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 = 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
|
syl6ibr |
⊢ ( ( 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 |
|
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 ) ≤ - 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
|
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 ) ) ) ) |