Step |
Hyp |
Ref |
Expression |
1 |
|
matbas.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
matbas.g |
⊢ 𝐺 = ( 𝑅 freeLMod ( 𝑁 × 𝑁 ) ) |
3 |
|
baseid |
⊢ Base = Slot ( Base ‘ ndx ) |
4 |
|
basendxnmulrndx |
⊢ ( Base ‘ ndx ) ≠ ( .r ‘ ndx ) |
5 |
3 4
|
setsnid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ ( 𝐺 sSet 〈 ( .r ‘ ndx ) , ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) 〉 ) ) |
6 |
|
eqid |
⊢ ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) = ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) |
7 |
1 2 6
|
matval |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝐴 = ( 𝐺 sSet 〈 ( .r ‘ ndx ) , ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) 〉 ) ) |
8 |
7
|
fveq2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( Base ‘ 𝐴 ) = ( Base ‘ ( 𝐺 sSet 〈 ( .r ‘ ndx ) , ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) 〉 ) ) ) |
9 |
5 8
|
eqtr4id |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐴 ) ) |