Step |
Hyp |
Ref |
Expression |
1 |
|
ply1baspropd.b1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑅 ) ) |
2 |
|
ply1baspropd.b2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑆 ) ) |
3 |
|
ply1baspropd.p |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) |
4 |
1 2 3
|
mplbaspropd |
⊢ ( 𝜑 → ( Base ‘ ( 1o mPoly 𝑅 ) ) = ( Base ‘ ( 1o mPoly 𝑆 ) ) ) |
5 |
|
eqid |
⊢ ( Poly1 ‘ 𝑅 ) = ( Poly1 ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( PwSer1 ‘ 𝑅 ) = ( PwSer1 ‘ 𝑅 ) |
7 |
|
eqid |
⊢ ( Base ‘ ( Poly1 ‘ 𝑅 ) ) = ( Base ‘ ( Poly1 ‘ 𝑅 ) ) |
8 |
5 6 7
|
ply1bas |
⊢ ( Base ‘ ( Poly1 ‘ 𝑅 ) ) = ( Base ‘ ( 1o mPoly 𝑅 ) ) |
9 |
|
eqid |
⊢ ( Poly1 ‘ 𝑆 ) = ( Poly1 ‘ 𝑆 ) |
10 |
|
eqid |
⊢ ( PwSer1 ‘ 𝑆 ) = ( PwSer1 ‘ 𝑆 ) |
11 |
|
eqid |
⊢ ( Base ‘ ( Poly1 ‘ 𝑆 ) ) = ( Base ‘ ( Poly1 ‘ 𝑆 ) ) |
12 |
9 10 11
|
ply1bas |
⊢ ( Base ‘ ( Poly1 ‘ 𝑆 ) ) = ( Base ‘ ( 1o mPoly 𝑆 ) ) |
13 |
4 8 12
|
3eqtr4g |
⊢ ( 𝜑 → ( Base ‘ ( Poly1 ‘ 𝑅 ) ) = ( Base ‘ ( Poly1 ‘ 𝑆 ) ) ) |