Step |
Hyp |
Ref |
Expression |
1 |
|
algrf.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
algrf.2 |
⊢ 𝑅 = seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) |
3 |
|
seqfn |
⊢ ( 𝑀 ∈ ℤ → seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) Fn ( ℤ≥ ‘ 𝑀 ) ) |
4 |
3
|
adantr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) Fn ( ℤ≥ ‘ 𝑀 ) ) |
5 |
|
seqfn |
⊢ ( 𝑀 ∈ ℤ → seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) Fn ( ℤ≥ ‘ 𝑀 ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) Fn ( ℤ≥ ‘ 𝑀 ) ) |
7 |
|
fveq2 |
⊢ ( 𝑦 = 𝑀 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑀 ) ) |
8 |
|
fveq2 |
⊢ ( 𝑦 = 𝑀 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑀 ) ) |
9 |
7 8
|
eqeq12d |
⊢ ( 𝑦 = 𝑀 → ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑦 ) ↔ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑀 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑀 ) ) ) |
10 |
9
|
imbi2d |
⊢ ( 𝑦 = 𝑀 → ( ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑀 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑀 ) ) ) ) |
11 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) ) |
12 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) |
13 |
11 12
|
eqeq12d |
⊢ ( 𝑦 = 𝑥 → ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑦 ) ↔ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) ) |
14 |
13
|
imbi2d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) ) ) |
15 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝑥 + 1 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝑥 + 1 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) ) |
17 |
15 16
|
eqeq12d |
⊢ ( 𝑦 = ( 𝑥 + 1 ) → ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑦 ) ↔ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) ) ) |
18 |
17
|
imbi2d |
⊢ ( 𝑦 = ( 𝑥 + 1 ) → ( ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑦 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑦 ) ) ↔ ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) ) ) ) |
19 |
|
seq1 |
⊢ ( 𝑀 ∈ ℤ → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑀 ) = ( ( 𝑍 × { 𝐴 } ) ‘ 𝑀 ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑀 ) = ( ( 𝑍 × { 𝐴 } ) ‘ 𝑀 ) ) |
21 |
|
seq1 |
⊢ ( 𝑀 ∈ ℤ → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑀 ) = ( { 〈 𝑀 , 𝐴 〉 } ‘ 𝑀 ) ) |
22 |
21
|
adantr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑀 ) = ( { 〈 𝑀 , 𝐴 〉 } ‘ 𝑀 ) ) |
23 |
|
id |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉 ) |
24 |
|
uzid |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
25 |
24 1
|
eleqtrrdi |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ 𝑍 ) |
26 |
|
fvconst2g |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑀 ∈ 𝑍 ) → ( ( 𝑍 × { 𝐴 } ) ‘ 𝑀 ) = 𝐴 ) |
27 |
23 25 26
|
syl2anr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑍 × { 𝐴 } ) ‘ 𝑀 ) = 𝐴 ) |
28 |
|
fvsng |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → ( { 〈 𝑀 , 𝐴 〉 } ‘ 𝑀 ) = 𝐴 ) |
29 |
27 28
|
eqtr4d |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝑍 × { 𝐴 } ) ‘ 𝑀 ) = ( { 〈 𝑀 , 𝐴 〉 } ‘ 𝑀 ) ) |
30 |
22 29
|
eqtr4d |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑀 ) = ( ( 𝑍 × { 𝐴 } ) ‘ 𝑀 ) ) |
31 |
20 30
|
eqtr4d |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑀 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑀 ) ) |
32 |
31
|
ex |
⊢ ( 𝑀 ∈ ℤ → ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑀 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑀 ) ) ) |
33 |
|
fveq2 |
⊢ ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) → ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) ) = ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) ) |
34 |
|
seqp1 |
⊢ ( 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) = ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) ( 𝐹 ∘ 1st ) ( ( 𝑍 × { 𝐴 } ) ‘ ( 𝑥 + 1 ) ) ) ) |
35 |
|
fvex |
⊢ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) ∈ V |
36 |
|
fvex |
⊢ ( ( 𝑍 × { 𝐴 } ) ‘ ( 𝑥 + 1 ) ) ∈ V |
37 |
35 36
|
algrflem |
⊢ ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) ( 𝐹 ∘ 1st ) ( ( 𝑍 × { 𝐴 } ) ‘ ( 𝑥 + 1 ) ) ) = ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) ) |
38 |
34 37
|
eqtrdi |
⊢ ( 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) = ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) ) ) |
39 |
|
seqp1 |
⊢ ( 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) = ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ( 𝐹 ∘ 1st ) ( { 〈 𝑀 , 𝐴 〉 } ‘ ( 𝑥 + 1 ) ) ) ) |
40 |
|
fvex |
⊢ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ∈ V |
41 |
|
fvex |
⊢ ( { 〈 𝑀 , 𝐴 〉 } ‘ ( 𝑥 + 1 ) ) ∈ V |
42 |
40 41
|
algrflem |
⊢ ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ( 𝐹 ∘ 1st ) ( { 〈 𝑀 , 𝐴 〉 } ‘ ( 𝑥 + 1 ) ) ) = ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) |
43 |
39 42
|
eqtrdi |
⊢ ( 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) = ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) ) |
44 |
38 43
|
eqeq12d |
⊢ ( 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) ↔ ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) ) = ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) ) ) |
45 |
44
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) ↔ ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) ) = ( 𝐹 ‘ ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) ) ) |
46 |
33 45
|
syl5ibr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) ) ) |
47 |
46
|
expcom |
⊢ ( 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐴 ∈ 𝑉 → ( ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) ) ) ) |
48 |
47
|
a2d |
⊢ ( 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) → ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ ( 𝑥 + 1 ) ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ ( 𝑥 + 1 ) ) ) ) ) |
49 |
10 14 18 14 32 48
|
uzind4 |
⊢ ( 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐴 ∈ 𝑉 → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) ) |
50 |
49
|
impcom |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) |
51 |
50
|
adantll |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) ‘ 𝑥 ) = ( seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ‘ 𝑥 ) ) |
52 |
4 6 51
|
eqfnfvd |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → seq 𝑀 ( ( 𝐹 ∘ 1st ) , ( 𝑍 × { 𝐴 } ) ) = seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ) |
53 |
2 52
|
eqtrid |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ 𝑉 ) → 𝑅 = seq 𝑀 ( ( 𝐹 ∘ 1st ) , { 〈 𝑀 , 𝐴 〉 } ) ) |