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Metamath Proof Explorer

Theorem volsup2

Description: The volume of A is the supremum of the sequence vol*( A i^i ( -u n , n ) ) of volumes of bounded sets. (Contributed by Mario Carneiro, 30-Aug-2014)

Ref Expression
Assertion volsup2 ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ∃ 𝑛 ∈ ℕ 𝐵 < ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) )

Proof

Step Hyp Ref Expression
1 simp3 ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → 𝐵 < ( vol ‘ 𝐴 ) )
2 rexr ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* )
3 2 3ad2ant2 ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → 𝐵 ∈ ℝ* )
4 iccssxr ⊢ ( 0 [,] +∞ ) ⊆ ℝ*
5 volf ⊢ vol : dom vol ⟶ ( 0 [,] +∞ )
6 5 ffvelcdmi ⊢ ( 𝐴 ∈ dom vol → ( vol ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )
7 4 6 sselid ⊢ ( 𝐴 ∈ dom vol → ( vol ‘ 𝐴 ) ∈ ℝ* )
8 7 3ad2ant1 ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( vol ‘ 𝐴 ) ∈ ℝ* )
9 xrltnle ⊢ ( ( 𝐵 ∈ ℝ* ∧ ( vol ‘ 𝐴 ) ∈ ℝ* ) → ( 𝐵 < ( vol ‘ 𝐴 ) ↔ ¬ ( vol ‘ 𝐴 ) ≤ 𝐵 ) )
10 3 8 9 syl2anc ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( 𝐵 < ( vol ‘ 𝐴 ) ↔ ¬ ( vol ‘ 𝐴 ) ≤ 𝐵 ) )
11 1 10 mpbid ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ¬ ( vol ‘ 𝐴 ) ≤ 𝐵 )
12 negeq ⊢ ( 𝑚 = 𝑛 → - 𝑚 = - 𝑛 )
13 id ⊢ ( 𝑚 = 𝑛 → 𝑚 = 𝑛 )
14 12 13 oveq12d ⊢ ( 𝑚 = 𝑛 → ( - 𝑚 [,] 𝑚 ) = ( - 𝑛 [,] 𝑛 ) )
15 14 ineq2d ⊢ ( 𝑚 = 𝑛 → ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) = ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) )
16 eqid ⊢ ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) = ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) )
17 ovex ⊢ ( - 𝑛 [,] 𝑛 ) ∈ V
18 17 inex2 ⊢ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ∈ V
19 15 16 18 fvmpt ⊢ ( 𝑛 ∈ ℕ → ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) = ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) )
20 19 iuneq2i ⊢ ∪ 𝑛 ∈ ℕ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) = ∪ 𝑛 ∈ ℕ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) )
21 iunin2 ⊢ ∪ 𝑛 ∈ ℕ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) = ( 𝐴 ∩ ∪ 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) )
22 20 21 eqtri ⊢ ∪ 𝑛 ∈ ℕ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) = ( 𝐴 ∩ ∪ 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) )
23 simpl1 ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → 𝐴 ∈ dom vol )
24 nnre ⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ )
25 24 adantl ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℝ )
26 25 renegcld ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → - 𝑛 ∈ ℝ )
27 iccmbl ⊢ ( ( - 𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ ) → ( - 𝑛 [,] 𝑛 ) ∈ dom vol )
28 26 25 27 syl2anc ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( - 𝑛 [,] 𝑛 ) ∈ dom vol )
29 inmbl ⊢ ( ( 𝐴 ∈ dom vol ∧ ( - 𝑛 [,] 𝑛 ) ∈ dom vol ) → ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ∈ dom vol )
30 23 28 29 syl2anc ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ∈ dom vol )
31 15 cbvmptv ⊢ ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) )
32 30 31 fmptd ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) : ℕ ⟶ dom vol )
33 32 ffnd ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) Fn ℕ )
34 fniunfv ⊢ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) Fn ℕ → ∪ 𝑛 ∈ ℕ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) = ∪ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) )
35 33 34 syl ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ∪ 𝑛 ∈ ℕ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) = ∪ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) )
36 mblss ⊢ ( 𝐴 ∈ dom vol → 𝐴 ⊆ ℝ )
37 36 3ad2ant1 ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → 𝐴 ⊆ ℝ )
38 37 sselda ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℝ )
39 recn ⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ℂ )
40 39 abscld ⊢ ( 𝑥 ∈ ℝ → ( abs ‘ 𝑥 ) ∈ ℝ )
41 arch ⊢ ( ( abs ‘ 𝑥 ) ∈ ℝ → ∃ 𝑛 ∈ ℕ ( abs ‘ 𝑥 ) < 𝑛 )
42 40 41 syl ⊢ ( 𝑥 ∈ ℝ → ∃ 𝑛 ∈ ℕ ( abs ‘ 𝑥 ) < 𝑛 )
43 ltle ⊢ ( ( ( abs ‘ 𝑥 ) ∈ ℝ ∧ 𝑛 ∈ ℝ ) → ( ( abs ‘ 𝑥 ) < 𝑛 → ( abs ‘ 𝑥 ) ≤ 𝑛 ) )
44 40 24 43 syl2an ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( abs ‘ 𝑥 ) < 𝑛 → ( abs ‘ 𝑥 ) ≤ 𝑛 ) )
45 id ⊢ ( ( 𝑥 ∈ ℝ ∧ - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) → ( 𝑥 ∈ ℝ ∧ - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) )
46 45 3expib ⊢ ( 𝑥 ∈ ℝ → ( ( - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) → ( 𝑥 ∈ ℝ ∧ - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) )
47 46 adantr ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) → ( 𝑥 ∈ ℝ ∧ - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) )
48 absle ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ ) → ( ( abs ‘ 𝑥 ) ≤ 𝑛 ↔ ( - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) )
49 24 48 sylan2 ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( abs ‘ 𝑥 ) ≤ 𝑛 ↔ ( - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) )
50 24 adantl ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℝ )
51 50 renegcld ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → - 𝑛 ∈ ℝ )
52 elicc2 ⊢ ( ( - 𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ ) → ( 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) ↔ ( 𝑥 ∈ ℝ ∧ - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) )
53 51 50 52 syl2anc ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) ↔ ( 𝑥 ∈ ℝ ∧ - 𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) )
54 47 49 53 3imtr4d ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( abs ‘ 𝑥 ) ≤ 𝑛 → 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) ) )
55 44 54 syld ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( abs ‘ 𝑥 ) < 𝑛 → 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) ) )
56 55 reximdva ⊢ ( 𝑥 ∈ ℝ → ( ∃ 𝑛 ∈ ℕ ( abs ‘ 𝑥 ) < 𝑛 → ∃ 𝑛 ∈ ℕ 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) ) )
57 42 56 mpd ⊢ ( 𝑥 ∈ ℝ → ∃ 𝑛 ∈ ℕ 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) )
58 38 57 syl ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑛 ∈ ℕ 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) )
59 58 ex ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( 𝑥 ∈ 𝐴 → ∃ 𝑛 ∈ ℕ 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) ) )
60 eliun ⊢ ( 𝑥 ∈ ∪ 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) ↔ ∃ 𝑛 ∈ ℕ 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) )
61 59 60 imbitrrdi ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) ) )
62 61 ssrdv ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) )
63 dfss2 ⊢ ( 𝐴 ⊆ ∪ 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) ↔ ( 𝐴 ∩ ∪ 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) ) = 𝐴 )
64 62 63 sylib ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( 𝐴 ∩ ∪ 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) ) = 𝐴 )
65 22 35 64 3eqtr3a ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ∪ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) = 𝐴 )
66 65 fveq2d ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( vol ‘ ∪ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) = ( vol ‘ 𝐴 ) )
67 peano2re ⊢ ( 𝑛 ∈ ℝ → ( 𝑛 + 1 ) ∈ ℝ )
68 25 67 syl ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 + 1 ) ∈ ℝ )
69 68 renegcld ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → - ( 𝑛 + 1 ) ∈ ℝ )
70 25 lep1d ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ≤ ( 𝑛 + 1 ) )
71 25 68 lenegd ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 ≤ ( 𝑛 + 1 ) ↔ - ( 𝑛 + 1 ) ≤ - 𝑛 ) )
72 70 71 mpbid ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → - ( 𝑛 + 1 ) ≤ - 𝑛 )
73 iccss ⊢ ( ( ( - ( 𝑛 + 1 ) ∈ ℝ ∧ ( 𝑛 + 1 ) ∈ ℝ ) ∧ ( - ( 𝑛 + 1 ) ≤ - 𝑛 ∧ 𝑛 ≤ ( 𝑛 + 1 ) ) ) → ( - 𝑛 [,] 𝑛 ) ⊆ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) )
74 69 68 72 70 73 syl22anc ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( - 𝑛 [,] 𝑛 ) ⊆ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) )
75 sslin ⊢ ( ( - 𝑛 [,] 𝑛 ) ⊆ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) → ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ⊆ ( 𝐴 ∩ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) ) )
76 74 75 syl ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ⊆ ( 𝐴 ∩ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) ) )
77 19 adantl ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) = ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) )
78 peano2nn ⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ℕ )
79 78 adantl ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 + 1 ) ∈ ℕ )
80 negeq ⊢ ( 𝑚 = ( 𝑛 + 1 ) → - 𝑚 = - ( 𝑛 + 1 ) )
81 id ⊢ ( 𝑚 = ( 𝑛 + 1 ) → 𝑚 = ( 𝑛 + 1 ) )
82 80 81 oveq12d ⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( - 𝑚 [,] 𝑚 ) = ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) )
83 82 ineq2d ⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) = ( 𝐴 ∩ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) ) )
84 ovex ⊢ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) ∈ V
85 84 inex2 ⊢ ( 𝐴 ∩ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) ) ∈ V
86 83 16 85 fvmpt ⊢ ( ( 𝑛 + 1 ) ∈ ℕ → ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ ( 𝑛 + 1 ) ) = ( 𝐴 ∩ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) ) )
87 79 86 syl ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ ( 𝑛 + 1 ) ) = ( 𝐴 ∩ ( - ( 𝑛 + 1 ) [,] ( 𝑛 + 1 ) ) ) )
88 76 77 87 3sstr4d ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ⊆ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ ( 𝑛 + 1 ) ) )
89 88 ralrimiva ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ∀ 𝑛 ∈ ℕ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ⊆ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ ( 𝑛 + 1 ) ) )
90 volsup ⊢ ( ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) : ℕ ⟶ dom vol ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ⊆ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ ( 𝑛 + 1 ) ) ) → ( vol ‘ ∪ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) = sup ( ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) , ℝ* , < ) )
91 32 89 90 syl2anc ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( vol ‘ ∪ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) = sup ( ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) , ℝ* , < ) )
92 66 91 eqtr3d ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( vol ‘ 𝐴 ) = sup ( ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) , ℝ* , < ) )
93 92 breq1d ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( ( vol ‘ 𝐴 ) ≤ 𝐵 ↔ sup ( ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) , ℝ* , < ) ≤ 𝐵 ) )
94 imassrn ⊢ ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) ⊆ ran vol
95 frn ⊢ ( vol : dom vol ⟶ ( 0 [,] +∞ ) → ran vol ⊆ ( 0 [,] +∞ ) )
96 5 95 ax-mp ⊢ ran vol ⊆ ( 0 [,] +∞ )
97 96 4 sstri ⊢ ran vol ⊆ ℝ*
98 94 97 sstri ⊢ ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) ⊆ ℝ*
99 supxrleub ⊢ ( ( ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( sup ( ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) , ℝ* , < ) ≤ 𝐵 ↔ ∀ 𝑛 ∈ ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) 𝑛 ≤ 𝐵 ) )
100 98 3 99 sylancr ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( sup ( ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) , ℝ* , < ) ≤ 𝐵 ↔ ∀ 𝑛 ∈ ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) 𝑛 ≤ 𝐵 ) )
101 ffn ⊢ ( vol : dom vol ⟶ ( 0 [,] +∞ ) → vol Fn dom vol )
102 5 101 ax-mp ⊢ vol Fn dom vol
103 32 frnd ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ⊆ dom vol )
104 breq1 ⊢ ( 𝑛 = ( vol ‘ 𝑧 ) → ( 𝑛 ≤ 𝐵 ↔ ( vol ‘ 𝑧 ) ≤ 𝐵 ) )
105 104 ralima ⊢ ( ( vol Fn dom vol ∧ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ⊆ dom vol ) → ( ∀ 𝑛 ∈ ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) 𝑛 ≤ 𝐵 ↔ ∀ 𝑧 ∈ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ( vol ‘ 𝑧 ) ≤ 𝐵 ) )
106 102 103 105 sylancr ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( ∀ 𝑛 ∈ ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) 𝑛 ≤ 𝐵 ↔ ∀ 𝑧 ∈ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ( vol ‘ 𝑧 ) ≤ 𝐵 ) )
107 fveq2 ⊢ ( 𝑧 = ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) → ( vol ‘ 𝑧 ) = ( vol ‘ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ) )
108 107 breq1d ⊢ ( 𝑧 = ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) → ( ( vol ‘ 𝑧 ) ≤ 𝐵 ↔ ( vol ‘ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ) ≤ 𝐵 ) )
109 108 ralrn ⊢ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) Fn ℕ → ( ∀ 𝑧 ∈ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ( vol ‘ 𝑧 ) ≤ 𝐵 ↔ ∀ 𝑛 ∈ ℕ ( vol ‘ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ) ≤ 𝐵 ) )
110 33 109 syl ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( ∀ 𝑧 ∈ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ( vol ‘ 𝑧 ) ≤ 𝐵 ↔ ∀ 𝑛 ∈ ℕ ( vol ‘ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ) ≤ 𝐵 ) )
111 19 fveq2d ⊢ ( 𝑛 ∈ ℕ → ( vol ‘ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ) = ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) )
112 111 breq1d ⊢ ( 𝑛 ∈ ℕ → ( ( vol ‘ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ) ≤ 𝐵 ↔ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 ) )
113 112 ralbiia ⊢ ( ∀ 𝑛 ∈ ℕ ( vol ‘ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ‘ 𝑛 ) ) ≤ 𝐵 ↔ ∀ 𝑛 ∈ ℕ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 )
114 110 113 bitrdi ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( ∀ 𝑧 ∈ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ( vol ‘ 𝑧 ) ≤ 𝐵 ↔ ∀ 𝑛 ∈ ℕ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 ) )
115 106 114 bitrd ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( ∀ 𝑛 ∈ ( vol “ ran ( 𝑚 ∈ ℕ ↦ ( 𝐴 ∩ ( - 𝑚 [,] 𝑚 ) ) ) ) 𝑛 ≤ 𝐵 ↔ ∀ 𝑛 ∈ ℕ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 ) )
116 93 100 115 3bitrd ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( ( vol ‘ 𝐴 ) ≤ 𝐵 ↔ ∀ 𝑛 ∈ ℕ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 ) )
117 11 116 mtbid ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ¬ ∀ 𝑛 ∈ ℕ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 )
118 rexnal ⊢ ( ∃ 𝑛 ∈ ℕ ¬ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 ↔ ¬ ∀ 𝑛 ∈ ℕ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 )
119 117 118 sylibr ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ∃ 𝑛 ∈ ℕ ¬ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 )
120 3 adantr ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → 𝐵 ∈ ℝ* )
121 5 ffvelcdmi ⊢ ( ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ∈ dom vol → ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ∈ ( 0 [,] +∞ ) )
122 4 121 sselid ⊢ ( ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ∈ dom vol → ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ∈ ℝ* )
123 30 122 syl ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ∈ ℝ* )
124 xrltnle ⊢ ( ( 𝐵 ∈ ℝ* ∧ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ∈ ℝ* ) → ( 𝐵 < ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ↔ ¬ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 ) )
125 120 123 124 syl2anc ⊢ ( ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝐵 < ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ↔ ¬ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 ) )
126 125 rexbidva ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( ∃ 𝑛 ∈ ℕ 𝐵 < ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ↔ ∃ 𝑛 ∈ ℕ ¬ ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) ≤ 𝐵 ) )
127 119 126 mpbird ⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ∃ 𝑛 ∈ ℕ 𝐵 < ( vol ‘ ( 𝐴 ∩ ( - 𝑛 [,] 𝑛 ) ) ) )