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Theorem uniioombllem3 23693
Description: Lemma for uniioombl 23697. (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 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 4028 . . . . 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 23686 . . . . . . 7 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
65simpld 489 . . . . . 6 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
7 ovolficcss 23577 . . . . . . 7 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
84, 7syl 17 . . . . . 6 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
96, 8sstrd 3808 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
103, 9sstrd 3808 . . . 4 (𝜑𝐸 ⊆ ℝ)
11 uniioombl.e . . . 4 (𝜑 → (vol*‘𝐸) ∈ ℝ)
12 ovolsscl 23594 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
132, 10, 11, 12syl3anc 1491 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
14 difssd 3936 . . . 4 (𝜑 → (𝐸𝐴) ⊆ 𝐸)
15 ovolsscl 23594 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
1614, 10, 11, 15syl3anc 1491 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
17 inss1 4028 . . . . . 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 23692 . . . . . . 7 (𝜑 → (𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ∧ (vol*‘𝐾) ∈ ℝ))
3029simpld 489 . . . . . 6 (𝜑𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
31 inss2 4029 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
32 elfznn 12624 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
33 ffvelrn 6583 . . . . . . . . . . . . . 14 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
344, 32, 33syl2an 590 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
3531, 34sseldi 3796 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ (ℝ × ℝ))
36 1st2nd2 7440 . . . . . . . . . . . 12 ((𝐺𝑗) ∈ (ℝ × ℝ) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
3735, 36syl 17 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
3837fveq2d 6415 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
39 df-ov 6881 . . . . . . . . . 10 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
4038, 39syl6eqr 2851 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
41 ioossre 12484 . . . . . . . . 9 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
4240, 41syl6eqss 3851 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
4342ralrimiva 3147 . . . . . . 7 (𝜑 → ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
44 iunss 4751 . . . . . . 7 ( 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ ↔ ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
4543, 44sylibr 226 . . . . . 6 (𝜑 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
4630, 45eqsstrd 3835 . . . . 5 (𝜑𝐾 ⊆ ℝ)
4729simprd 490 . . . . 5 (𝜑 → (vol*‘𝐾) ∈ ℝ)
48 ovolsscl 23594 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4918, 46, 47, 48syl3anc 1491 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
5023rpred 12117 . . . 4 (𝜑𝐶 ∈ ℝ)
5149, 50readdcld 10358 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + 𝐶) ∈ ℝ)
52 difssd 3936 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
53 ovolsscl 23594 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
5452, 46, 47, 53syl3anc 1491 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
5554, 50readdcld 10358 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + 𝐶) ∈ ℝ)
56 ssun2 3975 . . . . . . 7 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
57 ioof 12521 . . . . . . . . . . . . . . 15 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
58 rexpssxrxp 10373 . . . . . . . . . . . . . . . . 17 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5931, 58sstri 3807 . . . . . . . . . . . . . . . 16 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
60 fss 6269 . . . . . . . . . . . . . . . 16 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
614, 59, 60sylancl 581 . . . . . . . . . . . . . . 15 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
62 fco 6273 . . . . . . . . . . . . . . 15 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
6357, 61, 62sylancr 582 . . . . . . . . . . . . . 14 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
6463ffnd 6257 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
65 fnima 6221 . . . . . . . . . . . . 13 (((,) ∘ 𝐺) Fn ℕ → (((,) ∘ 𝐺) “ ℕ) = ran ((,) ∘ 𝐺))
6664, 65syl 17 . . . . . . . . . . . 12 (𝜑 → (((,) ∘ 𝐺) “ ℕ) = ran ((,) ∘ 𝐺))
67 nnuz 11967 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6826peano2nnd 11331 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 + 1) ∈ ℕ)
6968, 67syl6eleq 2888 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 + 1) ∈ (ℤ‘1))
70 uzsplit 12666 . . . . . . . . . . . . . . . 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 2845 . . . . . . . . . . . . . 14 (𝜑 → ℕ = ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))))
7326nncnd 11330 . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ ℂ)
74 ax-1cn 10282 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
75 pncan 10578 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
7673, 74, 75sylancl 581 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
7776oveq2d 6894 . . . . . . . . . . . . . . 15 (𝜑 → (1...((𝑀 + 1) − 1)) = (1...𝑀))
7877uneq1d 3964 . . . . . . . . . . . . . 14 (𝜑 → ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))) = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
7972, 78eqtrd 2833 . . . . . . . . . . . . 13 (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
8079imaeq2d 5683 . . . . . . . . . . . 12 (𝜑 → (((,) ∘ 𝐺) “ ℕ) = (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
8166, 80eqtr3d 2835 . . . . . . . . . . 11 (𝜑 → ran ((,) ∘ 𝐺) = (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
82 imaundi 5762 . . . . . . . . . . 11 (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8381, 82syl6eq 2849 . . . . . . . . . 10 (𝜑 → ran ((,) ∘ 𝐺) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8483unieqd 4638 . . . . . . . . 9 (𝜑 ran ((,) ∘ 𝐺) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
85 uniun 4649 . . . . . . . . 9 ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8684, 85syl6eq 2849 . . . . . . . 8 (𝜑 ran ((,) ∘ 𝐺) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8728uneq1i 3961 . . . . . . . 8 (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8886, 87syl6eqr 2851 . . . . . . 7 (𝜑 ran ((,) ∘ 𝐺) = (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8956, 88syl5sseqr 3850 . . . . . 6 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ 𝐺))
9019, 20, 21, 22, 11, 23, 4, 3, 24, 25uniioombllem1 23689 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
91 ssid 3819 . . . . . . . 8 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
9224ovollb 23587 . . . . . . . 8 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
934, 91, 92sylancl 581 . . . . . . 7 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
94 ovollecl 23591 . . . . . . 7 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
959, 90, 93, 94syl3anc 1491 . . . . . 6 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
96 ovolsscl 23594 . . . . . 6 (( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)
9789, 9, 95, 96syl3anc 1491 . . . . 5 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)
9849, 97readdcld 10358 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
99 unss1 3980 . . . . . . . 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, 88syl5sseqr 3850 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺))
102 ovolsscl 23594 . . . . . 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 1491 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
1043, 88sseqtrd 3837 . . . . . . . 8 (𝜑𝐸 ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
105104ssrind 4035 . . . . . . 7 (𝜑 → (𝐸𝐴) ⊆ ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴))
106 indir 4076 . . . . . . . 8 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴) = ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴))
107 inss1 4028 . . . . . . . . 9 ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))
108 unss2 3982 . . . . . . . . 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 3831 . . . . . . 7 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
111105, 110syl6ss 3810 . . . . . 6 (𝜑 → (𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
112101, 9sstrd 3808 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ)
113 ovolss 23593 . . . . . 6 (((𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∧ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ) → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
114111, 112, 113syl2anc 580 . . . . 5 (𝜑 → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
11518, 46sstrd 3808 . . . . . 6 (𝜑 → (𝐾𝐴) ⊆ ℝ)
11689, 9sstrd 3808 . . . . . 6 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ℝ)
117 ovolun 23607 . . . . . 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 868 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
11913, 103, 98, 114, 118letrd 10484 . . . 4 (𝜑 → (vol*‘(𝐸𝐴)) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
120 rge0ssre 12531 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
121 eqid 2799 . . . . . . . . . . 11 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
122121, 24ovolsf 23580 . . . . . . . . . 10 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
1234, 122syl 17 . . . . . . . . 9 (𝜑𝑇:ℕ⟶(0[,)+∞))
124123, 26ffvelrnd 6586 . . . . . . . 8 (𝜑 → (𝑇𝑀) ∈ (0[,)+∞))
125120, 124sseldi 3796 . . . . . . 7 (𝜑 → (𝑇𝑀) ∈ ℝ)
12690, 125resubcld 10750 . . . . . 6 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ∈ ℝ)
12797rexrd 10378 . . . . . . 7 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ*)
128 id 22 . . . . . . . . . . . . . 14 (𝑧 ∈ ℕ → 𝑧 ∈ ℕ)
129 nnaddcl 11336 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑧 + 𝑀) ∈ ℕ)
130128, 26, 129syl2anr 591 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ℕ) → (𝑧 + 𝑀) ∈ ℕ)
1314ffvelrnda 6585 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 + 𝑀) ∈ ℕ) → (𝐺‘(𝑧 + 𝑀)) ∈ ( ≤ ∩ (ℝ × ℝ)))
132130, 131syldan 586 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℕ) → (𝐺‘(𝑧 + 𝑀)) ∈ ( ≤ ∩ (ℝ × ℝ)))
133132fmpttd 6611 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
134 eqid 2799 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))) = ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
135 eqid 2799 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) = seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
136134, 135ovolsf 23580 . . . . . . . . . . 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 6263 . . . . . . . . 9 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ (0[,)+∞))
139 icossxr 12507 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ*
140138, 139syl6ss 3810 . . . . . . . 8 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ ℝ*)
141 supxrcl 12394 . . . . . . . 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 10378 . . . . . . 7 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ∈ ℝ*)
144 1zzd 11698 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 1 ∈ ℤ)
14526nnzd 11771 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℤ)
146145adantr 473 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
147 addcom 10512 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑀 + 1) = (1 + 𝑀))
14873, 74, 147sylancl 581 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 + 1) = (1 + 𝑀))
149148fveq2d 6415 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (ℤ‘(𝑀 + 1)) = (ℤ‘(1 + 𝑀)))
150149eleq2d 2864 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑥 ∈ (ℤ‘(𝑀 + 1)) ↔ 𝑥 ∈ (ℤ‘(1 + 𝑀))))
151150biimpa 469 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ‘(1 + 𝑀)))
152 eluzsub 11960 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ (ℤ‘(1 + 𝑀))) → (𝑥𝑀) ∈ (ℤ‘1))
153144, 146, 151, 152syl3anc 1491 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝑥𝑀) ∈ (ℤ‘1))
154153, 67syl6eleqr 2889 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝑥𝑀) ∈ ℕ)
155 eluzelz 11940 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (ℤ‘(𝑀 + 1)) → 𝑥 ∈ ℤ)
156155adantl 474 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ℤ)
157156zcnd 11773 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ℂ)
15873adantr 473 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℂ)
159157, 158npcand 10688 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → ((𝑥𝑀) + 𝑀) = 𝑥)
160159eqcomd 2805 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 = ((𝑥𝑀) + 𝑀))
161 oveq1 6885 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑥𝑀) → (𝑧 + 𝑀) = ((𝑥𝑀) + 𝑀))
162161rspceeqv 3515 . . . . . . . . . . . . . . . . 17 (((𝑥𝑀) ∈ ℕ ∧ 𝑥 = ((𝑥𝑀) + 𝑀)) → ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
163154, 160, 162syl2anc 580 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
164 vex 3388 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
165 eqid 2799 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) = (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))
166165elrnmpt 5576 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ V → (𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) ↔ ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀)))
167164, 166ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) ↔ ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
168163, 167sylibr 226 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
169168ex 402 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ (ℤ‘(𝑀 + 1)) → 𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
170169ssrdv 3804 . . . . . . . . . . . . 13 (𝜑 → (ℤ‘(𝑀 + 1)) ⊆ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
171 imass2 5718 . . . . . . . . . . . . 13 ((ℤ‘(𝑀 + 1)) ⊆ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
172170, 171syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
173 rnco2 5861 . . . . . . . . . . . . 13 ran (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
174 eqidd 2800 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) = (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
1754feqmptd 6474 . . . . . . . . . . . . . . 15 (𝜑𝐺 = (𝑤 ∈ ℕ ↦ (𝐺𝑤)))
176 fveq2 6411 . . . . . . . . . . . . . . 15 (𝑤 = (𝑧 + 𝑀) → (𝐺𝑤) = (𝐺‘(𝑧 + 𝑀)))
177130, 174, 175, 176fmptco 6623 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
178177rneqd 5556 . . . . . . . . . . . . 13 (𝜑 → ran (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
179173, 178syl5eqr 2847 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
180172, 179sseqtrd 3837 . . . . . . . . . . 11 (𝜑 → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
181 imass2 5718 . . . . . . . . . . 11 ((𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))) → ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1)))) ⊆ ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
182180, 181syl 17 . . . . . . . . . 10 (𝜑 → ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1)))) ⊆ ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
183 imaco 5859 . . . . . . . . . 10 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) = ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1))))
184 rnco2 5861 . . . . . . . . . 10 ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))) = ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
185182, 183, 1843sstr4g 3842 . . . . . . . . 9 (𝜑 → (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
186185unissd 4654 . . . . . . . 8 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
187135ovollb 23587 . . . . . . . 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*, < ))
188133, 186, 187syl2anc 580 . . . . . . 7 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ))
189123frnd 6263 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
190189, 139syl6ss 3810 . . . . . . . . . . . . 13 (𝜑 → ran 𝑇 ⊆ ℝ*)
191190adantr 473 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ran 𝑇 ⊆ ℝ*)
19224fveq1i 6412 . . . . . . . . . . . . . 14 (𝑇‘(𝑀 + 𝑛)) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛))
19326nnred 11329 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℝ)
194193ltp1d 11246 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 < (𝑀 + 1))
195 fzdisj 12622 . . . . . . . . . . . . . . . . . 18 (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
196194, 195syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
197196adantr 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
198 nnnn0 11588 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
199 nn0addge1 11628 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℝ ∧ 𝑛 ∈ ℕ0) → 𝑀 ≤ (𝑀 + 𝑛))
200193, 198, 199syl2an 590 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑀 ≤ (𝑀 + 𝑛))
20126adantr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℕ)
202201, 67syl6eleq 2888 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ (ℤ‘1))
203 nnaddcl 11336 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℕ)
20426, 203sylan 576 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℕ)
205204nnzd 11771 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℤ)
206 elfz5 12588 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ (ℤ‘1) ∧ (𝑀 + 𝑛) ∈ ℤ) → (𝑀 ∈ (1...(𝑀 + 𝑛)) ↔ 𝑀 ≤ (𝑀 + 𝑛)))
207202, 205, 206syl2anc 580 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (𝑀 ∈ (1...(𝑀 + 𝑛)) ↔ 𝑀 ≤ (𝑀 + 𝑛)))
208200, 207mpbird 249 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ (1...(𝑀 + 𝑛)))
209 fzsplit 12621 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (1...(𝑀 + 𝑛)) → (1...(𝑀 + 𝑛)) = ((1...𝑀) ∪ ((𝑀 + 1)...(𝑀 + 𝑛))))
210208, 209syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1...(𝑀 + 𝑛)) = ((1...𝑀) ∪ ((𝑀 + 1)...(𝑀 + 𝑛))))
211 fzfid 13027 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1...(𝑀 + 𝑛)) ∈ Fin)
2124adantr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
213 elfznn 12624 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (1...(𝑀 + 𝑛)) → 𝑗 ∈ ℕ)
214 ovolfcl 23574 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
215212, 213, 214syl2an 590 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
216215simp2d 1174 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
217215simp1d 1173 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (1st ‘(𝐺𝑗)) ∈ ℝ)
218216, 217resubcld 10750 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
219218recnd 10357 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
220197, 210, 211, 219fsumsplit 14812 . . . . . . . . . . . . . . 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)))))
221121ovolfsval 23578 . . . . . . . . . . . . . . . . 17 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
222212, 213, 221syl2an 590 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
223204, 67syl6eleq 2888 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ (ℤ‘1))
224222, 223, 219fsumser 14802 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛)))
2254ad2antrr 718 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22632adantl 474 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
227225, 226, 221syl2anc 580 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
2284, 32, 214syl2an 590 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (1...𝑀)) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
229228simp2d 1174 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
230228simp1d 1173 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
231229, 230resubcld 10750 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
232231adantlr 707 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
233232recnd 10357 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
234227, 202, 233fsumser 14802 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑀))
23524fveq1i 6412 . . . . . . . . . . . . . . . . 17 (𝑇𝑀) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑀)
236234, 235syl6eqr 2851 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (𝑇𝑀))
237201nnzd 11771 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℤ)
238237peano2zd 11775 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 1) ∈ ℤ)
2394ad2antrr 718 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
240201peano2nnd 11331 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 1) ∈ ℕ)
241 elfzuz 12592 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛)) → 𝑗 ∈ (ℤ‘(𝑀 + 1)))
242 eluznn 12003 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 + 1) ∈ ℕ ∧ 𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝑗 ∈ ℕ)
243240, 241, 242syl2an 590 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → 𝑗 ∈ ℕ)
244239, 243, 214syl2anc 580 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
245244simp2d 1174 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
246244simp1d 1173 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → (1st ‘(𝐺𝑗)) ∈ ℝ)
247245, 246resubcld 10750 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
248247recnd 10357 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
249 2fveq3 6416 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 𝑀) → (2nd ‘(𝐺𝑗)) = (2nd ‘(𝐺‘(𝑘 + 𝑀))))
250 2fveq3 6416 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 𝑀) → (1st ‘(𝐺𝑗)) = (1st ‘(𝐺‘(𝑘 + 𝑀))))
251249, 250oveq12d 6896 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑘 + 𝑀) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
252237, 238, 205, 248, 251fsumshftm 14851 . . . . . . . . . . . . . . . . 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)))))
253201nncnd 11330 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℂ)
254 pncan2 10579 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 𝑀) = 1)
255253, 74, 254sylancl 581 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ((𝑀 + 1) − 𝑀) = 1)
256 nncn 11321 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
257256adantl 474 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
258253, 257pncan2d 10686 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ((𝑀 + 𝑛) − 𝑀) = 𝑛)
259255, 258oveq12d 6896 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (((𝑀 + 1) − 𝑀)...((𝑀 + 𝑛) − 𝑀)) = (1...𝑛))
260259sumeq1d 14772 . . . . . . . . . . . . . . . . 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)))))
261133adantr 473 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
262 elfznn 12624 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
263134ovolfsval 23578 . . . . . . . . . . . . . . . . . . . 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))))
264261, 262, 263syl2an 590 . . . . . . . . . . . . . . . . . . 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))))
265262adantl 474 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
266 fvoveq1 6901 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = 𝑘 → (𝐺‘(𝑧 + 𝑀)) = (𝐺‘(𝑘 + 𝑀)))
267 eqid 2799 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))) = (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))
268 fvex 6424 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺‘(𝑘 + 𝑀)) ∈ V
269266, 267, 268fvmpt 6507 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ → ((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘) = (𝐺‘(𝑘 + 𝑀)))
270265, 269syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘) = (𝐺‘(𝑘 + 𝑀)))
271270fveq2d 6415 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (2nd ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) = (2nd ‘(𝐺‘(𝑘 + 𝑀))))
272270fveq2d 6415 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1st ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) = (1st ‘(𝐺‘(𝑘 + 𝑀))))
273271, 272oveq12d 6896 . . . . . . . . . . . . . . . . . . 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)))))
274264, 273eqtrd 2833 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))‘𝑘) = ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
275 simpr 478 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
276275, 67syl6eleq 2888 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
2774ad2antrr 718 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
278 nnaddcl 11336 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑘 + 𝑀) ∈ ℕ)
279262, 201, 278syl2anr 591 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑘 + 𝑀) ∈ ℕ)
280 ovolfcl 23574 . . . . . . . . . . . . . . . . . . . . . 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)))))
281277, 279, 280syl2anc 580 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((1st ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ ∧ (2nd ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ ∧ (1st ‘(𝐺‘(𝑘 + 𝑀))) ≤ (2nd ‘(𝐺‘(𝑘 + 𝑀)))))
282281simp2d 1174 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (2nd ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ)
283281simp1d 1173 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1st ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ)
284282, 283resubcld 10750 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) ∈ ℝ)
285284recnd 10357 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) ∈ ℂ)
286274, 276, 285fsumser 14802 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛))
287252, 260, 2863eqtrd 2837 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛))
288236, 287oveq12d 6896 . . . . . . . . . . . . . . 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)))
289220, 224, 2883eqtr3d 2841 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛)) = ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)))
290192, 289syl5eq 2845 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) = ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)))
291123ffnd 6257 . . . . . . . . . . . . . . 15 (𝜑𝑇 Fn ℕ)
292291adantr 473 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑇 Fn ℕ)
293 fnfvelrn 6582 . . . . . . . . . . . . . 14 ((𝑇 Fn ℕ ∧ (𝑀 + 𝑛) ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) ∈ ran 𝑇)
294292, 204, 293syl2anc 580 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) ∈ ran 𝑇)
295290, 294eqeltrrd 2879 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ∈ ran 𝑇)
296 supxrub 12403 . . . . . . . . . . . 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*, < ))
297191, 295, 296syl2anc 580 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ≤ sup(ran 𝑇, ℝ*, < ))
298125adantr 473 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝑇𝑀) ∈ ℝ)
299137ffvelrnda 6585 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ∈ (0[,)+∞))
300120, 299sseldi 3796 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ∈ ℝ)
30190adantr 473 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
302298, 300, 301leaddsub2d 10921 . . . . . . . . . . 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))))
303297, 302mpbid 224 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
304303ralrimiva 3147 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
305137ffnd 6257 . . . . . . . . . 10 (𝜑 → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) Fn ℕ)
306 breq1 4846 . . . . . . . . . . 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))))
307306ralrn 6588 . . . . . . . . . 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))))
308305, 307syl 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))))
309304, 308mpbird 249 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))𝑥 ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
310 supxrleub 12405 . . . . . . . . 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))))
311140, 143, 310syl2anc 580 . . . . . . . 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))))
312309, 311mpbird 249 . . . . . . 7 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
313127, 142, 143, 188, 312xrletrd 12242 . . . . . 6 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
314125, 90, 50absdifltd 14513 . . . . . . . . 9 (𝜑 → ((abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶 ↔ ((sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀) ∧ (𝑇𝑀) < (sup(ran 𝑇, ℝ*, < ) + 𝐶))))
31527, 314mpbid 224 . . . . . . . 8 (𝜑 → ((sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀) ∧ (𝑇𝑀) < (sup(ran 𝑇, ℝ*, < ) + 𝐶)))
316315simpld 489 . . . . . . 7 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀))
31790, 50, 125, 316ltsub23d 10924 . . . . . 6 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) < 𝐶)
31897, 126, 50, 313, 317lelttrd 10485 . . . . 5 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) < 𝐶)
31997, 50, 49, 318ltadd2dd 10486 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) < ((vol*‘(𝐾𝐴)) + 𝐶))
32013, 98, 51, 119, 319lelttrd 10485 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) < ((vol*‘(𝐾𝐴)) + 𝐶))
32154, 97readdcld 10358 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
322 difss 3935 . . . . . . . 8 (𝐾𝐴) ⊆ 𝐾
323 unss1 3980 . . . . . . . 8 ((𝐾𝐴) ⊆ 𝐾 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
324322, 323ax-mp 5 . . . . . . 7 ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
325324, 88syl5sseqr 3850 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺))
326 ovolsscl 23594 . . . . . 6 ((((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
327325, 9, 95, 326syl3anc 1491 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
328104ssdifd 3944 . . . . . . 7 (𝜑 → (𝐸𝐴) ⊆ ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴))
329 difundir 4081 . . . . . . . 8 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴) = ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴))
330 difss 3935 . . . . . . . . 9 ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))
331 unss2 3982 . . . . . . . . 9 (( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) → ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
332330, 331ax-mp 5 . . . . . . . 8 ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
333329, 332eqsstri 3831 . . . . . . 7 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
334328, 333syl6ss 3810 . . . . . 6 (𝜑 → (𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
335325, 9sstrd 3808 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ)
336 ovolss 23593 . . . . . 6 (((𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∧ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ) → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
337334, 335, 336syl2anc 580 . . . . 5 (𝜑 → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
33852, 46sstrd 3808 . . . . . 6 (𝜑 → (𝐾𝐴) ⊆ ℝ)
339 ovolun 23607 . . . . . 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))))))
340338, 54, 116, 97, 339syl22anc 868 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
34116, 327, 321, 337, 340letrd 10484 . . . 4 (𝜑 → (vol*‘(𝐸𝐴)) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
34297, 50, 54, 318ltadd2dd 10486 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) < ((vol*‘(𝐾𝐴)) + 𝐶))
34316, 321, 55, 341, 342lelttrd 10485 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) < ((vol*‘(𝐾𝐴)) + 𝐶))
34413, 16, 51, 55, 320, 343lt2addd 10942 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + 𝐶) + ((vol*‘(𝐾𝐴)) + 𝐶)))
34549recnd 10357 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℂ)
34650recnd 10357 . . 3 (𝜑𝐶 ∈ ℂ)
34754recnd 10357 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℂ)
348345, 346, 347, 346add4d 10554 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + 𝐶) + ((vol*‘(𝐾𝐴)) + 𝐶)) = (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
349344, 348breqtrd 4869 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  wrex 3090  Vcvv 3385  cdif 3766  cun 3767  cin 3768  wss 3769  c0 4115  𝒫 cpw 4349  cop 4374   cuni 4628   ciun 4710  Disj wdisj 4811   class class class wbr 4843  cmpt 4922   × cxp 5310  ran crn 5313  cima 5315  ccom 5316   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  supcsup 8588  cc 10222  cr 10223  0cc0 10224  1c1 10225   + caddc 10227  +∞cpnf 10360  *cxr 10362   < clt 10363  cle 10364  cmin 10556  cn 11312  0cn0 11580  cz 11666  cuz 11930  +crp 12074  (,)cioo 12424  [,)cico 12426  [,]cicc 12427  ...cfz 12580  seqcseq 13055  abscabs 14315  Σcsu 14757  vol*covol 23570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-of 7131  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fi 8559  df-sup 8590  df-inf 8591  df-oi 8657  df-card 9051  df-acn 9054  df-cda 9278  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-nn 11313  df-2 11376  df-3 11377  df-n0 11581  df-z 11667  df-uz 11931  df-q 12034  df-rp 12075  df-xneg 12193  df-xadd 12194  df-xmul 12195  df-ioo 12428  df-ico 12430  df-icc 12431  df-fz 12581  df-fzo 12721  df-fl 12848  df-seq 13056  df-exp 13115  df-hash 13371  df-cj 14180  df-re 14181  df-im 14182  df-sqrt 14316  df-abs 14317  df-clim 14560  df-rlim 14561  df-sum 14758  df-rest 16398  df-topgen 16419  df-psmet 20060  df-xmet 20061  df-met 20062  df-bl 20063  df-mopn 20064  df-top 21027  df-topon 21044  df-bases 21079  df-cmp 21519  df-ovol 23572  df-vol 23573
This theorem is referenced by:  uniioombllem5  23695
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