Step |
Hyp |
Ref |
Expression |
1 |
|
cpm2mfval.i |
⊢ 𝐼 = ( 𝑁 cPolyMatToMat 𝑅 ) |
2 |
|
cpm2mfval.s |
⊢ 𝑆 = ( 𝑁 ConstPolyMat 𝑅 ) |
3 |
1 2
|
cpm2mfval |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝐼 = ( 𝑚 ∈ 𝑆 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ) |
4 |
3
|
3adant3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆 ) → 𝐼 = ( 𝑚 ∈ 𝑆 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ) |
5 |
|
oveq |
⊢ ( 𝑚 = 𝑀 → ( 𝑥 𝑚 𝑦 ) = ( 𝑥 𝑀 𝑦 ) ) |
6 |
5
|
fveq2d |
⊢ ( 𝑚 = 𝑀 → ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) = ( coe1 ‘ ( 𝑥 𝑀 𝑦 ) ) ) |
7 |
6
|
fveq1d |
⊢ ( 𝑚 = 𝑀 → ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) = ( ( coe1 ‘ ( 𝑥 𝑀 𝑦 ) ) ‘ 0 ) ) |
8 |
7
|
mpoeq3dv |
⊢ ( 𝑚 = 𝑀 → ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑀 𝑦 ) ) ‘ 0 ) ) ) |
9 |
8
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆 ) ∧ 𝑚 = 𝑀 ) → ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑀 𝑦 ) ) ‘ 0 ) ) ) |
10 |
|
simp3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆 ) → 𝑀 ∈ 𝑆 ) |
11 |
|
simp1 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆 ) → 𝑁 ∈ Fin ) |
12 |
|
mpoexga |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ) → ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑀 𝑦 ) ) ‘ 0 ) ) ∈ V ) |
13 |
11 11 12
|
syl2anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆 ) → ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑀 𝑦 ) ) ‘ 0 ) ) ∈ V ) |
14 |
4 9 10 13
|
fvmptd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆 ) → ( 𝐼 ‘ 𝑀 ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑀 𝑦 ) ) ‘ 0 ) ) ) |