Step |
Hyp |
Ref |
Expression |
1 |
|
cpm2mfval.i |
⊢ 𝐼 = ( 𝑁 cPolyMatToMat 𝑅 ) |
2 |
|
cpm2mfval.s |
⊢ 𝑆 = ( 𝑁 ConstPolyMat 𝑅 ) |
3 |
|
df-cpmat2mat |
⊢ cPolyMatToMat = ( 𝑛 ∈ Fin , 𝑟 ∈ V ↦ ( 𝑚 ∈ ( 𝑛 ConstPolyMat 𝑟 ) ↦ ( 𝑥 ∈ 𝑛 , 𝑦 ∈ 𝑛 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ) |
4 |
3
|
a1i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → cPolyMatToMat = ( 𝑛 ∈ Fin , 𝑟 ∈ V ↦ ( 𝑚 ∈ ( 𝑛 ConstPolyMat 𝑟 ) ↦ ( 𝑥 ∈ 𝑛 , 𝑦 ∈ 𝑛 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ) ) |
5 |
|
oveq12 |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑛 ConstPolyMat 𝑟 ) = ( 𝑁 ConstPolyMat 𝑅 ) ) |
6 |
5 2
|
eqtr4di |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑛 ConstPolyMat 𝑟 ) = 𝑆 ) |
7 |
|
simpl |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → 𝑛 = 𝑁 ) |
8 |
|
eqidd |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) = ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) |
9 |
7 7 8
|
mpoeq123dv |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑥 ∈ 𝑛 , 𝑦 ∈ 𝑛 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) |
10 |
6 9
|
mpteq12dv |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑚 ∈ ( 𝑛 ConstPolyMat 𝑟 ) ↦ ( 𝑥 ∈ 𝑛 , 𝑦 ∈ 𝑛 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) = ( 𝑚 ∈ 𝑆 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ) |
11 |
10
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) ∧ ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) ) → ( 𝑚 ∈ ( 𝑛 ConstPolyMat 𝑟 ) ↦ ( 𝑥 ∈ 𝑛 , 𝑦 ∈ 𝑛 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) = ( 𝑚 ∈ 𝑆 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ) |
12 |
|
simpl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝑁 ∈ Fin ) |
13 |
|
elex |
⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ∈ V ) |
14 |
13
|
adantl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝑅 ∈ V ) |
15 |
2
|
ovexi |
⊢ 𝑆 ∈ V |
16 |
|
mptexg |
⊢ ( 𝑆 ∈ V → ( 𝑚 ∈ 𝑆 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ∈ V ) |
17 |
15 16
|
mp1i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑚 ∈ 𝑆 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ∈ V ) |
18 |
4 11 12 14 17
|
ovmpod |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑁 cPolyMatToMat 𝑅 ) = ( 𝑚 ∈ 𝑆 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ) |
19 |
1 18
|
syl5eq |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝐼 = ( 𝑚 ∈ 𝑆 ↦ ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ ( ( coe1 ‘ ( 𝑥 𝑚 𝑦 ) ) ‘ 0 ) ) ) ) |