Step |
Hyp |
Ref |
Expression |
1 |
|
ovn0lem.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
ovn0lem.n0 |
⊢ ( 𝜑 → 𝑋 ≠ ∅ ) |
3 |
|
ovn0lem.m |
⊢ 𝑀 = { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) } |
4 |
|
ovn0lem.infm |
⊢ ( 𝜑 → inf ( 𝑀 , ℝ* , < ) ∈ ( 0 [,] +∞ ) ) |
5 |
|
ovn0lem.i |
⊢ 𝐼 = ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ) |
6 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
7 |
6 4
|
sselid |
⊢ ( 𝜑 → inf ( 𝑀 , ℝ* , < ) ∈ ℝ* ) |
8 |
|
0xr |
⊢ 0 ∈ ℝ* |
9 |
8
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℝ* ) |
10 |
|
ssrab2 |
⊢ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) } ⊆ ℝ* |
11 |
3 10
|
eqsstri |
⊢ 𝑀 ⊆ ℝ* |
12 |
11
|
a1i |
⊢ ( 𝜑 → 𝑀 ⊆ ℝ* ) |
13 |
|
1re |
⊢ 1 ∈ ℝ |
14 |
|
0re |
⊢ 0 ∈ ℝ |
15 |
13 14
|
pm3.2i |
⊢ ( 1 ∈ ℝ ∧ 0 ∈ ℝ ) |
16 |
|
opelxp |
⊢ ( 〈 1 , 0 〉 ∈ ( ℝ × ℝ ) ↔ ( 1 ∈ ℝ ∧ 0 ∈ ℝ ) ) |
17 |
15 16
|
mpbir |
⊢ 〈 1 , 0 〉 ∈ ( ℝ × ℝ ) |
18 |
17
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝑋 ) → 〈 1 , 0 〉 ∈ ( ℝ × ℝ ) ) |
19 |
|
eqid |
⊢ ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) = ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) |
20 |
18 19
|
fmptd |
⊢ ( 𝜑 → ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
21 |
|
reex |
⊢ ℝ ∈ V |
22 |
21 21
|
xpex |
⊢ ( ℝ × ℝ ) ∈ V |
23 |
22
|
a1i |
⊢ ( 𝜑 → ( ℝ × ℝ ) ∈ V ) |
24 |
|
elmapg |
⊢ ( ( ( ℝ × ℝ ) ∈ V ∧ 𝑋 ∈ Fin ) → ( ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↔ ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
25 |
23 1 24
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↔ ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
26 |
20 25
|
mpbird |
⊢ ( 𝜑 → ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
28 |
27 5
|
fmptd |
⊢ ( 𝜑 → 𝐼 : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
29 |
|
ovexd |
⊢ ( 𝜑 → ( ( ℝ × ℝ ) ↑m 𝑋 ) ∈ V ) |
30 |
|
nnex |
⊢ ℕ ∈ V |
31 |
30
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
32 |
|
elmapg |
⊢ ( ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ∈ V ∧ ℕ ∈ V ) → ( 𝐼 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ↔ 𝐼 : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) ) |
33 |
29 31 32
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ↔ 𝐼 : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) ) |
34 |
28 33
|
mpbird |
⊢ ( 𝜑 → 𝐼 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
35 |
|
n0 |
⊢ ( 𝑋 ≠ ∅ ↔ ∃ 𝑙 𝑙 ∈ 𝑋 ) |
36 |
2 35
|
sylib |
⊢ ( 𝜑 → ∃ 𝑙 𝑙 ∈ 𝑋 ) |
37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∃ 𝑙 𝑙 ∈ 𝑋 ) |
38 |
|
nfv |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑙 ∈ 𝑋 ) |
39 |
|
nfcv |
⊢ Ⅎ 𝑘 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑙 ) ) |
40 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑙 ∈ 𝑋 ) → 𝑋 ∈ Fin ) |
41 |
28
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐼 ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
42 |
|
elmapi |
⊢ ( ( 𝐼 ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) → ( 𝐼 ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
43 |
41 42
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐼 ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
44 |
43
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐼 ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
45 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ 𝑋 ) |
46 |
44 45
|
fvovco |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
47 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ ) |
48 |
27
|
elexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ∈ V ) |
49 |
5
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ℕ ∧ ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ∈ V ) → ( 𝐼 ‘ 𝑗 ) = ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ) |
50 |
47 48 49
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐼 ‘ 𝑗 ) = ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ) |
51 |
50
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐼 ‘ 𝑗 ) = ( 𝑙 ∈ 𝑋 ↦ 〈 1 , 0 〉 ) ) |
52 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) ∧ 𝑙 = 𝑘 ) → 〈 1 , 0 〉 = 〈 1 , 0 〉 ) |
53 |
17
|
elexi |
⊢ 〈 1 , 0 〉 ∈ V |
54 |
53
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → 〈 1 , 0 〉 ∈ V ) |
55 |
51 52 45 54
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) = 〈 1 , 0 〉 ) |
56 |
55
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( 1st ‘ 〈 1 , 0 〉 ) ) |
57 |
13
|
elexi |
⊢ 1 ∈ V |
58 |
8
|
elexi |
⊢ 0 ∈ V |
59 |
57 58
|
op1st |
⊢ ( 1st ‘ 〈 1 , 0 〉 ) = 1 |
60 |
59
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ 〈 1 , 0 〉 ) = 1 ) |
61 |
56 60
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) = 1 ) |
62 |
55
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( 2nd ‘ 〈 1 , 0 〉 ) ) |
63 |
57 58
|
op2nd |
⊢ ( 2nd ‘ 〈 1 , 0 〉 ) = 0 |
64 |
63
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ 〈 1 , 0 〉 ) = 0 ) |
65 |
62 64
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) = 0 ) |
66 |
61 65
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( 1 [,) 0 ) ) |
67 |
|
0le1 |
⊢ 0 ≤ 1 |
68 |
|
1xr |
⊢ 1 ∈ ℝ* |
69 |
|
ico0 |
⊢ ( ( 1 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( ( 1 [,) 0 ) = ∅ ↔ 0 ≤ 1 ) ) |
70 |
68 8 69
|
mp2an |
⊢ ( ( 1 [,) 0 ) = ∅ ↔ 0 ≤ 1 ) |
71 |
67 70
|
mpbir |
⊢ ( 1 [,) 0 ) = ∅ |
72 |
71
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1 [,) 0 ) = ∅ ) |
73 |
46 66 72
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) = ∅ ) |
74 |
73
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ∅ ) ) |
75 |
|
vol0 |
⊢ ( vol ‘ ∅ ) = 0 |
76 |
75
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ∅ ) = 0 ) |
77 |
74 76
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = 0 ) |
78 |
|
0cn |
⊢ 0 ∈ ℂ |
79 |
78
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → 0 ∈ ℂ ) |
80 |
77 79
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ∈ ℂ ) |
81 |
80
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑙 ∈ 𝑋 ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ∈ ℂ ) |
82 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑙 → ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑙 ) ) ) |
83 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑙 ∈ 𝑋 ) → 𝑙 ∈ 𝑋 ) |
84 |
|
eleq1w |
⊢ ( 𝑘 = 𝑙 → ( 𝑘 ∈ 𝑋 ↔ 𝑙 ∈ 𝑋 ) ) |
85 |
84
|
anbi2d |
⊢ ( 𝑘 = 𝑙 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑙 ∈ 𝑋 ) ) ) |
86 |
82
|
eqeq1d |
⊢ ( 𝑘 = 𝑙 → ( ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = 0 ↔ ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑙 ) ) = 0 ) ) |
87 |
85 86
|
imbi12d |
⊢ ( 𝑘 = 𝑙 → ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = 0 ) ↔ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑙 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑙 ) ) = 0 ) ) ) |
88 |
87 77
|
chvarvv |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑙 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑙 ) ) = 0 ) |
89 |
38 39 40 81 82 83 88
|
fprod0 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑙 ∈ 𝑋 ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = 0 ) |
90 |
89
|
ex |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑙 ∈ 𝑋 → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = 0 ) ) |
91 |
90
|
exlimdv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ∃ 𝑙 𝑙 ∈ 𝑋 → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = 0 ) ) |
92 |
37 91
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = 0 ) |
93 |
92
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑗 ∈ ℕ ↦ 0 ) ) |
94 |
93
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ 0 ) ) ) |
95 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
96 |
95 31
|
sge0z |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ 0 ) ) = 0 ) |
97 |
|
eqidd |
⊢ ( 𝜑 → 0 = 0 ) |
98 |
94 96 97
|
3eqtrrd |
⊢ ( 𝜑 → 0 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
99 |
|
fveq1 |
⊢ ( 𝑖 = 𝐼 → ( 𝑖 ‘ 𝑗 ) = ( 𝐼 ‘ 𝑗 ) ) |
100 |
99
|
coeq2d |
⊢ ( 𝑖 = 𝐼 → ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) = ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ) |
101 |
100
|
fveq1d |
⊢ ( 𝑖 = 𝐼 → ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
102 |
101
|
fveq2d |
⊢ ( 𝑖 = 𝐼 → ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
103 |
102
|
ralrimivw |
⊢ ( 𝑖 = 𝐼 → ∀ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
104 |
103
|
prodeq2d |
⊢ ( 𝑖 = 𝐼 → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
105 |
104
|
mpteq2dv |
⊢ ( 𝑖 = 𝐼 → ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
106 |
105
|
fveq2d |
⊢ ( 𝑖 = 𝐼 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
107 |
106
|
rspceeqv |
⊢ ( ( 𝐼 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ 0 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) → ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) 0 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
108 |
34 98 107
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) 0 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
109 |
9 108
|
jca |
⊢ ( 𝜑 → ( 0 ∈ ℝ* ∧ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) 0 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
110 |
|
eqeq1 |
⊢ ( 𝑧 = 0 → ( 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ↔ 0 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
111 |
110
|
rexbidv |
⊢ ( 𝑧 = 0 → ( ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ↔ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) 0 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
112 |
111 3
|
elrab2 |
⊢ ( 0 ∈ 𝑀 ↔ ( 0 ∈ ℝ* ∧ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) 0 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
113 |
109 112
|
sylibr |
⊢ ( 𝜑 → 0 ∈ 𝑀 ) |
114 |
|
infxrlb |
⊢ ( ( 𝑀 ⊆ ℝ* ∧ 0 ∈ 𝑀 ) → inf ( 𝑀 , ℝ* , < ) ≤ 0 ) |
115 |
12 113 114
|
syl2anc |
⊢ ( 𝜑 → inf ( 𝑀 , ℝ* , < ) ≤ 0 ) |
116 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
117 |
116
|
a1i |
⊢ ( 𝜑 → +∞ ∈ ℝ* ) |
118 |
|
iccgelb |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ inf ( 𝑀 , ℝ* , < ) ∈ ( 0 [,] +∞ ) ) → 0 ≤ inf ( 𝑀 , ℝ* , < ) ) |
119 |
9 117 4 118
|
syl3anc |
⊢ ( 𝜑 → 0 ≤ inf ( 𝑀 , ℝ* , < ) ) |
120 |
7 9 115 119
|
xrletrid |
⊢ ( 𝜑 → inf ( 𝑀 , ℝ* , < ) = 0 ) |