Step |
Hyp |
Ref |
Expression |
1 |
|
mply1topmat.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
mply1topmat.q |
⊢ 𝑄 = ( Poly1 ‘ 𝐴 ) |
3 |
|
mply1topmat.l |
⊢ 𝐿 = ( Base ‘ 𝑄 ) |
4 |
|
mply1topmat.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
5 |
|
mply1topmat.m |
⊢ · = ( ·𝑠 ‘ 𝑃 ) |
6 |
|
mply1topmat.e |
⊢ 𝐸 = ( .g ‘ ( mulGrp ‘ 𝑃 ) ) |
7 |
|
mply1topmat.y |
⊢ 𝑌 = ( var1 ‘ 𝑅 ) |
8 |
|
mply1topmat.i |
⊢ 𝐼 = ( 𝑝 ∈ 𝐿 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑝 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) ) ) ) |
9 |
|
fveq2 |
⊢ ( 𝑝 = 𝑂 → ( coe1 ‘ 𝑝 ) = ( coe1 ‘ 𝑂 ) ) |
10 |
9
|
fveq1d |
⊢ ( 𝑝 = 𝑂 → ( ( coe1 ‘ 𝑝 ) ‘ 𝑘 ) = ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) ) |
11 |
10
|
oveqd |
⊢ ( 𝑝 = 𝑂 → ( 𝑖 ( ( coe1 ‘ 𝑝 ) ‘ 𝑘 ) 𝑗 ) = ( 𝑖 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) 𝑗 ) ) |
12 |
11
|
oveq1d |
⊢ ( 𝑝 = 𝑂 → ( ( 𝑖 ( ( coe1 ‘ 𝑝 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) = ( ( 𝑖 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) |
13 |
12
|
mpteq2dv |
⊢ ( 𝑝 = 𝑂 → ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑝 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) = ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) ) |
14 |
13
|
oveq2d |
⊢ ( 𝑝 = 𝑂 → ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑝 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) ) = ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) ) ) |
15 |
14
|
mpoeq3dv |
⊢ ( 𝑝 = 𝑂 → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑝 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) ) ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) ) ) ) |
16 |
|
simpr |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿 ) → 𝑂 ∈ 𝐿 ) |
17 |
|
simpl |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿 ) → 𝑁 ∈ 𝑉 ) |
18 |
|
mpoexga |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) ) ) ∈ V ) |
19 |
17 18
|
syldan |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿 ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) ) ) ∈ V ) |
20 |
8 15 16 19
|
fvmptd3 |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿 ) → ( 𝐼 ‘ 𝑂 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ ( 𝑃 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑖 ( ( coe1 ‘ 𝑂 ) ‘ 𝑘 ) 𝑗 ) · ( 𝑘 𝐸 𝑌 ) ) ) ) ) ) |