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 |
3 4
|
ply1bas |
⊢ 𝐵 = ( Base ‘ ( 1o mPoly 𝐻 ) ) |
10 |
|
1on |
⊢ 1o ∈ On |
11 |
10
|
a1i |
⊢ ( 𝜑 → 1o ∈ On ) |
12 |
|
eqid |
⊢ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) = ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) |
13 |
7 2 8 9 11 5 12
|
ressmpladd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( +g ‘ ( 1o mPoly 𝐻 ) ) 𝑌 ) = ( 𝑋 ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) 𝑌 ) ) |
14 |
|
eqid |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) |
15 |
3 8 14
|
ply1plusg |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ ( 1o mPoly 𝐻 ) ) |
16 |
15
|
oveqi |
⊢ ( 𝑋 ( +g ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( +g ‘ ( 1o mPoly 𝐻 ) ) 𝑌 ) |
17 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
18 |
1 7 17
|
ply1plusg |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ ( 1o mPoly 𝑅 ) ) |
19 |
4
|
fvexi |
⊢ 𝐵 ∈ V |
20 |
6 17
|
ressplusg |
⊢ ( 𝐵 ∈ V → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑃 ) ) |
21 |
19 20
|
ax-mp |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑃 ) |
22 |
|
eqid |
⊢ ( +g ‘ ( 1o mPoly 𝑅 ) ) = ( +g ‘ ( 1o mPoly 𝑅 ) ) |
23 |
12 22
|
ressplusg |
⊢ ( 𝐵 ∈ V → ( +g ‘ ( 1o mPoly 𝑅 ) ) = ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) ) |
24 |
19 23
|
ax-mp |
⊢ ( +g ‘ ( 1o mPoly 𝑅 ) ) = ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) |
25 |
18 21 24
|
3eqtr3i |
⊢ ( +g ‘ 𝑃 ) = ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) |
26 |
25
|
oveqi |
⊢ ( 𝑋 ( +g ‘ 𝑃 ) 𝑌 ) = ( 𝑋 ( +g ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) 𝑌 ) |
27 |
13 16 26
|
3eqtr4g |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( +g ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( +g ‘ 𝑃 ) 𝑌 ) ) |