Step |
Hyp |
Ref |
Expression |
1 |
|
ply1mpl0.m |
⊢ 𝑀 = ( 1o mPoly 𝑅 ) |
2 |
|
ply1mpl0.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
ply1mpl0.z |
⊢ 0 = ( 0g ‘ 𝑃 ) |
4 |
|
eqidd |
⊢ ( ⊤ → ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
6 |
2 5
|
ply1bas |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ ( 1o mPoly 𝑅 ) ) |
7 |
1
|
fveq2i |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ ( 1o mPoly 𝑅 ) ) |
8 |
6 7
|
eqtr4i |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑀 ) |
9 |
8
|
a1i |
⊢ ( ⊤ → ( Base ‘ 𝑃 ) = ( Base ‘ 𝑀 ) ) |
10 |
|
eqid |
⊢ ( +g ‘ 𝑃 ) = ( +g ‘ 𝑃 ) |
11 |
2 1 10
|
ply1plusg |
⊢ ( +g ‘ 𝑃 ) = ( +g ‘ 𝑀 ) |
12 |
11
|
a1i |
⊢ ( ⊤ → ( +g ‘ 𝑃 ) = ( +g ‘ 𝑀 ) ) |
13 |
12
|
oveqdr |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ( Base ‘ 𝑃 ) ∧ 𝑦 ∈ ( Base ‘ 𝑃 ) ) ) → ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ) |
14 |
4 9 13
|
grpidpropd |
⊢ ( ⊤ → ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑀 ) ) |
15 |
14
|
mptru |
⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑀 ) |
16 |
3 15
|
eqtri |
⊢ 0 = ( 0g ‘ 𝑀 ) |