Step |
Hyp |
Ref |
Expression |
1 |
|
matbas.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
matbas.g |
⊢ 𝐺 = ( 𝑅 freeLMod ( 𝑁 × 𝑁 ) ) |
3 |
|
vscaid |
⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx ) |
4 |
|
vscandx |
⊢ ( ·𝑠 ‘ ndx ) = 6 |
5 |
|
3re |
⊢ 3 ∈ ℝ |
6 |
|
3lt6 |
⊢ 3 < 6 |
7 |
5 6
|
gtneii |
⊢ 6 ≠ 3 |
8 |
|
mulrndx |
⊢ ( .r ‘ ndx ) = 3 |
9 |
7 8
|
neeqtrri |
⊢ 6 ≠ ( .r ‘ ndx ) |
10 |
4 9
|
eqnetri |
⊢ ( ·𝑠 ‘ ndx ) ≠ ( .r ‘ ndx ) |
11 |
3 10
|
setsnid |
⊢ ( ·𝑠 ‘ 𝐺 ) = ( ·𝑠 ‘ ( 𝐺 sSet 〈 ( .r ‘ ndx ) , ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) 〉 ) ) |
12 |
|
eqid |
⊢ ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) = ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) |
13 |
1 2 12
|
matval |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝐴 = ( 𝐺 sSet 〈 ( .r ‘ ndx ) , ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) 〉 ) ) |
14 |
13
|
fveq2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ ( 𝐺 sSet 〈 ( .r ‘ ndx ) , ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) 〉 ) ) ) |
15 |
11 14
|
eqtr4id |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( ·𝑠 ‘ 𝐺 ) = ( ·𝑠 ‘ 𝐴 ) ) |