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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  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304  cn 11058  0cn0 11330  cz 11415  cuz 11725  +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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