Step |
Hyp |
Ref |
Expression |
1 |
|
hoicvrrex.fi |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
hoicvrrex.y |
⊢ ( 𝜑 → 𝑌 ⊆ ( ℝ ↑m 𝑋 ) ) |
3 |
|
nnre |
⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℝ ) |
4 |
3
|
renegcld |
⊢ ( 𝑗 ∈ ℕ → - 𝑗 ∈ ℝ ) |
5 |
|
opelxpi |
⊢ ( ( - 𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ ) → 〈 - 𝑗 , 𝑗 〉 ∈ ( ℝ × ℝ ) ) |
6 |
4 3 5
|
syl2anc |
⊢ ( 𝑗 ∈ ℕ → 〈 - 𝑗 , 𝑗 〉 ∈ ( ℝ × ℝ ) ) |
7 |
6
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → 〈 - 𝑗 , 𝑗 〉 ∈ ( ℝ × ℝ ) ) |
8 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) = ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) |
9 |
7 8
|
fmptd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
10 |
|
reex |
⊢ ℝ ∈ V |
11 |
10 10
|
xpex |
⊢ ( ℝ × ℝ ) ∈ V |
12 |
11
|
a1i |
⊢ ( 𝜑 → ( ℝ × ℝ ) ∈ V ) |
13 |
|
elmapg |
⊢ ( ( ( ℝ × ℝ ) ∈ V ∧ 𝑋 ∈ Fin ) → ( ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
14 |
12 1 13
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
16 |
9 15
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
17 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) |
18 |
16 17
|
fmptd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
19 |
|
ovex |
⊢ ( ( ℝ × ℝ ) ↑m 𝑋 ) ∈ V |
20 |
|
nnex |
⊢ ℕ ∈ V |
21 |
19 20
|
elmap |
⊢ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ↔ ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
22 |
18 21
|
sylibr |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
23 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) |
24 |
23 1
|
hoicvr |
⊢ ( 𝜑 → ( ℝ ↑m 𝑋 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
25 |
|
eqidd |
⊢ ( 𝑙 = 𝑘 → 〈 - 𝑗 , 𝑗 〉 = 〈 - 𝑗 , 𝑗 〉 ) |
26 |
25
|
cbvmptv |
⊢ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) = ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) |
27 |
26
|
mpteq2i |
⊢ ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) |
28 |
27
|
a1i |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ) |
29 |
28
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) = ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) |
30 |
29
|
coeq2d |
⊢ ( 𝜑 → ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) = ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ) |
31 |
30
|
fveq1d |
⊢ ( 𝜑 → ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
32 |
31
|
ixpeq2dv |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
33 |
32
|
iuneq2d |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑙 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
34 |
24 33
|
sseqtrd |
⊢ ( 𝜑 → ( ℝ ↑m 𝑋 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
35 |
2 34
|
sstrd |
⊢ ( 𝜑 → 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
36 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ ) |
37 |
16
|
elexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ∈ V ) |
38 |
17
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ℕ ∧ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ∈ V ) → ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) |
39 |
36 37 38
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) |
40 |
39 7
|
fmpt3d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
41 |
40
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
42 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ 𝑋 ) |
43 |
41 42
|
fvovco |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( 1st ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
44 |
39
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ‘ 𝑘 ) ) |
45 |
44
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ‘ 𝑘 ) ) |
46 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ 𝑋 ) |
47 |
|
opex |
⊢ 〈 - 𝑗 , 𝑗 〉 ∈ V |
48 |
47
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 〈 - 𝑗 , 𝑗 〉 ∈ V ) |
49 |
8
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝑋 ∧ 〈 - 𝑗 , 𝑗 〉 ∈ V ) → ( ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ‘ 𝑘 ) = 〈 - 𝑗 , 𝑗 〉 ) |
50 |
46 48 49
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ‘ 𝑘 ) = 〈 - 𝑗 , 𝑗 〉 ) |
51 |
50
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ‘ 𝑘 ) = 〈 - 𝑗 , 𝑗 〉 ) |
52 |
45 51
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) = 〈 - 𝑗 , 𝑗 〉 ) |
53 |
52
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) ) = ( 1st ‘ 〈 - 𝑗 , 𝑗 〉 ) ) |
54 |
|
negex |
⊢ - 𝑗 ∈ V |
55 |
|
vex |
⊢ 𝑗 ∈ V |
56 |
54 55
|
op1st |
⊢ ( 1st ‘ 〈 - 𝑗 , 𝑗 〉 ) = - 𝑗 |
57 |
56
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ 〈 - 𝑗 , 𝑗 〉 ) = - 𝑗 ) |
58 |
53 57
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) ) = - 𝑗 ) |
59 |
52
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) ) = ( 2nd ‘ 〈 - 𝑗 , 𝑗 〉 ) ) |
60 |
54 55
|
op2nd |
⊢ ( 2nd ‘ 〈 - 𝑗 , 𝑗 〉 ) = 𝑗 |
61 |
60
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ 〈 - 𝑗 , 𝑗 〉 ) = 𝑗 ) |
62 |
59 61
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) ) = 𝑗 ) |
63 |
58 62
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 1st ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( - 𝑗 [,) 𝑗 ) ) |
64 |
43 63
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) = ( - 𝑗 [,) 𝑗 ) ) |
65 |
64
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( - 𝑗 [,) 𝑗 ) ) ) |
66 |
|
volico |
⊢ ( ( - 𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ ) → ( vol ‘ ( - 𝑗 [,) 𝑗 ) ) = if ( - 𝑗 < 𝑗 , ( 𝑗 − - 𝑗 ) , 0 ) ) |
67 |
4 3 66
|
syl2anc |
⊢ ( 𝑗 ∈ ℕ → ( vol ‘ ( - 𝑗 [,) 𝑗 ) ) = if ( - 𝑗 < 𝑗 , ( 𝑗 − - 𝑗 ) , 0 ) ) |
68 |
|
nnrp |
⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℝ+ ) |
69 |
|
neglt |
⊢ ( 𝑗 ∈ ℝ+ → - 𝑗 < 𝑗 ) |
70 |
68 69
|
syl |
⊢ ( 𝑗 ∈ ℕ → - 𝑗 < 𝑗 ) |
71 |
70
|
iftrued |
⊢ ( 𝑗 ∈ ℕ → if ( - 𝑗 < 𝑗 , ( 𝑗 − - 𝑗 ) , 0 ) = ( 𝑗 − - 𝑗 ) ) |
72 |
3
|
recnd |
⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℂ ) |
73 |
72 72
|
subnegd |
⊢ ( 𝑗 ∈ ℕ → ( 𝑗 − - 𝑗 ) = ( 𝑗 + 𝑗 ) ) |
74 |
72
|
2timesd |
⊢ ( 𝑗 ∈ ℕ → ( 2 · 𝑗 ) = ( 𝑗 + 𝑗 ) ) |
75 |
73 74
|
eqtr4d |
⊢ ( 𝑗 ∈ ℕ → ( 𝑗 − - 𝑗 ) = ( 2 · 𝑗 ) ) |
76 |
67 71 75
|
3eqtrd |
⊢ ( 𝑗 ∈ ℕ → ( vol ‘ ( - 𝑗 [,) 𝑗 ) ) = ( 2 · 𝑗 ) ) |
77 |
76
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( - 𝑗 [,) 𝑗 ) ) = ( 2 · 𝑗 ) ) |
78 |
65 77
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( 2 · 𝑗 ) ) |
79 |
78
|
prodeq2dv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑋 ( 2 · 𝑗 ) ) |
80 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑋 ∈ Fin ) |
81 |
|
2cnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 2 ∈ ℂ ) |
82 |
72
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℂ ) |
83 |
81 82
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 2 · 𝑗 ) ∈ ℂ ) |
84 |
|
fprodconst |
⊢ ( ( 𝑋 ∈ Fin ∧ ( 2 · 𝑗 ) ∈ ℂ ) → ∏ 𝑘 ∈ 𝑋 ( 2 · 𝑗 ) = ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) |
85 |
80 83 84
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( 2 · 𝑗 ) = ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) |
86 |
79 85
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) |
87 |
86
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) ) |
88 |
87
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) ) ) |
89 |
20
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
90 |
68
|
ssriv |
⊢ ℕ ⊆ ℝ+ |
91 |
|
ioorp |
⊢ ( 0 (,) +∞ ) = ℝ+ |
92 |
91
|
eqcomi |
⊢ ℝ+ = ( 0 (,) +∞ ) |
93 |
90 92
|
sseqtri |
⊢ ℕ ⊆ ( 0 (,) +∞ ) |
94 |
|
ioossicc |
⊢ ( 0 (,) +∞ ) ⊆ ( 0 [,] +∞ ) |
95 |
93 94
|
sstri |
⊢ ℕ ⊆ ( 0 [,] +∞ ) |
96 |
|
2nn |
⊢ 2 ∈ ℕ |
97 |
96
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 2 ∈ ℕ ) |
98 |
97 36
|
nnmulcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 2 · 𝑗 ) ∈ ℕ ) |
99 |
|
hashcl |
⊢ ( 𝑋 ∈ Fin → ( ♯ ‘ 𝑋 ) ∈ ℕ0 ) |
100 |
1 99
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝑋 ) ∈ ℕ0 ) |
101 |
100
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ♯ ‘ 𝑋 ) ∈ ℕ0 ) |
102 |
|
nnexpcl |
⊢ ( ( ( 2 · 𝑗 ) ∈ ℕ ∧ ( ♯ ‘ 𝑋 ) ∈ ℕ0 ) → ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ∈ ℕ ) |
103 |
98 101 102
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ∈ ℕ ) |
104 |
95 103
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ∈ ( 0 [,] +∞ ) ) |
105 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) |
106 |
104 105
|
fmptd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
107 |
89 106
|
sge0xrcl |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) ) ∈ ℝ* ) |
108 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
109 |
108
|
a1i |
⊢ ( 𝜑 → +∞ ∈ ℝ* ) |
110 |
|
1nn |
⊢ 1 ∈ ℕ |
111 |
95 110
|
sselii |
⊢ 1 ∈ ( 0 [,] +∞ ) |
112 |
111
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 1 ∈ ( 0 [,] +∞ ) ) |
113 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ 1 ) = ( 𝑗 ∈ ℕ ↦ 1 ) |
114 |
112 113
|
fmptd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ 1 ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
115 |
89 114
|
sge0xrcl |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ 1 ) ) ∈ ℝ* ) |
116 |
|
nnnfi |
⊢ ¬ ℕ ∈ Fin |
117 |
116
|
a1i |
⊢ ( 𝜑 → ¬ ℕ ∈ Fin ) |
118 |
|
1rp |
⊢ 1 ∈ ℝ+ |
119 |
118
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℝ+ ) |
120 |
89 117 119
|
sge0rpcpnf |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ 1 ) ) = +∞ ) |
121 |
120
|
eqcomd |
⊢ ( 𝜑 → +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ 1 ) ) ) |
122 |
109 121
|
xreqled |
⊢ ( 𝜑 → +∞ ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ 1 ) ) ) |
123 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
124 |
114
|
fvmptelrn |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 1 ∈ ( 0 [,] +∞ ) ) |
125 |
103
|
nnge1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 1 ≤ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) |
126 |
123 89 124 104 125
|
sge0lempt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ 1 ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) ) ) |
127 |
109 115 107 122 126
|
xrletrd |
⊢ ( 𝜑 → +∞ ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) ) ) |
128 |
107 127
|
xrgepnfd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 2 · 𝑗 ) ↑ ( ♯ ‘ 𝑋 ) ) ) ) = +∞ ) |
129 |
|
eqidd |
⊢ ( 𝜑 → +∞ = +∞ ) |
130 |
88 128 129
|
3eqtrrd |
⊢ ( 𝜑 → +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
131 |
35 130
|
jca |
⊢ ( 𝜑 → ( 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
132 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑖 |
133 |
|
nfmpt1 |
⊢ Ⅎ 𝑗 ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) |
134 |
132 133
|
nfeq |
⊢ Ⅎ 𝑗 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) |
135 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑖 |
136 |
|
nfcv |
⊢ Ⅎ 𝑘 ℕ |
137 |
|
nfmpt1 |
⊢ Ⅎ 𝑘 ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) |
138 |
136 137
|
nfmpt |
⊢ Ⅎ 𝑘 ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) |
139 |
135 138
|
nfeq |
⊢ Ⅎ 𝑘 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) |
140 |
|
fveq1 |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( 𝑖 ‘ 𝑗 ) = ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) |
141 |
140
|
coeq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) = ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ) |
142 |
141
|
fveq1d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
143 |
142
|
adantr |
⊢ ( ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
144 |
139 143
|
ixpeq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
145 |
144
|
adantr |
⊢ ( ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
146 |
134 145
|
iuneq2df |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
147 |
146
|
sseq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ↔ 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
148 |
142
|
fveq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
149 |
148
|
a1d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( 𝑘 ∈ 𝑋 → ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
150 |
139 149
|
ralrimi |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ∀ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
151 |
150
|
adantr |
⊢ ( ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ∧ 𝑗 ∈ ℕ ) → ∀ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
152 |
151
|
prodeq2d |
⊢ ( ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
153 |
134 152
|
mpteq2da |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
154 |
153
|
fveq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
155 |
154
|
eqeq2d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ↔ +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
156 |
147 155
|
anbi12d |
⊢ ( 𝑖 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) → ( ( 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ↔ ( 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) ) |
157 |
156
|
rspcev |
⊢ ( ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ ( 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ 〈 - 𝑗 , 𝑗 〉 ) ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) → ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |
158 |
22 131 157
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ +∞ = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) |