Step |
Hyp |
Ref |
Expression |
1 |
|
ressply1.s |
⊢ 𝑆 = ( Poly1 ‘ 𝑅 ) |
2 |
|
ressply1.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
3 |
|
ressply1.u |
⊢ 𝑈 = ( Poly1 ‘ 𝐻 ) |
4 |
|
ressply1.b |
⊢ 𝐵 = ( Base ‘ 𝑈 ) |
5 |
|
ressply1.2 |
⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) |
6 |
|
ressply1.p |
⊢ 𝑃 = ( 𝑆 ↾s 𝐵 ) |
7 |
|
eqid |
⊢ ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 ) |
8 |
|
eqid |
⊢ ( 1o mPoly 𝐻 ) = ( 1o mPoly 𝐻 ) |
9 |
|
eqid |
⊢ ( PwSer1 ‘ 𝐻 ) = ( PwSer1 ‘ 𝐻 ) |
10 |
3 9 4
|
ply1bas |
⊢ 𝐵 = ( Base ‘ ( 1o mPoly 𝐻 ) ) |
11 |
|
1on |
⊢ 1o ∈ On |
12 |
11
|
a1i |
⊢ ( 𝜑 → 1o ∈ On ) |
13 |
|
eqid |
⊢ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) = ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) |
14 |
7 2 8 10 12 5 13
|
ressmplvsca |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( ·𝑠 ‘ ( 1o mPoly 𝐻 ) ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) 𝑌 ) ) |
15 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 ) |
16 |
3 8 15
|
ply1vsca |
⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ ( 1o mPoly 𝐻 ) ) |
17 |
16
|
oveqi |
⊢ ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ ( 1o mPoly 𝐻 ) ) 𝑌 ) |
18 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 ) |
19 |
1 7 18
|
ply1vsca |
⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) |
20 |
4
|
fvexi |
⊢ 𝐵 ∈ V |
21 |
6 18
|
ressvsca |
⊢ ( 𝐵 ∈ V → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑃 ) ) |
22 |
20 21
|
ax-mp |
⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑃 ) |
23 |
|
eqid |
⊢ ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) = ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) |
24 |
13 23
|
ressvsca |
⊢ ( 𝐵 ∈ V → ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) = ( ·𝑠 ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) ) |
25 |
20 24
|
ax-mp |
⊢ ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) = ( ·𝑠 ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) |
26 |
19 22 25
|
3eqtr3i |
⊢ ( ·𝑠 ‘ 𝑃 ) = ( ·𝑠 ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) |
27 |
26
|
oveqi |
⊢ ( 𝑋 ( ·𝑠 ‘ 𝑃 ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) 𝑌 ) |
28 |
14 17 27
|
3eqtr4g |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) 𝑌 ) ) |