ*)
128 id 22 . . . . . . . . . . . . . 14 (𝑧 ∈ ℕ → 𝑧 ∈ ℕ)
129 nnaddcl 11659 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑧 + 𝑀) ∈ ℕ)
130128, 26, 129syl2anr 598 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ℕ) → (𝑧 + 𝑀) ∈ ℕ)
1314ffvelrnda 6850 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 + 𝑀) ∈ ℕ) → (𝐺‘(𝑧 + 𝑀)) ∈ ( ≤ ∩ (ℝ × ℝ)))
132130, 131syldan 593 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℕ) → (𝐺‘(𝑧 + 𝑀)) ∈ ( ≤ ∩ (ℝ × ℝ)))
133132fmpttd 6878 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
134 eqid 2821 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))) = ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
135 eqid 2821 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) = seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
136134, 135ovolsf 24072 . . . . . . . . . . 11 ((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))):ℕ⟶(0[,)+∞))
137133, 136syl 17 . . . . . . . . . 10 (𝜑 → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))):ℕ⟶(0[,)+∞))
138137frnd 6520 . . . . . . . . 9 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ (0[,)+∞))
139 icossxr 12820 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ*
140138, 139sstrdi 3978 . . . . . . . 8 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ ℝ*)
141 supxrcl 12707 . . . . . . . 8 (ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ∈ ℝ*)
142140, 141syl 17 . . . . . . 7 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ∈ ℝ*)
143126rexrd 10690 . . . . . . 7 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ∈ ℝ*)
144 1zzd 12012 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 1 ∈ ℤ)
14526nnzd 12085 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℤ)
146145adantr 483 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
147 addcom 10825 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑀 + 1) = (1 + 𝑀))
14873, 74, 147sylancl 588 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 + 1) = (1 + 𝑀))
149148fveq2d 6673 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (ℤ‘(𝑀 + 1)) = (ℤ‘(1 + 𝑀)))
150149eleq2d 2898 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑥 ∈ (ℤ‘(𝑀 + 1)) ↔ 𝑥 ∈ (ℤ‘(1 + 𝑀))))
151150biimpa 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ‘(1 + 𝑀)))
152 eluzsub 12273 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ (ℤ‘(1 + 𝑀))) → (𝑥𝑀) ∈ (ℤ‘1))
153144, 146, 151, 152syl3anc 1367 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝑥𝑀) ∈ (ℤ‘1))
154153, 67eleqtrrdi 2924 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝑥𝑀) ∈ ℕ)
155 eluzelz 12252 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (ℤ‘(𝑀 + 1)) → 𝑥 ∈ ℤ)
156155adantl 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ℤ)
157156zcnd 12087 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ℂ)
15873adantr 483 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℂ)
159157, 158npcand 11000 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → ((𝑥𝑀) + 𝑀) = 𝑥)
160159eqcomd 2827 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 = ((𝑥𝑀) + 𝑀))
161 oveq1 7162 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑥𝑀) → (𝑧 + 𝑀) = ((𝑥𝑀) + 𝑀))
162161rspceeqv 3637 . . . . . . . . . . . . . . . . 17 (((𝑥𝑀) ∈ ℕ ∧ 𝑥 = ((𝑥𝑀) + 𝑀)) → ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
163154, 160, 162syl2anc 586 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
164 eqid 2821 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) = (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))
165164elrnmpt 5827 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ V → (𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) ↔ ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀)))
166165elv 3499 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) ↔ ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
167163, 166sylibr 236 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
168167ex 415 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ (ℤ‘(𝑀 + 1)) → 𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
169168ssrdv 3972 . . . . . . . . . . . . 13 (𝜑 → (ℤ‘(𝑀 + 1)) ⊆ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
170 imass2 5964 . . . . . . . . . . . . 13 ((ℤ‘(𝑀 + 1)) ⊆ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
171169, 170syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
172 rnco2 6105 . . . . . . . . . . . . 13 ran (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
1734, 130cofmpt 6893 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
174173rneqd 5807 . . . . . . . . . . . . 13 (𝜑 → ran (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
175172, 174syl5eqr 2870 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
176171, 175sseqtrd 4006 . . . . . . . . . . 11 (𝜑 → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
177 imass2 5964 . . . . . . . . . . 11 ((𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))) → ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1)))) ⊆ ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
178176, 177syl 17 . . . . . . . . . 10 (𝜑 → ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1)))) ⊆ ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
179 imaco 6103 . . . . . . . . . 10 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) = ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1))))
180 rnco2 6105 . . . . . . . . . 10 ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))) = ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
181178, 179, 1803sstr4g 4011 . . . . . . . . 9 (𝜑 → (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
182181unissd 4847 . . . . . . . 8 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
183135ovollb 24079 . . . . . . . 8 (((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ))
184133, 182, 183syl2anc 586 . . . . . . 7 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ))
185123frnd 6520 . . . . . . . . . . . . 13 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
186185, 139sstrdi 3978 . . . . . . . . . . . 12 (𝜑 → ran 𝑇 ⊆ ℝ*)
18724fveq1i 6670 . . . . . . . . . . . . . 14 (𝑇‘(𝑀 + 𝑛)) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛))
18826nnred 11652 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℝ)
189188ltp1d 11569 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 < (𝑀 + 1))
190 fzdisj 12933 . . . . . . . . . . . . . . . . . 18 (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
191189, 190syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
192191adantr 483 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
193 nnnn0 11903 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
194 nn0addge1 11942 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℝ ∧ 𝑛 ∈ ℕ0) → 𝑀 ≤ (𝑀 + 𝑛))
195188, 193, 194syl2an 597 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑀 ≤ (𝑀 + 𝑛))
19626adantr 483 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℕ)
197196, 67eleqtrdi 2923 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ (ℤ‘1))
198 nnaddcl 11659 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℕ)
19926, 198sylan 582 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℕ)
200199nnzd 12085 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℤ)
201 elfz5 12899 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ (ℤ‘1) ∧ (𝑀 + 𝑛) ∈ ℤ) → (𝑀 ∈ (1...(𝑀 + 𝑛)) ↔ 𝑀 ≤ (𝑀 + 𝑛)))
202197, 200, 201syl2anc 586 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (𝑀 ∈ (1...(𝑀 + 𝑛)) ↔ 𝑀 ≤ (𝑀 + 𝑛)))
203195, 202mpbird 259 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ (1...(𝑀 + 𝑛)))
204 fzsplit 12932 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (1...(𝑀 + 𝑛)) → (1...(𝑀 + 𝑛)) = ((1...𝑀) ∪ ((𝑀 + 1)...(𝑀 + 𝑛))))
205203, 204syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1...(𝑀 + 𝑛)) = ((1...𝑀) ∪ ((𝑀 + 1)...(𝑀 + 𝑛))))
206 fzfid 13340 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1...(𝑀 + 𝑛)) ∈ Fin)
2074adantr 483 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
208 elfznn 12935 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (1...(𝑀 + 𝑛)) → 𝑗 ∈ ℕ)
209 ovolfcl 24066 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
210207, 208, 209syl2an 597 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
211210simp2d 1139 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
212210simp1d 1138 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (1st ‘(𝐺𝑗)) ∈ ℝ)
213211, 212resubcld 11067 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
214213recnd 10668 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
215192, 205, 206, 214fsumsplit 15096 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) + Σ𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗)))))
216121ovolfsval 24070 . . . . . . . . . . . . . . . . 17 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
217207, 208, 216syl2an 597 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
218199, 67eleqtrdi 2923 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ (ℤ‘1))
219217, 218, 214fsumser 15086 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛)))
2204ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22132adantl 484 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
222220, 221, 216syl2anc 586 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
2234, 32, 209syl2an 597 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (1...𝑀)) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
224223simp2d 1139 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
225223simp1d 1138 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
226224, 225resubcld 11067 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
227226adantlr 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
228227recnd 10668 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
229222, 197, 228fsumser 15086 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑀))
23024fveq1i 6670 . . . . . . . . . . . . . . . . 17 (𝑇𝑀) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑀)
231229, 230syl6eqr 2874 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (𝑇𝑀))
232196nnzd 12085 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℤ)
233232peano2zd 12089 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 1) ∈ ℤ)
2344ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
235196peano2nnd 11654 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 1) ∈ ℕ)
236 elfzuz 12903 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛)) → 𝑗 ∈ (ℤ‘(𝑀 + 1)))
237 eluznn 12317 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 + 1) ∈ ℕ ∧ 𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝑗 ∈ ℕ)
238235, 236, 237syl2an 597 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → 𝑗 ∈ ℕ)
239234, 238, 209syl2anc 586 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
240239simp2d 1139 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
241239simp1d 1138 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → (1st ‘(𝐺𝑗)) ∈ ℝ)
242240, 241resubcld 11067 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
243242recnd 10668 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
244 2fveq3 6674 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 𝑀) → (2nd ‘(𝐺𝑗)) = (2nd ‘(𝐺‘(𝑘 + 𝑀))))
245 2fveq3 6674 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 𝑀) → (1st ‘(𝐺𝑗)) = (1st ‘(𝐺‘(𝑘 + 𝑀))))
246244, 245oveq12d 7173 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑘 + 𝑀) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
247232, 233, 200, 243, 246fsumshftm 15135 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = Σ𝑘 ∈ (((𝑀 + 1) − 𝑀)...((𝑀 + 𝑛) − 𝑀))((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
248196nncnd 11653 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℂ)
249 pncan2 10892 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 𝑀) = 1)
250248, 74, 249sylancl 588 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ((𝑀 + 1) − 𝑀) = 1)
251 nncn 11645 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
252251adantl 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
253248, 252pncan2d 10998 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ((𝑀 + 𝑛) − 𝑀) = 𝑛)
254250, 253oveq12d 7173 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (((𝑀 + 1) − 𝑀)...((𝑀 + 𝑛) − 𝑀)) = (1...𝑛))
255254sumeq1d 15057 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (((𝑀 + 1) − 𝑀)...((𝑀 + 𝑛) − 𝑀))((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) = Σ𝑘 ∈ (1...𝑛)((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
256133adantr 483 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
257 elfznn 12935 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
258134ovolfsval 24070 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑘 ∈ ℕ) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))‘𝑘) = ((2nd ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) − (1st ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘))))
259256, 257, 258syl2an 597 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))‘𝑘) = ((2nd ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) − (1st ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘))))
260257adantl 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
261 fvoveq1 7178 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = 𝑘 → (𝐺‘(𝑧 + 𝑀)) = (𝐺‘(𝑘 + 𝑀)))
262 eqid 2821 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))) = (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))
263 fvex 6682 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺‘(𝑘 + 𝑀)) ∈ V
264261, 262, 263fvmpt 6767 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ → ((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘) = (𝐺‘(𝑘 + 𝑀)))
265260, 264syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘) = (𝐺‘(𝑘 + 𝑀)))
266265fveq2d 6673 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (2nd ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) = (2nd ‘(𝐺‘(𝑘 + 𝑀))))
267265fveq2d 6673 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1st ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) = (1st ‘(𝐺‘(𝑘 + 𝑀))))
268266, 267oveq12d 7173 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((2nd ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) − (1st ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘))) = ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
269259, 268eqtrd 2856 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))‘𝑘) = ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
270 simpr 487 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
271270, 67eleqtrdi 2923 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
2724ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
273 nnaddcl 11659 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑘 + 𝑀) ∈ ℕ)
274257, 196, 273syl2anr 598 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑘 + 𝑀) ∈ ℕ)
275 ovolfcl 24066 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝑘 + 𝑀) ∈ ℕ) → ((1st ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ ∧ (2nd ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ ∧ (1st ‘(𝐺‘(𝑘 + 𝑀))) ≤ (2nd ‘(𝐺‘(𝑘 + 𝑀)))))
276272, 274, 275syl2anc 586 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((1st ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ ∧ (2nd ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ ∧ (1st ‘(𝐺‘(𝑘 + 𝑀))) ≤ (2nd ‘(𝐺‘(𝑘 + 𝑀)))))
277276simp2d 1139 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (2nd ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ)
278276simp1d 1138 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1st ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ)
279277, 278resubcld 11067 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) ∈ ℝ)
280279recnd 10668 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) ∈ ℂ)
281269, 271, 280fsumser 15086 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛))
282247, 255, 2813eqtrd 2860 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛))
283231, 282oveq12d 7173 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) + Σ𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗)))) = ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)))
284215, 219, 2833eqtr3d 2864 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛)) = ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)))
285187, 284syl5eq 2868 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) = ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)))
286123ffnd 6514 . . . . . . . . . . . . . 14 (𝜑𝑇 Fn ℕ)
287 fnfvelrn 6847 . . . . . . . . . . . . . 14 ((𝑇 Fn ℕ ∧ (𝑀 + 𝑛) ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) ∈ ran 𝑇)
288286, 199, 287syl2an2r 683 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) ∈ ran 𝑇)
289285, 288eqeltrrd 2914 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ∈ ran 𝑇)
290 supxrub 12716 . . . . . . . . . . . 12 ((ran 𝑇 ⊆ ℝ* ∧ ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ∈ ran 𝑇) → ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ≤ sup(ran 𝑇, ℝ*, < ))
291186, 289, 290syl2an2r 683 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ≤ sup(ran 𝑇, ℝ*, < ))
292125adantr 483 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝑇𝑀) ∈ ℝ)
293137ffvelrnda 6850 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ∈ (0[,)+∞))
294120, 293sseldi 3964 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ∈ ℝ)
29590adantr 483 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
296292, 294, 295leaddsub2d 11241 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ≤ sup(ran 𝑇, ℝ*, < ) ↔ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀))))
297291, 296mpbid 234 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
298297ralrimiva 3182 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
299137ffnd 6514 . . . . . . . . . 10 (𝜑 → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) Fn ℕ)
300 breq1 5068 . . . . . . . . . . 11 (𝑥 = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) → (𝑥 ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ↔ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀))))
301300ralrn 6853 . . . . . . . . . 10 (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) Fn ℕ → (∀𝑥 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))𝑥 ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ↔ ∀𝑛 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀))))
302299, 301syl 17 . . . . . . . . 9 (𝜑 → (∀𝑥 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))𝑥 ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ↔ ∀𝑛 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀))))
303298, 302mpbird 259 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))𝑥 ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
304 supxrleub 12718 . . . . . . . . 9 ((ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ ℝ* ∧ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ∈ ℝ*) → (sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ↔ ∀𝑥 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))𝑥 ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀))))
305140, 143, 304syl2anc 586 . . . . . . . 8 (𝜑 → (sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ↔ ∀𝑥 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))𝑥 ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀))))
306303, 305mpbird 259 . . . . . . 7 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
307127, 142, 143, 184, 306xrletrd 12554 . . . . . 6 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
308125, 90, 50absdifltd 14792 . . . . . . . . 9 (𝜑 → ((abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶 ↔ ((sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀) ∧ (𝑇𝑀) < (sup(ran 𝑇, ℝ*, < ) + 𝐶))))
30927, 308mpbid 234 . . . . . . . 8 (𝜑 → ((sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀) ∧ (𝑇𝑀) < (sup(ran 𝑇, ℝ*, < ) + 𝐶)))
310309simpld 497 . . . . . . 7 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀))
31190, 50, 125, 310ltsub23d 11244 . . . . . 6 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) < 𝐶)
31297, 126, 50, 307, 311lelttrd 10797 . . . . 5 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) < 𝐶)
31397, 50, 49, 312ltadd2dd 10798 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) < ((vol*‘(𝐾𝐴)) + 𝐶))
31413, 98, 51, 119, 313lelttrd 10797 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) < ((vol*‘(𝐾𝐴)) + 𝐶))
31554, 97readdcld 10669 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
316 difss 4107 . . . . . . . 8 (𝐾𝐴) ⊆ 𝐾
317 unss1 4154 . . . . . . . 8 ((𝐾𝐴) ⊆ 𝐾 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
318316, 317ax-mp 5 . . . . . . 7 ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
319318, 88sseqtrrid 4019 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺))
320 ovolsscl 24086 . . . . . 6 ((((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
321319, 9, 95, 320syl3anc 1367 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
322104ssdifd 4116 . . . . . . 7 (𝜑 → (𝐸𝐴) ⊆ ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴))
323 difundir 4256 . . . . . . . 8 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴) = ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴))
324 difss 4107 . . . . . . . . 9 ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))
325 unss2 4156 . . . . . . . . 9 (( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) → ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
326324, 325ax-mp 5 . . . . . . . 8 ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
327323, 326eqsstri 4000 . . . . . . 7 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
328322, 327sstrdi 3978 . . . . . 6 (𝜑 → (𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
329319, 9sstrd 3976 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ)
330 ovolss 24085 . . . . . 6 (((𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∧ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ) → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
331328, 329, 330syl2anc 586 . . . . 5 (𝜑 → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
33252, 46sstrd 3976 . . . . . 6 (𝜑 → (𝐾𝐴) ⊆ ℝ)
333 ovolun 24099 . . . . . 6 ((((𝐾𝐴) ⊆ ℝ ∧ (vol*‘(𝐾𝐴)) ∈ ℝ) ∧ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ℝ ∧ (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)) → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
334332, 54, 116, 97, 333syl22anc 836 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
33516, 321, 315, 331, 334letrd 10796 . . . 4 (𝜑 → (vol*‘(𝐸𝐴)) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
33697, 50, 54, 312ltadd2dd 10798 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) < ((vol*‘(𝐾𝐴)) + 𝐶))
33716, 315, 55, 335, 336lelttrd 10797 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) < ((vol*‘(𝐾𝐴)) + 𝐶))
33813, 16, 51, 55, 314, 337lt2addd 11262 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + 𝐶) + ((vol*‘(𝐾𝐴)) + 𝐶)))
33949recnd 10668 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℂ)
34050recnd 10668 . . 3 (𝜑𝐶 ∈ ℂ)
34154recnd 10668 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℂ)
342339, 340, 341, 340add4d 10867 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + 𝐶) + ((vol*‘(𝐾𝐴)) + 𝐶)) = (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
343338, 342breqtrd 5091 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  wrex 3139  Vcvv 3494  cdif 3932  cun 3933  cin 3934  wss 3935  c0 4290  𝒫 cpw 4538  cop 4572   cuni 4837   ciun 4918  Disj wdisj 5030   class class class wbr 5065  cmpt 5145   × cxp 5552  ran crn 5555  cima 5557  ccom 5558   Fn wfn 6349  wf 6350  cfv 6354  (class class class)co 7155  1st c1st 7686  2nd c2nd 7687  supcsup 8903  cc 10534  cr 10535  0cc0 10536  1c1 10537   + caddc 10539  +∞cpnf 10671  *cxr 10673   < clt 10674  cle 10675  cmin 10869  cn 11637  0cn0 11896  cz 11980  cuz 12242  +crp 12388  (,)cioo 12737  [,)cico 12739  [,]cicc 12740  ...cfz 12891  seqcseq 13368  abscabs 14592  Σcsu 15041  vol*covol 24062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-er 8288  df-map 8407  df-pm 8408  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-fi 8874  df-sup 8905  df-inf 8906  df-oi 8973  df-dju 9329  df-card 9367  df-acn 9370  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-q 12348  df-rp 12389  df-xneg 12506  df-xadd 12507  df-xmul 12508  df-ioo 12741  df-ico 12743  df-icc 12744  df-fz 12892  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13429  df-hash 13690  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844  df-rlim 14845  df-sum 15042  df-rest 16695  df-topgen 16716  df-psmet 20536  df-xmet 20537  df-met 20538  df-bl 20539  df-mopn 20540  df-top 21501  df-topon 21518  df-bases 21553  df-cmp 21994  df-ovol 24064  df-vol 24065
This theorem is referenced by:  uniioombllem5  24187
  Copyright terms: Public domain W3C validator