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 ffvelrni ( 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 = 𝑛𝑚 = 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 ) ) ∧ 𝑥𝐴 ) → 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 ∧ - 𝑛𝑥𝑥𝑛 ) → ( 𝑥 ∈ ℝ ∧ - 𝑛𝑥𝑥𝑛 ) )
46 45 3expib ( 𝑥 ∈ ℝ → ( ( - 𝑛𝑥𝑥𝑛 ) → ( 𝑥 ∈ ℝ ∧ - 𝑛𝑥𝑥𝑛 ) ) )
47 46 adantr ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( - 𝑛𝑥𝑥𝑛 ) → ( 𝑥 ∈ ℝ ∧ - 𝑛𝑥𝑥𝑛 ) ) )
48 absle ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ ) → ( ( abs ‘ 𝑥 ) ≤ 𝑛 ↔ ( - 𝑛𝑥𝑥𝑛 ) ) )
49 24 48 sylan2 ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( abs ‘ 𝑥 ) ≤ 𝑛 ↔ ( - 𝑛𝑥𝑥𝑛 ) ) )
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 ∧ - 𝑛𝑥𝑥𝑛 ) ) )
53 51 50 52 syl2anc ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) ↔ ( 𝑥 ∈ ℝ ∧ - 𝑛𝑥𝑥𝑛 ) ) )
54 47 49 53 3imtr4d ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( abs ‘ 𝑥 ) ≤ 𝑛𝑥 ∈ ( - 𝑛 [,] 𝑛 ) ) )
55 44 54 syld ( ( 𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( abs ‘ 𝑥 ) < 𝑛𝑥 ∈ ( - 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 ) ) ∧ 𝑥𝐴 ) → ∃ 𝑛 ∈ ℕ 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) )
59 58 ex ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( 𝑥𝐴 → ∃ 𝑛 ∈ ℕ 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) ) )
60 eliun ( 𝑥 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) ↔ ∃ 𝑛 ∈ ℕ 𝑥 ∈ ( - 𝑛 [,] 𝑛 ) )
61 59 60 syl6ibr ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → ( 𝑥𝐴𝑥 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) ) )
62 61 ssrdv ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < ( vol ‘ 𝐴 ) ) → 𝐴 𝑛 ∈ ℕ ( - 𝑛 [,] 𝑛 ) )
63 df-ss ( 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 ) ≤ - 𝑛𝑛 ≤ ( 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 ) ) ) ) 𝑛𝐵 ) )
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 ) ) ) ) 𝑛𝐵 ) )
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 ) → ( 𝑛𝐵 ↔ ( 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 ) ) ) ) 𝑛𝐵 ↔ ∀ 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 ) ) ) ) 𝑛𝐵 ↔ ∀ 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 ) ) ) ) 𝑛𝐵 ↔ ∀ 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 ffvelrni ( ( 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 ) ) ) )