Step |
Hyp |
Ref |
Expression |
1 |
|
mat2pmatfval.t |
⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 ) |
2 |
|
mat2pmatfval.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
3 |
|
mat2pmatfval.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
4 |
|
mat2pmatfval.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
5 |
|
mat2pmatfval.s |
⊢ 𝑆 = ( algSc ‘ 𝑃 ) |
6 |
1 2 3 4 5
|
mat2pmatfval |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝑇 = ( 𝑚 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( 𝑆 ‘ ( 𝑥 𝑚 𝑦 ) ) ) ) ) |
7 |
6
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) → 𝑇 = ( 𝑚 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( 𝑆 ‘ ( 𝑥 𝑚 𝑦 ) ) ) ) ) |
8 |
|
oveq |
⊢ ( 𝑚 = 𝑀 → ( 𝑥 𝑚 𝑦 ) = ( 𝑥 𝑀 𝑦 ) ) |
9 |
8
|
fveq2d |
⊢ ( 𝑚 = 𝑀 → ( 𝑆 ‘ ( 𝑥 𝑚 𝑦 ) ) = ( 𝑆 ‘ ( 𝑥 𝑀 𝑦 ) ) ) |
10 |
9
|
mpoeq3dv |
⊢ ( 𝑚 = 𝑀 → ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( 𝑆 ‘ ( 𝑥 𝑚 𝑦 ) ) ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( 𝑆 ‘ ( 𝑥 𝑀 𝑦 ) ) ) ) |
11 |
10
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑚 = 𝑀 ) → ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( 𝑆 ‘ ( 𝑥 𝑚 𝑦 ) ) ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( 𝑆 ‘ ( 𝑥 𝑀 𝑦 ) ) ) ) |
12 |
|
simp3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) → 𝑀 ∈ 𝐵 ) |
13 |
|
simp1 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) → 𝑁 ∈ Fin ) |
14 |
|
mpoexga |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) → ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( 𝑆 ‘ ( 𝑥 𝑀 𝑦 ) ) ) ∈ V ) |
15 |
13 13 14
|
syl2anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) → ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( 𝑆 ‘ ( 𝑥 𝑀 𝑦 ) ) ) ∈ V ) |
16 |
7 11 12 15
|
fvmptd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑀 ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( 𝑆 ‘ ( 𝑥 𝑀 𝑦 ) ) ) ) |