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Theorem uniioombllem3 23399
 Description: Lemma for uniioombl 23403. (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 3866 . . . . 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 23392 . . . . . . 7 (𝜑 → ( ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺) ∧ (vol*‘( ran ([,] ∘ 𝐺) ∖ ran ((,) ∘ 𝐺))) = 0))
65simpld 474 . . . . . 6 (𝜑 ran ((,) ∘ 𝐺) ⊆ ran ([,] ∘ 𝐺))
7 ovolficcss 23284 . . . . . . 7 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐺) ⊆ ℝ)
84, 7syl 17 . . . . . 6 (𝜑 ran ([,] ∘ 𝐺) ⊆ ℝ)
96, 8sstrd 3646 . . . . 5 (𝜑 ran ((,) ∘ 𝐺) ⊆ ℝ)
103, 9sstrd 3646 . . . 4 (𝜑𝐸 ⊆ ℝ)
11 uniioombl.e . . . 4 (𝜑 → (vol*‘𝐸) ∈ ℝ)
12 ovolsscl 23300 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
132, 10, 11, 12syl3anc 1366 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
14 difssd 3771 . . . 4 (𝜑 → (𝐸𝐴) ⊆ 𝐸)
15 ovolsscl 23300 . . . 4 (((𝐸𝐴) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸𝐴)) ∈ ℝ)
1614, 10, 11, 15syl3anc 1366 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) ∈ ℝ)
17 inss1 3866 . . . . . 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 23398 . . . . . . 7 (𝜑 → (𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ∧ (vol*‘𝐾) ∈ ℝ))
3029simpld 474 . . . . . 6 (𝜑𝐾 = 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)))
31 inss2 3867 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
32 elfznn 12408 . . . . . . . . . . . . . 14 (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
33 ffvelrn 6397 . . . . . . . . . . . . . 14 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
344, 32, 33syl2an 493 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
3531, 34sseldi 3634 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) ∈ (ℝ × ℝ))
36 1st2nd2 7249 . . . . . . . . . . . 12 ((𝐺𝑗) ∈ (ℝ × ℝ) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
3735, 36syl 17 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (1...𝑀)) → (𝐺𝑗) = ⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
3837fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩))
39 df-ov 6693 . . . . . . . . . 10 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) = ((,)‘⟨(1st ‘(𝐺𝑗)), (2nd ‘(𝐺𝑗))⟩)
4038, 39syl6eqr 2703 . . . . . . . . 9 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) = ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))))
41 ioossre 12273 . . . . . . . . 9 ((1st ‘(𝐺𝑗))(,)(2nd ‘(𝐺𝑗))) ⊆ ℝ
4240, 41syl6eqss 3688 . . . . . . . 8 ((𝜑𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺𝑗)) ⊆ ℝ)
4342ralrimiva 2995 . . . . . . 7 (𝜑 → ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
44 iunss 4593 . . . . . . 7 ( 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ ↔ ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
4543, 44sylibr 224 . . . . . 6 (𝜑 𝑗 ∈ (1...𝑀)((,)‘(𝐺𝑗)) ⊆ ℝ)
4630, 45eqsstrd 3672 . . . . 5 (𝜑𝐾 ⊆ ℝ)
4729simprd 478 . . . . 5 (𝜑 → (vol*‘𝐾) ∈ ℝ)
48 ovolsscl 23300 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
4918, 46, 47, 48syl3anc 1366 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
5023rpred 11910 . . . 4 (𝜑𝐶 ∈ ℝ)
5149, 50readdcld 10107 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + 𝐶) ∈ ℝ)
52 difssd 3771 . . . . 5 (𝜑 → (𝐾𝐴) ⊆ 𝐾)
53 ovolsscl 23300 . . . . 5 (((𝐾𝐴) ⊆ 𝐾𝐾 ⊆ ℝ ∧ (vol*‘𝐾) ∈ ℝ) → (vol*‘(𝐾𝐴)) ∈ ℝ)
5452, 46, 47, 53syl3anc 1366 . . . 4 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℝ)
5554, 50readdcld 10107 . . 3 (𝜑 → ((vol*‘(𝐾𝐴)) + 𝐶) ∈ ℝ)
56 ssun2 3810 . . . . . . 7 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
57 ioof 12309 . . . . . . . . . . . . . . 15 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
58 rexpssxrxp 10122 . . . . . . . . . . . . . . . . 17 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
5931, 58sstri 3645 . . . . . . . . . . . . . . . 16 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
60 fss 6094 . . . . . . . . . . . . . . . 16 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
614, 59, 60sylancl 695 . . . . . . . . . . . . . . 15 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
62 fco 6096 . . . . . . . . . . . . . . 15 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
6357, 61, 62sylancr 696 . . . . . . . . . . . . . 14 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
64 ffn 6083 . . . . . . . . . . . . . 14 (((,) ∘ 𝐺):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐺) Fn ℕ)
6563, 64syl 17 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
66 fnima 6048 . . . . . . . . . . . . 13 (((,) ∘ 𝐺) Fn ℕ → (((,) ∘ 𝐺) “ ℕ) = ran ((,) ∘ 𝐺))
6765, 66syl 17 . . . . . . . . . . . 12 (𝜑 → (((,) ∘ 𝐺) “ ℕ) = ran ((,) ∘ 𝐺))
68 nnuz 11761 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6926peano2nnd 11075 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 + 1) ∈ ℕ)
7069, 68syl6eleq 2740 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑀 + 1) ∈ (ℤ‘1))
71 uzsplit 12450 . . . . . . . . . . . . . . . 16 ((𝑀 + 1) ∈ (ℤ‘1) → (ℤ‘1) = ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))))
7270, 71syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (ℤ‘1) = ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))))
7368, 72syl5eq 2697 . . . . . . . . . . . . . 14 (𝜑 → ℕ = ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))))
7426nncnd 11074 . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ ℂ)
75 ax-1cn 10032 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
76 pncan 10325 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
7774, 75, 76sylancl 695 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
7877oveq2d 6706 . . . . . . . . . . . . . . 15 (𝜑 → (1...((𝑀 + 1) − 1)) = (1...𝑀))
7978uneq1d 3799 . . . . . . . . . . . . . 14 (𝜑 → ((1...((𝑀 + 1) − 1)) ∪ (ℤ‘(𝑀 + 1))) = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
8073, 79eqtrd 2685 . . . . . . . . . . . . 13 (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
8180imaeq2d 5501 . . . . . . . . . . . 12 (𝜑 → (((,) ∘ 𝐺) “ ℕ) = (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
8267, 81eqtr3d 2687 . . . . . . . . . . 11 (𝜑 → ran ((,) ∘ 𝐺) = (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
83 imaundi 5580 . . . . . . . . . . 11 (((,) ∘ 𝐺) “ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8482, 83syl6eq 2701 . . . . . . . . . 10 (𝜑 → ran ((,) ∘ 𝐺) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8584unieqd 4478 . . . . . . . . 9 (𝜑 ran ((,) ∘ 𝐺) = ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
86 uniun 4488 . . . . . . . . 9 ((((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8785, 86syl6eq 2701 . . . . . . . 8 (𝜑 ran ((,) ∘ 𝐺) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
8828uneq1i 3796 . . . . . . . 8 (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) = ( (((,) ∘ 𝐺) “ (1...𝑀)) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
8987, 88syl6eqr 2703 . . . . . . 7 (𝜑 ran ((,) ∘ 𝐺) = (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
9056, 89syl5sseqr 3687 . . . . . 6 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ 𝐺))
9119, 20, 21, 22, 11, 23, 4, 3, 24, 25uniioombllem1 23395 . . . . . . 7 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
92 ssid 3657 . . . . . . . 8 ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)
9324ovollb 23293 . . . . . . . 8 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝐺) ⊆ ran ((,) ∘ 𝐺)) → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
944, 92, 93sylancl 695 . . . . . . 7 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < ))
95 ovollecl 23297 . . . . . . 7 (( ran ((,) ∘ 𝐺) ⊆ ℝ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ≤ sup(ran 𝑇, ℝ*, < )) → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
969, 91, 94, 95syl3anc 1366 . . . . . 6 (𝜑 → (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ)
97 ovolsscl 23300 . . . . . 6 (( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)
9890, 9, 96, 97syl3anc 1366 . . . . 5 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ)
9949, 98readdcld 10107 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
100 unss1 3815 . . . . . . . 8 ((𝐾𝐴) ⊆ 𝐾 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
10117, 100ax-mp 5 . . . . . . 7 ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
102101, 89syl5sseqr 3687 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺))
103 ovolsscl 23300 . . . . . 6 ((((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
104102, 9, 96, 103syl3anc 1366 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
1053, 89sseqtrd 3674 . . . . . . . 8 (𝜑𝐸 ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
106 ssrin 3871 . . . . . . . 8 (𝐸 ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) → (𝐸𝐴) ⊆ ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴))
107105, 106syl 17 . . . . . . 7 (𝜑 → (𝐸𝐴) ⊆ ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴))
108 indir 3908 . . . . . . . 8 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴) = ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴))
109 inss1 3866 . . . . . . . . 9 ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))
110 unss2 3817 . . . . . . . . 9 (( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) → ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
111109, 110ax-mp 5 . . . . . . . 8 ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∩ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
112108, 111eqsstri 3668 . . . . . . 7 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∩ 𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
113107, 112syl6ss 3648 . . . . . 6 (𝜑 → (𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
114102, 9sstrd 3646 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ)
115 ovolss 23299 . . . . . 6 (((𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∧ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ) → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
116113, 114, 115syl2anc 694 . . . . 5 (𝜑 → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
11718, 46sstrd 3646 . . . . . 6 (𝜑 → (𝐾𝐴) ⊆ ℝ)
11890, 9sstrd 3646 . . . . . 6 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ℝ)
119 ovolun 23313 . . . . . 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))))))
120117, 49, 118, 98, 119syl22anc 1367 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
12113, 104, 99, 116, 120letrd 10232 . . . 4 (𝜑 → (vol*‘(𝐸𝐴)) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
122 rge0ssre 12318 . . . . . . . 8 (0[,)+∞) ⊆ ℝ
123 eqid 2651 . . . . . . . . . . 11 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
124123, 24ovolsf 23287 . . . . . . . . . 10 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
1254, 124syl 17 . . . . . . . . 9 (𝜑𝑇:ℕ⟶(0[,)+∞))
126125, 26ffvelrnd 6400 . . . . . . . 8 (𝜑 → (𝑇𝑀) ∈ (0[,)+∞))
127122, 126sseldi 3634 . . . . . . 7 (𝜑 → (𝑇𝑀) ∈ ℝ)
12891, 127resubcld 10496 . . . . . 6 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ∈ ℝ)
12998rexrd 10127 . . . . . . 7 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∈ ℝ*)
130 id 22 . . . . . . . . . . . . . 14 (𝑧 ∈ ℕ → 𝑧 ∈ ℕ)
131 nnaddcl 11080 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑧 + 𝑀) ∈ ℕ)
132130, 26, 131syl2anr 494 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ ℕ) → (𝑧 + 𝑀) ∈ ℕ)
1334ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧 + 𝑀) ∈ ℕ) → (𝐺‘(𝑧 + 𝑀)) ∈ ( ≤ ∩ (ℝ × ℝ)))
134132, 133syldan 486 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℕ) → (𝐺‘(𝑧 + 𝑀)) ∈ ( ≤ ∩ (ℝ × ℝ)))
135 eqid 2651 . . . . . . . . . . . 12 (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))) = (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))
136134, 135fmptd 6425 . . . . . . . . . . 11 (𝜑 → (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
137 eqid 2651 . . . . . . . . . . . 12 ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))) = ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
138 eqid 2651 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) = seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
139137, 138ovolsf 23287 . . . . . . . . . . 11 ((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))):ℕ⟶(0[,)+∞))
140136, 139syl 17 . . . . . . . . . 10 (𝜑 → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))):ℕ⟶(0[,)+∞))
141 frn 6091 . . . . . . . . . 10 (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ (0[,)+∞))
142140, 141syl 17 . . . . . . . . 9 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ (0[,)+∞))
143 icossxr 12296 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ*
144142, 143syl6ss 3648 . . . . . . . 8 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ ℝ*)
145 supxrcl 12183 . . . . . . . 8 (ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ∈ ℝ*)
146144, 145syl 17 . . . . . . 7 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ∈ ℝ*)
147128rexrd 10127 . . . . . . 7 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) ∈ ℝ*)
148 1zzd 11446 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 1 ∈ ℤ)
14926nnzd 11519 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℤ)
150149adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
151 addcom 10260 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑀 + 1) = (1 + 𝑀))
15274, 75, 151sylancl 695 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 + 1) = (1 + 𝑀))
153152fveq2d 6233 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (ℤ‘(𝑀 + 1)) = (ℤ‘(1 + 𝑀)))
154153eleq2d 2716 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑥 ∈ (ℤ‘(𝑀 + 1)) ↔ 𝑥 ∈ (ℤ‘(1 + 𝑀))))
155154biimpa 500 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ (ℤ‘(1 + 𝑀)))
156 eluzsub 11755 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ (ℤ‘(1 + 𝑀))) → (𝑥𝑀) ∈ (ℤ‘1))
157148, 150, 155, 156syl3anc 1366 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝑥𝑀) ∈ (ℤ‘1))
158157, 68syl6eleqr 2741 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → (𝑥𝑀) ∈ ℕ)
159 eluzelz 11735 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (ℤ‘(𝑀 + 1)) → 𝑥 ∈ ℤ)
160159adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ℤ)
161160zcnd 11521 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ℂ)
16274adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℂ)
163161, 162npcand 10434 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → ((𝑥𝑀) + 𝑀) = 𝑥)
164163eqcomd 2657 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 = ((𝑥𝑀) + 𝑀))
165 oveq1 6697 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑥𝑀) → (𝑧 + 𝑀) = ((𝑥𝑀) + 𝑀))
166165eqeq2d 2661 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑥𝑀) → (𝑥 = (𝑧 + 𝑀) ↔ 𝑥 = ((𝑥𝑀) + 𝑀)))
167166rspcev 3340 . . . . . . . . . . . . . . . . 17 (((𝑥𝑀) ∈ ℕ ∧ 𝑥 = ((𝑥𝑀) + 𝑀)) → ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
168158, 164, 167syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
169 vex 3234 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
170 eqid 2651 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) = (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))
171170elrnmpt 5404 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ V → (𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) ↔ ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀)))
172169, 171ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) ↔ ∃𝑧 ∈ ℕ 𝑥 = (𝑧 + 𝑀))
173168, 172sylibr 224 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 1))) → 𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
174173ex 449 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ (ℤ‘(𝑀 + 1)) → 𝑥 ∈ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
175174ssrdv 3642 . . . . . . . . . . . . 13 (𝜑 → (ℤ‘(𝑀 + 1)) ⊆ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
176 imass2 5536 . . . . . . . . . . . . 13 ((ℤ‘(𝑀 + 1)) ⊆ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
177175, 176syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))))
178 rnco2 5680 . . . . . . . . . . . . 13 ran (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
179 eqidd 2652 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)) = (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀)))
1804feqmptd 6288 . . . . . . . . . . . . . . 15 (𝜑𝐺 = (𝑤 ∈ ℕ ↦ (𝐺𝑤)))
181 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑤 = (𝑧 + 𝑀) → (𝐺𝑤) = (𝐺‘(𝑧 + 𝑀)))
182132, 179, 180, 181fmptco 6436 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
183182rneqd 5385 . . . . . . . . . . . . 13 (𝜑 → ran (𝐺 ∘ (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
184178, 183syl5eqr 2699 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ ran (𝑧 ∈ ℕ ↦ (𝑧 + 𝑀))) = ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
185177, 184sseqtrd 3674 . . . . . . . . . . 11 (𝜑 → (𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
186 imass2 5536 . . . . . . . . . . 11 ((𝐺 “ (ℤ‘(𝑀 + 1))) ⊆ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))) → ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1)))) ⊆ ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
187185, 186syl 17 . . . . . . . . . 10 (𝜑 → ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1)))) ⊆ ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
188 imaco 5678 . . . . . . . . . 10 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) = ((,) “ (𝐺 “ (ℤ‘(𝑀 + 1))))
189 rnco2 5680 . . . . . . . . . 10 ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))) = ((,) “ ran (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))
190187, 188, 1893sstr4g 3679 . . . . . . . . 9 (𝜑 → (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
191190unissd 4494 . . . . . . . 8 (𝜑 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ⊆ ran ((,) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))
192138ovollb 23293 . . . . . . . 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*, < ))
193136, 191, 192syl2anc 694 . . . . . . 7 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ))
194 frn 6091 . . . . . . . . . . . . . . 15 (𝑇:ℕ⟶(0[,)+∞) → ran 𝑇 ⊆ (0[,)+∞))
195125, 194syl 17 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
196195, 143syl6ss 3648 . . . . . . . . . . . . 13 (𝜑 → ran 𝑇 ⊆ ℝ*)
197196adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ran 𝑇 ⊆ ℝ*)
19824fveq1i 6230 . . . . . . . . . . . . . 14 (𝑇‘(𝑀 + 𝑛)) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛))
19926nnred 11073 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℝ)
200199ltp1d 10992 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 < (𝑀 + 1))
201 fzdisj 12406 . . . . . . . . . . . . . . . . . 18 (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
202200, 201syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
203202adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → ((1...𝑀) ∩ ((𝑀 + 1)...(𝑀 + 𝑛))) = ∅)
204 nnnn0 11337 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
205 nn0addge1 11377 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℝ ∧ 𝑛 ∈ ℕ0) → 𝑀 ≤ (𝑀 + 𝑛))
206199, 204, 205syl2an 493 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑀 ≤ (𝑀 + 𝑛))
20726adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℕ)
208207, 68syl6eleq 2740 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ (ℤ‘1))
209 nnaddcl 11080 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℕ)
21026, 209sylan 487 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℕ)
211210nnzd 11519 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ ℤ)
212 elfz5 12372 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ (ℤ‘1) ∧ (𝑀 + 𝑛) ∈ ℤ) → (𝑀 ∈ (1...(𝑀 + 𝑛)) ↔ 𝑀 ≤ (𝑀 + 𝑛)))
213208, 211, 212syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (𝑀 ∈ (1...(𝑀 + 𝑛)) ↔ 𝑀 ≤ (𝑀 + 𝑛)))
214206, 213mpbird 247 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ (1...(𝑀 + 𝑛)))
215 fzsplit 12405 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (1...(𝑀 + 𝑛)) → (1...(𝑀 + 𝑛)) = ((1...𝑀) ∪ ((𝑀 + 1)...(𝑀 + 𝑛))))
216214, 215syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1...(𝑀 + 𝑛)) = ((1...𝑀) ∪ ((𝑀 + 1)...(𝑀 + 𝑛))))
217 fzfid 12812 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (1...(𝑀 + 𝑛)) ∈ Fin)
2184adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
219 elfznn 12408 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (1...(𝑀 + 𝑛)) → 𝑗 ∈ ℕ)
220 ovolfcl 23281 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
221218, 219, 220syl2an 493 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
222221simp2d 1094 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
223221simp1d 1093 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (1st ‘(𝐺𝑗)) ∈ ℝ)
224222, 223resubcld 10496 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
225224recnd 10106 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
226203, 216, 217, 225fsumsplit 14515 . . . . . . . . . . . . . . 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)))))
227123ovolfsval 23285 . . . . . . . . . . . . . . . . 17 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
228218, 219, 227syl2an 493 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...(𝑀 + 𝑛))) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
229210, 68syl6eleq 2740 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 𝑛) ∈ (ℤ‘1))
230228, 229, 225fsumser 14505 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛)))
2314ad2antrr 762 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
23232adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
233231, 232, 227syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → (((abs ∘ − ) ∘ 𝐺)‘𝑗) = ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))))
2344, 32, 220syl2an 493 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (1...𝑀)) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
235234simp2d 1094 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
236234simp1d 1093 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (1...𝑀)) → (1st ‘(𝐺𝑗)) ∈ ℝ)
237235, 236resubcld 10496 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
238237adantlr 751 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
239238recnd 10106 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
240233, 208, 239fsumser 14505 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑀))
24124fveq1i 6230 . . . . . . . . . . . . . . . . 17 (𝑇𝑀) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑀)
242240, 241syl6eqr 2703 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ (1...𝑀)((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (𝑇𝑀))
243207nnzd 11519 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℤ)
244243peano2zd 11523 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 1) ∈ ℤ)
2454ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
246207peano2nnd 11075 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ ℕ) → (𝑀 + 1) ∈ ℕ)
247 elfzuz 12376 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛)) → 𝑗 ∈ (ℤ‘(𝑀 + 1)))
248 eluznn 11796 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 + 1) ∈ ℕ ∧ 𝑗 ∈ (ℤ‘(𝑀 + 1))) → 𝑗 ∈ ℕ)
249246, 247, 248syl2an 493 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → 𝑗 ∈ ℕ)
250245, 249, 220syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((1st ‘(𝐺𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺𝑗)) ∈ ℝ ∧ (1st ‘(𝐺𝑗)) ≤ (2nd ‘(𝐺𝑗))))
251250simp2d 1094 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → (2nd ‘(𝐺𝑗)) ∈ ℝ)
252250simp1d 1093 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → (1st ‘(𝐺𝑗)) ∈ ℝ)
253251, 252resubcld 10496 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℝ)
254253recnd 10106 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) ∈ ℂ)
255 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 𝑀) → (𝐺𝑗) = (𝐺‘(𝑘 + 𝑀)))
256255fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 𝑀) → (2nd ‘(𝐺𝑗)) = (2nd ‘(𝐺‘(𝑘 + 𝑀))))
257255fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 𝑀) → (1st ‘(𝐺𝑗)) = (1st ‘(𝐺‘(𝑘 + 𝑀))))
258256, 257oveq12d 6708 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑘 + 𝑀) → ((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
259243, 244, 211, 254, 258fsumshftm 14557 . . . . . . . . . . . . . . . . 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)))))
260207nncnd 11074 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑀 ∈ ℂ)
261 pncan2 10326 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 𝑀) = 1)
262260, 75, 261sylancl 695 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ((𝑀 + 1) − 𝑀) = 1)
263 nncn 11066 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
264263adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
265260, 264pncan2d 10432 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ((𝑀 + 𝑛) − 𝑀) = 𝑛)
266262, 265oveq12d 6708 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → (((𝑀 + 1) − 𝑀)...((𝑀 + 𝑛) − 𝑀)) = (1...𝑛))
267266sumeq1d 14475 . . . . . . . . . . . . . . . . 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)))))
268136adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
269 elfznn 12408 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
270137ovolfsval 23285 . . . . . . . . . . . . . . . . . . . 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))))
271268, 269, 270syl2an 493 . . . . . . . . . . . . . . . . . . 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))))
272269adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
273 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = 𝑘 → (𝑧 + 𝑀) = (𝑘 + 𝑀))
274273fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = 𝑘 → (𝐺‘(𝑧 + 𝑀)) = (𝐺‘(𝑘 + 𝑀)))
275 fvex 6239 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺‘(𝑘 + 𝑀)) ∈ V
276274, 135, 275fvmpt 6321 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ → ((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘) = (𝐺‘(𝑘 + 𝑀)))
277272, 276syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘) = (𝐺‘(𝑘 + 𝑀)))
278277fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (2nd ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) = (2nd ‘(𝐺‘(𝑘 + 𝑀))))
279277fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1st ‘((𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))‘𝑘)) = (1st ‘(𝐺‘(𝑘 + 𝑀))))
280278, 279oveq12d 6708 . . . . . . . . . . . . . . . . . . 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)))))
281271, 280eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))‘𝑘) = ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))))
282 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
283282, 68syl6eleq 2740 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
2844ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
285 nnaddcl 11080 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑘 + 𝑀) ∈ ℕ)
286269, 207, 285syl2anr 494 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑘 + 𝑀) ∈ ℕ)
287 ovolfcl 23281 . . . . . . . . . . . . . . . . . . . . . 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)))))
288284, 286, 287syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((1st ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ ∧ (2nd ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ ∧ (1st ‘(𝐺‘(𝑘 + 𝑀))) ≤ (2nd ‘(𝐺‘(𝑘 + 𝑀)))))
289288simp2d 1094 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (2nd ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ)
290288simp1d 1093 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (1st ‘(𝐺‘(𝑘 + 𝑀))) ∈ ℝ)
291289, 290resubcld 10496 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) ∈ ℝ)
292291recnd 10106 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) ∈ ℂ)
293281, 283, 292fsumser 14505 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)((2nd ‘(𝐺‘(𝑘 + 𝑀))) − (1st ‘(𝐺‘(𝑘 + 𝑀)))) = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛))
294259, 267, 2933eqtrd 2689 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → Σ𝑗 ∈ ((𝑀 + 1)...(𝑀 + 𝑛))((2nd ‘(𝐺𝑗)) − (1st ‘(𝐺𝑗))) = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛))
295242, 294oveq12d 6708 . . . . . . . . . . . . . . 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)))
296226, 230, 2953eqtr3d 2693 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘(𝑀 + 𝑛)) = ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)))
297198, 296syl5eq 2697 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) = ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)))
298 ffn 6083 . . . . . . . . . . . . . . . 16 (𝑇:ℕ⟶(0[,)+∞) → 𝑇 Fn ℕ)
299125, 298syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑇 Fn ℕ)
300299adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑇 Fn ℕ)
301 fnfvelrn 6396 . . . . . . . . . . . . . 14 ((𝑇 Fn ℕ ∧ (𝑀 + 𝑛) ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) ∈ ran 𝑇)
302300, 210, 301syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝑇‘(𝑀 + 𝑛)) ∈ ran 𝑇)
303297, 302eqeltrrd 2731 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ∈ ran 𝑇)
304 supxrub 12192 . . . . . . . . . . . 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*, < ))
305197, 303, 304syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((𝑇𝑀) + (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛)) ≤ sup(ran 𝑇, ℝ*, < ))
306127adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝑇𝑀) ∈ ℝ)
307140ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ∈ (0[,)+∞))
308122, 307sseldi 3634 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ∈ ℝ)
30991adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
310306, 308, 309leaddsub2d 10667 . . . . . . . . . . 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))))
311305, 310mpbid 222 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
312311ralrimiva 2995 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))‘𝑛) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
313 ffn 6083 . . . . . . . . . . 11 (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))):ℕ⟶(0[,)+∞) → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) Fn ℕ)
314140, 313syl 17 . . . . . . . . . 10 (𝜑 → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))) Fn ℕ)
315 breq1 4688 . . . . . . . . . . 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))))
316315ralrn 6402 . . . . . . . . . 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))))
317314, 316syl 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))))
318312, 317mpbird 247 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀)))))𝑥 ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
319 supxrleub 12194 . . . . . . . . 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))))
320144, 147, 319syl2anc 694 . . . . . . . 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))))
321318, 320mpbird 247 . . . . . . 7 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ (𝐺‘(𝑧 + 𝑀))))), ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
322129, 146, 147, 193, 321xrletrd 12031 . . . . . 6 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ≤ (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)))
323127, 91, 50absdifltd 14216 . . . . . . . . 9 (𝜑 → ((abs‘((𝑇𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶 ↔ ((sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀) ∧ (𝑇𝑀) < (sup(ran 𝑇, ℝ*, < ) + 𝐶))))
32427, 323mpbid 222 . . . . . . . 8 (𝜑 → ((sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀) ∧ (𝑇𝑀) < (sup(ran 𝑇, ℝ*, < ) + 𝐶)))
325324simpld 474 . . . . . . 7 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − 𝐶) < (𝑇𝑀))
32691, 50, 127, 325ltsub23d 10670 . . . . . 6 (𝜑 → (sup(ran 𝑇, ℝ*, < ) − (𝑇𝑀)) < 𝐶)
32798, 128, 50, 322, 326lelttrd 10233 . . . . 5 (𝜑 → (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) < 𝐶)
32898, 50, 49, 327ltadd2dd 10234 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) < ((vol*‘(𝐾𝐴)) + 𝐶))
32913, 99, 51, 121, 328lelttrd 10233 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) < ((vol*‘(𝐾𝐴)) + 𝐶))
33054, 98readdcld 10107 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
331 difss 3770 . . . . . . . 8 (𝐾𝐴) ⊆ 𝐾
332 unss1 3815 . . . . . . . 8 ((𝐾𝐴) ⊆ 𝐾 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
333331, 332ax-mp 5 . . . . . . 7 ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ (𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
334333, 89syl5sseqr 3687 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺))
335 ovolsscl 23300 . . . . . 6 ((((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ran ((,) ∘ 𝐺) ∧ ran ((,) ∘ 𝐺) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝐺)) ∈ ℝ) → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
336334, 9, 96, 335syl3anc 1366 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ∈ ℝ)
337105ssdifd 3779 . . . . . . 7 (𝜑 → (𝐸𝐴) ⊆ ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴))
338 difundir 3913 . . . . . . . 8 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴) = ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴))
339 difss 3770 . . . . . . . . 9 ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))
340 unss2 3817 . . . . . . . . 9 (( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴) ⊆ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) → ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
341339, 340ax-mp 5 . . . . . . . 8 ((𝐾𝐴) ∪ ( (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))) ∖ 𝐴)) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
342338, 341eqsstri 3668 . . . . . . 7 ((𝐾 (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∖ 𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))
343337, 342syl6ss 3648 . . . . . 6 (𝜑 → (𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))))
344334, 9sstrd 3646 . . . . . 6 (𝜑 → ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ)
345 ovolss 23299 . . . . . 6 (((𝐸𝐴) ⊆ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ∧ ((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1)))) ⊆ ℝ) → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
346343, 344, 345syl2anc 694 . . . . 5 (𝜑 → (vol*‘(𝐸𝐴)) ≤ (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
34752, 46sstrd 3646 . . . . . 6 (𝜑 → (𝐾𝐴) ⊆ ℝ)
348 ovolun 23313 . . . . . 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))))))
349347, 54, 118, 98, 348syl22anc 1367 . . . . 5 (𝜑 → (vol*‘((𝐾𝐴) ∪ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
35016, 336, 330, 346, 349letrd 10232 . . . 4 (𝜑 → (vol*‘(𝐸𝐴)) ≤ ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))))
35198, 50, 54, 327ltadd2dd 10234 . . . 4 (𝜑 → ((vol*‘(𝐾𝐴)) + (vol*‘ (((,) ∘ 𝐺) “ (ℤ‘(𝑀 + 1))))) < ((vol*‘(𝐾𝐴)) + 𝐶))
35216, 330, 55, 350, 351lelttrd 10233 . . 3 (𝜑 → (vol*‘(𝐸𝐴)) < ((vol*‘(𝐾𝐴)) + 𝐶))
35313, 16, 51, 55, 329, 352lt2addd 10688 . 2 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + 𝐶) + ((vol*‘(𝐾𝐴)) + 𝐶)))
35449recnd 10106 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℂ)
35550recnd 10106 . . 3 (𝜑𝐶 ∈ ℂ)
35654recnd 10106 . . 3 (𝜑 → (vol*‘(𝐾𝐴)) ∈ ℂ)
357354, 355, 356, 355add4d 10302 . 2 (𝜑 → (((vol*‘(𝐾𝐴)) + 𝐶) + ((vol*‘(𝐾𝐴)) + 𝐶)) = (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
358353, 357breqtrd 4711 1 (𝜑 → ((vol*‘(𝐸𝐴)) + (vol*‘(𝐸𝐴))) < (((vol*‘(𝐾𝐴)) + (vol*‘(𝐾𝐴))) + (𝐶 + 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  𝒫 cpw 4191  ⟨cop 4216  ∪ cuni 4468  ∪ ciun 4552  Disj wdisj 4652   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  ran crn 5144   “ cima 5146   ∘ ccom 5147   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  supcsup 8387  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112   ≤ cle 10113   − cmin 10304  ℕcn 11058  ℕ0cn0 11330  ℤcz 11415  ℤ≥cuz 11725  ℝ+crp 11870  (,)cioo 12213  [,)cico 12215  [,]cicc 12216  ...cfz 12364  seqcseq 12841  abscabs 14018  Σcsu 14460  vol*covol 23277 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280 This theorem is referenced by:  uniioombllem5  23401
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