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

Theorem uniioombllem3 25620
Description: Lemma for uniioombl 25624. (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 4237 . . . . 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 25613 . . . . . . 7 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
65simpld 494 . . . . . 6 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
7 ovolficcss 25504 . . . . . . 7 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
84, 7syl 17 . . . . . 6 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
96, 8sstrd 3994 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
103, 9sstrd 3994 . . . 4 (𝜑𝐸 ⊆ ℝ)
11 uniioombl.e . . . 4 (𝜑 → (vol*‘𝐸) ∈ ℝ)
12 ovolsscl 25521 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
132, 10, 11, 12syl3anc 1373 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
14 difssd 4137 . . . 4 (𝜑 → (𝐸𝐴) ⊆ 𝐸)
15 ovolsscl 25521 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
1614, 10, 11, 15syl3anc 1373 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
17 inss1 4237 . . . . . 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 25619 . . . . . . 7 (𝜑 → (𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ∧ (vol*‘𝐾) ∈ ℝ))
3029simpld 494 . . . . . 6 (𝜑𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
31 inss2 4238 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
32 elfznn 13593 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
33 ffvelcdm 7101 . . . . . . . . . . . . . 14 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
344, 32, 33syl2an 596 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
3531, 34sselid 3981 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ (ℝ × ℝ))
36 1st2nd2 8053 . . . . . . . . . . . 12 ((𝐺𝑗) ∈ (ℝ × ℝ) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
3735, 36syl 17 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
3837fveq2d 6910 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
39 df-ov 7434 . . . . . . . . . 10 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
4038, 39eqtr4di 2795 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
41 ioossre 13448 . . . . . . . . 9 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
4240, 41eqsstrdi 4028 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
4342ralrimiva 3146 . . . . . . 7 (𝜑 → ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
44 iunss 5045 . . . . . . 7 ( 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ ↔ ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
4543, 44sylibr 234 . . . . . 6 (𝜑 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
4630, 45eqsstrd 4018 . . . . 5 (𝜑𝐾 ⊆ ℝ)
4729simprd 495 . . . . 5 (𝜑 → (vol*‘𝐾) ∈ ℝ)
48 ovolsscl 25521 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4918, 46, 47, 48syl3anc 1373 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
5023rpred 13077 . . . 4 (𝜑𝐶 ∈ ℝ)
5149, 50readdcld 11290 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + 𝐶) ∈ ℝ)
52 difssd 4137 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
53 ovolsscl 25521 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
5452, 46, 47, 53syl3anc 1373 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
5554, 50readdcld 11290 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + 𝐶) ∈ ℝ)
56 ssun2 4179 . . . . . . 7 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
57 ioof 13487 . . . . . . . . . . . . . . 15 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
58 rexpssxrxp 11306 . . . . . . . . . . . . . . . . 17 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5931, 58sstri 3993 . . . . . . . . . . . . . . . 16 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
60 fss 6752 . . . . . . . . . . . . . . . 16 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
614, 59, 60sylancl 586 . . . . . . . . . . . . . . 15 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
62 fco 6760 . . . . . . . . . . . . . . 15 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
6357, 61, 62sylancr 587 . . . . . . . . . . . . . 14 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
6463ffnd 6737 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
65 fnima 6698 . . . . . . . . . . . . 13 (((,) ∘ 𝐺) Fn ℕ → (((,) ∘ 𝐺) “ ℕ) = ran ((,) ∘ 𝐺))
6664, 65syl 17 . . . . . . . . . . . 12 (𝜑 → (((,) ∘ 𝐺) “ ℕ) = ran ((,) ∘ 𝐺))
67 nnuz 12921 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6826peano2nnd 12283 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 + 1) ∈ ℕ)
6968, 67eleqtrdi 2851 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 + 1) ∈ (ℤ‘1))
70 uzsplit 13636 . . . . . . . . . . . . . . . 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, 71eqtrid 2789 . . . . . . . . . . . . . 14 (𝜑 → ℕ = ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))))
7326nncnd 12282 . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ ℂ)
74 ax-1cn 11213 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
75 pncan 11514 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
7673, 74, 75sylancl 586 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
7776oveq2d 7447 . . . . . . . . . . . . . . 15 (𝜑 → (1...((𝑀 + 1) − 1)) = (1...𝑀))
7877uneq1d 4167 . . . . . . . . . . . . . 14 (𝜑 → ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))) = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
7972, 78eqtrd 2777 . . . . . . . . . . . . 13 (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
8079imaeq2d 6078 . . . . . . . . . . . 12 (𝜑 → (((,) ∘ 𝐺) “ ℕ) = (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
8166, 80eqtr3d 2779 . . . . . . . . . . 11 (𝜑 → ran ((,) ∘ 𝐺) = (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
82 imaundi 6169 . . . . . . . . . . 11 (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8381, 82eqtrdi 2793 . . . . . . . . . 10 (𝜑 → ran ((,) ∘ 𝐺) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8483unieqd 4920 . . . . . . . . 9 (𝜑 ran ((,) ∘ 𝐺) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
85 uniun 4930 . . . . . . . . 9 ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8684, 85eqtrdi 2793 . . . . . . . 8 (𝜑 ran ((,) ∘ 𝐺) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8728uneq1i 4164 . . . . . . . 8 (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8886, 87eqtr4di 2795 . . . . . . 7 (𝜑 ran ((,) ∘ 𝐺) = (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8956, 88sseqtrrid 4027 . . . . . 6 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ 𝐺))
9019, 20, 21, 22, 11, 23, 4, 3, 24, 25uniioombllem1 25616 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
91 ssid 4006 . . . . . . . 8 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
9224ovollb 25514 . . . . . . . 8 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
934, 91, 92sylancl 586 . . . . . . 7 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
94 ovollecl 25518 . . . . . . 7 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
959, 90, 93, 94syl3anc 1373 . . . . . 6 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
96 ovolsscl 25521 . . . . . 6 (( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)
9789, 9, 95, 96syl3anc 1373 . . . . 5 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)
9849, 97readdcld 11290 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
99 unss1 4185 . . . . . . . 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 4027 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺))
102 ovolsscl 25521 . . . . . 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 1373 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
1043, 88sseqtrd 4020 . . . . . . . 8 (𝜑𝐸 ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
105104ssrind 4244 . . . . . . 7 (𝜑 → (𝐸𝐴) ⊆ ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴))
106 indir 4286 . . . . . . . 8 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴) = ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴))
107 inss1 4237 . . . . . . . . 9 ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))
108 unss2 4187 . . . . . . . . 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 4030 . . . . . . 7 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
111105, 110sstrdi 3996 . . . . . 6 (𝜑 → (𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
112101, 9sstrd 3994 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ)
113 ovolss 25520 . . . . . 6 (((𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∧ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ) → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
114111, 112, 113syl2anc 584 . . . . 5 (𝜑 → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
11518, 46sstrd 3994 . . . . . 6 (𝜑 → (𝐾𝐴) ⊆ ℝ)
11689, 9sstrd 3994 . . . . . 6 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ℝ)
117 ovolun 25534 . . . . . 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 839 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
11913, 103, 98, 114, 118letrd 11418 . . . 4 (𝜑 → (vol*‘(𝐸𝐴)) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
120 rge0ssre 13496 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
121 eqid 2737 . . . . . . . . . . 11 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
122121, 24ovolsf 25507 . . . . . . . . . 10 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
1234, 122syl 17 . . . . . . . . 9 (𝜑𝑇:ℕ⟶(0[,)+∞))
124123, 26ffvelcdmd 7105 . . . . . . . 8 (𝜑 → (𝑇𝑀) ∈ (0[,)+∞))
125120, 124sselid 3981 . . . . . . 7 (𝜑 → (𝑇𝑀) ∈ ℝ)
12690, 125resubcld 11691 . . . . . 6 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ∈ ℝ)
12797rexrd 11311 . . . . . . 7 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ*)
128 id 22 . . . . . . . . . . . . . 14 (𝑧 ∈ ℕ → 𝑧 ∈ ℕ)
129 nnaddcl 12289 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑧 + 𝑀) ∈ ℕ)
130128, 26, 129syl2anr 597 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ℕ) → (𝑧 + 𝑀) ∈ ℕ)
1314ffvelcdmda 7104 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 + 𝑀) ∈ ℕ) → (𝐺‘(𝑧 + 𝑀)) ∈ ( ≤ ∩ (ℝ × ℝ)))
132130, 131syldan 591 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℕ) → (𝐺‘(𝑧 + 𝑀)) ∈ ( ≤ ∩ (ℝ × ℝ)))
133132fmpttd 7135 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
134 eqid 2737 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))) = ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
135 eqid 2737 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) = seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
136134, 135ovolsf 25507 . . . . . . . . . . 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 6744 . . . . . . . . 9 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ (0[,)+∞))
139 icossxr 13472 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ*
140138, 139sstrdi 3996 . . . . . . . 8 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ ℝ*)
141 supxrcl 13357 . . . . . . . 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 11311 . . . . . . 7 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ∈ ℝ*)
144 1zzd 12648 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 1 ∈ ℤ)
14526nnzd 12640 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℤ)
146145adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
147 addcom 11447 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑀 + 1) = (1 + 𝑀))
14873, 74, 147sylancl 586 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 + 1) = (1 + 𝑀))
149148fveq2d 6910 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (ℤ‘(𝑀 + 1)) = (ℤ‘(1 + 𝑀)))
150149eleq2d 2827 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑥 ∈ (ℤ‘(𝑀 + 1)) ↔ 𝑥 ∈ (ℤ‘(1 + 𝑀))))
151150biimpa 476 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ‘(1 + 𝑀)))
152 eluzsub 12908 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ (ℤ‘(1 + 𝑀))) → (𝑥𝑀) ∈ (ℤ‘1))
153144, 146, 151, 152syl3anc 1373 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝑥𝑀) ∈ (ℤ‘1))
154153, 67eleqtrrdi 2852 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝑥𝑀) ∈ ℕ)
155 eluzelz 12888 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (ℤ‘(𝑀 + 1)) → 𝑥 ∈ ℤ)
156155adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ℤ)
157156zcnd 12723 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ℂ)
15873adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℂ)
159157, 158npcand 11624 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → ((𝑥𝑀) + 𝑀) = 𝑥)
160159eqcomd 2743 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 = ((𝑥𝑀) + 𝑀))
161 oveq1 7438 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑥𝑀) → (𝑧 + 𝑀) = ((𝑥𝑀) + 𝑀))
162161rspceeqv 3645 . . . . . . . . . . . . . . . . 17 (((𝑥𝑀) ∈ ℕ ∧ 𝑥 = ((𝑥𝑀) + 𝑀)) → ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
163154, 160, 162syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
164 eqid 2737 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) = (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))
165164elrnmpt 5969 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ V → (𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) ↔ ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀)))
166165elv 3485 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) ↔ ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
167163, 166sylibr 234 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
168167ex 412 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ (ℤ‘(𝑀 + 1)) → 𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
169168ssrdv 3989 . . . . . . . . . . . . 13 (𝜑 → (ℤ‘(𝑀 + 1)) ⊆ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
170 imass2 6120 . . . . . . . . . . . . 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 6273 . . . . . . . . . . . . 13 ran (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
1734, 130cofmpt 7152 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
174173rneqd 5949 . . . . . . . . . . . . 13 (𝜑 → ran (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
175172, 174eqtr3id 2791 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
176171, 175sseqtrd 4020 . . . . . . . . . . 11 (𝜑 → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
177 imass2 6120 . . . . . . . . . . 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 6271 . . . . . . . . . 10 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) = ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1))))
180 rnco2 6273 . . . . . . . . . 10 ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))) = ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
181178, 179, 1803sstr4g 4037 . . . . . . . . 9 (𝜑 → (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
182181unissd 4917 . . . . . . . 8 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
183135ovollb 25514 . . . . . . . 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 584 . . . . . . 7 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ))
185123frnd 6744 . . . . . . . . . . . . 13 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
186185, 139sstrdi 3996 . . . . . . . . . . . 12 (𝜑 → ran 𝑇 ⊆ ℝ*)
18724fveq1i 6907 . . . . . . . . . . . . . 14 (𝑇‘(𝑀 + 𝑛)) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛))
18826nnred 12281 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℝ)
189188ltp1d 12198 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 < (𝑀 + 1))
190 fzdisj 13591 . . . . . . . . . . . . . . . . . 18 (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
191189, 190syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
192191adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
193 nnnn0 12533 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
194 nn0addge1 12572 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℝ ∧ 𝑛 ∈ ℕ0) → 𝑀 ≤ (𝑀 + 𝑛))
195188, 193, 194syl2an 596 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑀 ≤ (𝑀 + 𝑛))
19626adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℕ)
197196, 67eleqtrdi 2851 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ (ℤ‘1))
198 nnaddcl 12289 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℕ)
19926, 198sylan 580 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℕ)
200199nnzd 12640 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℤ)
201 elfz5 13556 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ (ℤ‘1) ∧ (𝑀 + 𝑛) ∈ ℤ) → (𝑀 ∈ (1...(𝑀 + 𝑛)) ↔ 𝑀 ≤ (𝑀 + 𝑛)))
202197, 200, 201syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (𝑀 ∈ (1...(𝑀 + 𝑛)) ↔ 𝑀 ≤ (𝑀 + 𝑛)))
203195, 202mpbird 257 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ (1...(𝑀 + 𝑛)))
204 fzsplit 13590 . . . . . . . . . . . . . . . . 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 14014 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1...(𝑀 + 𝑛)) ∈ Fin)
2074adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
208 elfznn 13593 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (1...(𝑀 + 𝑛)) → 𝑗 ∈ ℕ)
209 ovolfcl 25501 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
210207, 208, 209syl2an 596 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
211210simp2d 1144 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
212210simp1d 1143 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (1st ‘(𝐺𝑗)) ∈ ℝ)
213211, 212resubcld 11691 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
214213recnd 11289 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
215192, 205, 206, 214fsumsplit 15777 . . . . . . . . . . . . . . 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 25505 . . . . . . . . . . . . . . . . 17 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
217207, 208, 216syl2an 596 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
218199, 67eleqtrdi 2851 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ (ℤ‘1))
219217, 218, 214fsumser 15766 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛)))
2204ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22132adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
222220, 221, 216syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
2234, 32, 209syl2an 596 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (1...𝑀)) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
224223simp2d 1144 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
225223simp1d 1143 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
226224, 225resubcld 11691 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
227226adantlr 715 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
228227recnd 11289 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
229222, 197, 228fsumser 15766 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑀))
23024fveq1i 6907 . . . . . . . . . . . . . . . . 17 (𝑇𝑀) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑀)
231229, 230eqtr4di 2795 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (𝑇𝑀))
232196nnzd 12640 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℤ)
233232peano2zd 12725 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 1) ∈ ℤ)
2344ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
235196peano2nnd 12283 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 1) ∈ ℕ)
236 elfzuz 13560 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛)) → 𝑗 ∈ (ℤ‘(𝑀 + 1)))
237 eluznn 12960 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 + 1) ∈ ℕ ∧ 𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝑗 ∈ ℕ)
238235, 236, 237syl2an 596 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → 𝑗 ∈ ℕ)
239234, 238, 209syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
240239simp2d 1144 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
241239simp1d 1143 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → (1st ‘(𝐺𝑗)) ∈ ℝ)
242240, 241resubcld 11691 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
243242recnd 11289 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
244 2fveq3 6911 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 𝑀) → (2nd ‘(𝐺𝑗)) = (2nd ‘(𝐺‘(𝑘 + 𝑀))))
245 2fveq3 6911 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 𝑀) → (1st ‘(𝐺𝑗)) = (1st ‘(𝐺‘(𝑘 + 𝑀))))
246244, 245oveq12d 7449 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑘 + 𝑀) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
247232, 233, 200, 243, 246fsumshftm 15817 . . . . . . . . . . . . . . . . 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 12282 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℂ)
249 pncan2 11515 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 𝑀) = 1)
250248, 74, 249sylancl 586 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ((𝑀 + 1) − 𝑀) = 1)
251 nncn 12274 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
252251adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
253248, 252pncan2d 11622 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ((𝑀 + 𝑛) − 𝑀) = 𝑛)
254250, 253oveq12d 7449 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (((𝑀 + 1) − 𝑀)...((𝑀 + 𝑛) − 𝑀)) = (1...𝑛))
255254sumeq1d 15736 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
257 elfznn 13593 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
258134ovolfsval 25505 . . . . . . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
261 fvoveq1 7454 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = 𝑘 → (𝐺‘(𝑧 + 𝑀)) = (𝐺‘(𝑘 + 𝑀)))
262 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))) = (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))
263 fvex 6919 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺‘(𝑘 + 𝑀)) ∈ V
264261, 262, 263fvmpt 7016 . . . . . . . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (2nd ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) = (2nd ‘(𝐺‘(𝑘 + 𝑀))))
267265fveq2d 6910 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1st ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) = (1st ‘(𝐺‘(𝑘 + 𝑀))))
268266, 267oveq12d 7449 . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))‘𝑘) = ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
270 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
271270, 67eleqtrdi 2851 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
2724ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
273 nnaddcl 12289 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑘 + 𝑀) ∈ ℕ)
274257, 196, 273syl2anr 597 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑘 + 𝑀) ∈ ℕ)
275 ovolfcl 25501 . . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . . 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 1144 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (2nd ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ)
278276simp1d 1143 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1st ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ)
279277, 278resubcld 11691 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) ∈ ℝ)
280279recnd 11289 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) ∈ ℂ)
281269, 271, 280fsumser 15766 . . . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛))
283231, 282oveq12d 7449 . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛)) = ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)))
285187, 284eqtrid 2789 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) = ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)))
286123ffnd 6737 . . . . . . . . . . . . . 14 (𝜑𝑇 Fn ℕ)
287 fnfvelrn 7100 . . . . . . . . . . . . . 14 ((𝑇 Fn ℕ ∧ (𝑀 + 𝑛) ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) ∈ ran 𝑇)
288286, 199, 287syl2an2r 685 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) ∈ ran 𝑇)
289285, 288eqeltrrd 2842 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ∈ ran 𝑇)
290 supxrub 13366 . . . . . . . . . . . 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 685 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ≤ sup(ran 𝑇, ℝ*, < ))
292125adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝑇𝑀) ∈ ℝ)
293137ffvelcdmda 7104 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ∈ (0[,)+∞))
294120, 293sselid 3981 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ∈ ℝ)
29590adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
296292, 294, 295leaddsub2d 11865 . . . . . . . . . . 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 232 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
298297ralrimiva 3146 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
299137ffnd 6737 . . . . . . . . . 10 (𝜑 → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) Fn ℕ)
300 breq1 5146 . . . . . . . . . . 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 7108 . . . . . . . . . 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 257 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))𝑥 ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
304 supxrleub 13368 . . . . . . . . 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 584 . . . . . . . 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 257 . . . . . . 7 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
307127, 142, 143, 184, 306xrletrd 13204 . . . . . 6 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
308125, 90, 50absdifltd 15472 . . . . . . . . 9 (𝜑 → ((abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶 ↔ ((sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀) ∧ (𝑇𝑀) < (sup(ran 𝑇, ℝ*, < ) + 𝐶))))
30927, 308mpbid 232 . . . . . . . 8 (𝜑 → ((sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀) ∧ (𝑇𝑀) < (sup(ran 𝑇, ℝ*, < ) + 𝐶)))
310309simpld 494 . . . . . . 7 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀))
31190, 50, 125, 310ltsub23d 11868 . . . . . 6 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) < 𝐶)
31297, 126, 50, 307, 311lelttrd 11419 . . . . 5 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) < 𝐶)
31397, 50, 49, 312ltadd2dd 11420 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) < ((vol*‘(𝐾𝐴)) + 𝐶))
31413, 98, 51, 119, 313lelttrd 11419 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) < ((vol*‘(𝐾𝐴)) + 𝐶))
31554, 97readdcld 11290 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
316 difss 4136 . . . . . . . 8 (𝐾𝐴) ⊆ 𝐾
317 unss1 4185 . . . . . . . 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 4027 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺))
320 ovolsscl 25521 . . . . . 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 1373 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
322104ssdifd 4145 . . . . . . 7 (𝜑 → (𝐸𝐴) ⊆ ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴))
323 difundir 4291 . . . . . . . 8 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴) = ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴))
324 difss 4136 . . . . . . . . 9 ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))
325 unss2 4187 . . . . . . . . 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 4030 . . . . . . 7 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
328322, 327sstrdi 3996 . . . . . 6 (𝜑 → (𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
329319, 9sstrd 3994 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ)
330 ovolss 25520 . . . . . 6 (((𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∧ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ) → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
331328, 329, 330syl2anc 584 . . . . 5 (𝜑 → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
33252, 46sstrd 3994 . . . . . 6 (𝜑 → (𝐾𝐴) ⊆ ℝ)
333 ovolun 25534 . . . . . 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 839 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
33516, 321, 315, 331, 334letrd 11418 . . . 4 (𝜑 → (vol*‘(𝐸𝐴)) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
33697, 50, 54, 312ltadd2dd 11420 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) < ((vol*‘(𝐾𝐴)) + 𝐶))
33716, 315, 55, 335, 336lelttrd 11419 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) < ((vol*‘(𝐾𝐴)) + 𝐶))
33813, 16, 51, 55, 314, 337lt2addd 11886 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + 𝐶) + ((vol*‘(𝐾𝐴)) + 𝐶)))
33949recnd 11289 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℂ)
34050recnd 11289 . . 3 (𝜑𝐶 ∈ ℂ)
34154recnd 11289 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℂ)
342339, 340, 341, 340add4d 11490 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + 𝐶) + ((vol*‘(𝐾𝐴)) + 𝐶)) = (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
343338, 342breqtrd 5169 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  cop 4632   cuni 4907   ciun 4991  Disj wdisj 5110   class class class wbr 5143  cmpt 5225   × cxp 5683  ran crn 5686  cima 5688  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  supcsup 9480  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cmin 11492  cn 12266  0cn0 12526  cz 12613  cuz 12878  +crp 13034  (,)cioo 13387  [,)cico 13389  [,]cicc 13390  ...cfz 13547  seqcseq 14042  abscabs 15273  Σcsu 15722  vol*covol 25497
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-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-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:  uniioombllem5  25622
  Copyright terms: Public domain W3C validator