Step |
Hyp |
Ref |
Expression |
1 |
|
ovoliun.t |
⊢ 𝑇 = seq 1 ( + , 𝐺 ) |
2 |
|
ovoliun.g |
⊢ 𝐺 = ( 𝑛 ∈ ℕ ↦ ( vol* ‘ 𝐴 ) ) |
3 |
|
ovoliun.a |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐴 ⊆ ℝ ) |
4 |
|
ovoliun.v |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( vol* ‘ 𝐴 ) ∈ ℝ ) |
5 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
6 |
5
|
a1i |
⊢ ( 𝜑 → -∞ ∈ ℝ* ) |
7 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
8 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
9 |
4 2
|
fmptd |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ ℝ ) |
10 |
9
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐺 ‘ 𝑘 ) ∈ ℝ ) |
11 |
7 8 10
|
serfre |
⊢ ( 𝜑 → seq 1 ( + , 𝐺 ) : ℕ ⟶ ℝ ) |
12 |
1
|
feq1i |
⊢ ( 𝑇 : ℕ ⟶ ℝ ↔ seq 1 ( + , 𝐺 ) : ℕ ⟶ ℝ ) |
13 |
11 12
|
sylibr |
⊢ ( 𝜑 → 𝑇 : ℕ ⟶ ℝ ) |
14 |
|
1nn |
⊢ 1 ∈ ℕ |
15 |
|
ffvelrn |
⊢ ( ( 𝑇 : ℕ ⟶ ℝ ∧ 1 ∈ ℕ ) → ( 𝑇 ‘ 1 ) ∈ ℝ ) |
16 |
13 14 15
|
sylancl |
⊢ ( 𝜑 → ( 𝑇 ‘ 1 ) ∈ ℝ ) |
17 |
16
|
rexrd |
⊢ ( 𝜑 → ( 𝑇 ‘ 1 ) ∈ ℝ* ) |
18 |
13
|
frnd |
⊢ ( 𝜑 → ran 𝑇 ⊆ ℝ ) |
19 |
|
ressxr |
⊢ ℝ ⊆ ℝ* |
20 |
18 19
|
sstrdi |
⊢ ( 𝜑 → ran 𝑇 ⊆ ℝ* ) |
21 |
|
supxrcl |
⊢ ( ran 𝑇 ⊆ ℝ* → sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ* ) |
22 |
20 21
|
syl |
⊢ ( 𝜑 → sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ* ) |
23 |
16
|
mnfltd |
⊢ ( 𝜑 → -∞ < ( 𝑇 ‘ 1 ) ) |
24 |
13
|
ffnd |
⊢ ( 𝜑 → 𝑇 Fn ℕ ) |
25 |
|
fnfvelrn |
⊢ ( ( 𝑇 Fn ℕ ∧ 1 ∈ ℕ ) → ( 𝑇 ‘ 1 ) ∈ ran 𝑇 ) |
26 |
24 14 25
|
sylancl |
⊢ ( 𝜑 → ( 𝑇 ‘ 1 ) ∈ ran 𝑇 ) |
27 |
|
supxrub |
⊢ ( ( ran 𝑇 ⊆ ℝ* ∧ ( 𝑇 ‘ 1 ) ∈ ran 𝑇 ) → ( 𝑇 ‘ 1 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ) |
28 |
20 26 27
|
syl2anc |
⊢ ( 𝜑 → ( 𝑇 ‘ 1 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ) |
29 |
6 17 22 23 28
|
xrltletrd |
⊢ ( 𝜑 → -∞ < sup ( ran 𝑇 , ℝ* , < ) ) |
30 |
|
xrrebnd |
⊢ ( sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ* → ( sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ↔ ( -∞ < sup ( ran 𝑇 , ℝ* , < ) ∧ sup ( ran 𝑇 , ℝ* , < ) < +∞ ) ) ) |
31 |
22 30
|
syl |
⊢ ( 𝜑 → ( sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ↔ ( -∞ < sup ( ran 𝑇 , ℝ* , < ) ∧ sup ( ran 𝑇 , ℝ* , < ) < +∞ ) ) ) |
32 |
29 31
|
mpbirand |
⊢ ( 𝜑 → ( sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ↔ sup ( ran 𝑇 , ℝ* , < ) < +∞ ) ) |
33 |
|
nfcv |
⊢ Ⅎ 𝑚 𝐴 |
34 |
|
nfcsb1v |
⊢ Ⅎ 𝑛 ⦋ 𝑚 / 𝑛 ⦌ 𝐴 |
35 |
|
csbeq1a |
⊢ ( 𝑛 = 𝑚 → 𝐴 = ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) |
36 |
33 34 35
|
cbviun |
⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 |
37 |
36
|
fveq2i |
⊢ ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) = ( vol* ‘ ∪ 𝑚 ∈ ℕ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) |
38 |
|
nfcv |
⊢ Ⅎ 𝑚 ( vol* ‘ 𝐴 ) |
39 |
|
nfcv |
⊢ Ⅎ 𝑛 vol* |
40 |
39 34
|
nffv |
⊢ Ⅎ 𝑛 ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) |
41 |
35
|
fveq2d |
⊢ ( 𝑛 = 𝑚 → ( vol* ‘ 𝐴 ) = ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ) |
42 |
38 40 41
|
cbvmpt |
⊢ ( 𝑛 ∈ ℕ ↦ ( vol* ‘ 𝐴 ) ) = ( 𝑚 ∈ ℕ ↦ ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ) |
43 |
2 42
|
eqtri |
⊢ 𝐺 = ( 𝑚 ∈ ℕ ↦ ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ) |
44 |
3
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ) |
45 |
|
nfv |
⊢ Ⅎ 𝑚 𝐴 ⊆ ℝ |
46 |
|
nfcv |
⊢ Ⅎ 𝑛 ℝ |
47 |
34 46
|
nfss |
⊢ Ⅎ 𝑛 ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ⊆ ℝ |
48 |
35
|
sseq1d |
⊢ ( 𝑛 = 𝑚 → ( 𝐴 ⊆ ℝ ↔ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ⊆ ℝ ) ) |
49 |
45 47 48
|
cbvralw |
⊢ ( ∀ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀ 𝑚 ∈ ℕ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ⊆ ℝ ) |
50 |
44 49
|
sylib |
⊢ ( 𝜑 → ∀ 𝑚 ∈ ℕ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ⊆ ℝ ) |
51 |
50
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) ∧ 𝑥 ∈ ℝ+ ) → ∀ 𝑚 ∈ ℕ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ⊆ ℝ ) |
52 |
51
|
r19.21bi |
⊢ ( ( ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑚 ∈ ℕ ) → ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ⊆ ℝ ) |
53 |
4
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( vol* ‘ 𝐴 ) ∈ ℝ ) |
54 |
38
|
nfel1 |
⊢ Ⅎ 𝑚 ( vol* ‘ 𝐴 ) ∈ ℝ |
55 |
40
|
nfel1 |
⊢ Ⅎ 𝑛 ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ∈ ℝ |
56 |
41
|
eleq1d |
⊢ ( 𝑛 = 𝑚 → ( ( vol* ‘ 𝐴 ) ∈ ℝ ↔ ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ∈ ℝ ) ) |
57 |
54 55 56
|
cbvralw |
⊢ ( ∀ 𝑛 ∈ ℕ ( vol* ‘ 𝐴 ) ∈ ℝ ↔ ∀ 𝑚 ∈ ℕ ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ∈ ℝ ) |
58 |
53 57
|
sylib |
⊢ ( 𝜑 → ∀ 𝑚 ∈ ℕ ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ∈ ℝ ) |
59 |
58
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) ∧ 𝑥 ∈ ℝ+ ) → ∀ 𝑚 ∈ ℕ ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ∈ ℝ ) |
60 |
59
|
r19.21bi |
⊢ ( ( ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑚 ∈ ℕ ) → ( vol* ‘ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ∈ ℝ ) |
61 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) ∧ 𝑥 ∈ ℝ+ ) → sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) |
62 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ+ ) |
63 |
1 43 52 60 61 62
|
ovoliunlem3 |
⊢ ( ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) ∧ 𝑥 ∈ ℝ+ ) → ( vol* ‘ ∪ 𝑚 ∈ ℕ ⦋ 𝑚 / 𝑛 ⦌ 𝐴 ) ≤ ( sup ( ran 𝑇 , ℝ* , < ) + 𝑥 ) ) |
64 |
37 63
|
eqbrtrid |
⊢ ( ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) ∧ 𝑥 ∈ ℝ+ ) → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ ( sup ( ran 𝑇 , ℝ* , < ) + 𝑥 ) ) |
65 |
64
|
ralrimiva |
⊢ ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) → ∀ 𝑥 ∈ ℝ+ ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ ( sup ( ran 𝑇 , ℝ* , < ) + 𝑥 ) ) |
66 |
|
iunss |
⊢ ( ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ) |
67 |
44 66
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ) |
68 |
|
ovolcl |
⊢ ( ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ∈ ℝ* ) |
69 |
67 68
|
syl |
⊢ ( 𝜑 → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ∈ ℝ* ) |
70 |
|
xralrple |
⊢ ( ( ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ∈ ℝ* ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) → ( ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ↔ ∀ 𝑥 ∈ ℝ+ ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ ( sup ( ran 𝑇 , ℝ* , < ) + 𝑥 ) ) ) |
71 |
69 70
|
sylan |
⊢ ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) → ( ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ↔ ∀ 𝑥 ∈ ℝ+ ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ ( sup ( ran 𝑇 , ℝ* , < ) + 𝑥 ) ) ) |
72 |
65 71
|
mpbird |
⊢ ( ( 𝜑 ∧ sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ) |
73 |
72
|
ex |
⊢ ( 𝜑 → ( sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ) ) |
74 |
32 73
|
sylbird |
⊢ ( 𝜑 → ( sup ( ran 𝑇 , ℝ* , < ) < +∞ → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ) ) |
75 |
|
nltpnft |
⊢ ( sup ( ran 𝑇 , ℝ* , < ) ∈ ℝ* → ( sup ( ran 𝑇 , ℝ* , < ) = +∞ ↔ ¬ sup ( ran 𝑇 , ℝ* , < ) < +∞ ) ) |
76 |
22 75
|
syl |
⊢ ( 𝜑 → ( sup ( ran 𝑇 , ℝ* , < ) = +∞ ↔ ¬ sup ( ran 𝑇 , ℝ* , < ) < +∞ ) ) |
77 |
|
pnfge |
⊢ ( ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ∈ ℝ* → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ +∞ ) |
78 |
69 77
|
syl |
⊢ ( 𝜑 → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ +∞ ) |
79 |
|
breq2 |
⊢ ( sup ( ran 𝑇 , ℝ* , < ) = +∞ → ( ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ↔ ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ +∞ ) ) |
80 |
78 79
|
syl5ibrcom |
⊢ ( 𝜑 → ( sup ( ran 𝑇 , ℝ* , < ) = +∞ → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ) ) |
81 |
76 80
|
sylbird |
⊢ ( 𝜑 → ( ¬ sup ( ran 𝑇 , ℝ* , < ) < +∞ → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ) ) |
82 |
74 81
|
pm2.61d |
⊢ ( 𝜑 → ( vol* ‘ ∪ 𝑛 ∈ ℕ 𝐴 ) ≤ sup ( ran 𝑇 , ℝ* , < ) ) |