Metamath Proof Explorer


Theorem ressmplmul

Description: A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015)

Ref Expression
Hypotheses ressmpl.s 𝑆 = ( 𝐼 mPoly 𝑅 )
ressmpl.h 𝐻 = ( 𝑅s 𝑇 )
ressmpl.u 𝑈 = ( 𝐼 mPoly 𝐻 )
ressmpl.b 𝐵 = ( Base ‘ 𝑈 )
ressmpl.1 ( 𝜑𝐼𝑉 )
ressmpl.2 ( 𝜑𝑇 ∈ ( SubRing ‘ 𝑅 ) )
ressmpl.p 𝑃 = ( 𝑆s 𝐵 )
Assertion ressmplmul ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 ( .r𝑈 ) 𝑌 ) = ( 𝑋 ( .r𝑃 ) 𝑌 ) )

Proof

Step Hyp Ref Expression
1 ressmpl.s 𝑆 = ( 𝐼 mPoly 𝑅 )
2 ressmpl.h 𝐻 = ( 𝑅s 𝑇 )
3 ressmpl.u 𝑈 = ( 𝐼 mPoly 𝐻 )
4 ressmpl.b 𝐵 = ( Base ‘ 𝑈 )
5 ressmpl.1 ( 𝜑𝐼𝑉 )
6 ressmpl.2 ( 𝜑𝑇 ∈ ( SubRing ‘ 𝑅 ) )
7 ressmpl.p 𝑃 = ( 𝑆s 𝐵 )
8 eqid ( 𝐼 mPwSer 𝐻 ) = ( 𝐼 mPwSer 𝐻 )
9 eqid ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝐻 ) )
10 3 8 4 9 mplbasss 𝐵 ⊆ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) )
11 10 sseli ( 𝑋𝐵𝑋 ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) )
12 10 sseli ( 𝑌𝐵𝑌 ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) )
13 11 12 anim12i ( ( 𝑋𝐵𝑌𝐵 ) → ( 𝑋 ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∧ 𝑌 ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) )
14 eqid ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 )
15 eqid ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) = ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) )
16 14 2 8 9 15 6 resspsrmul ( ( 𝜑 ∧ ( 𝑋 ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∧ 𝑌 ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) → ( 𝑋 ( .r ‘ ( 𝐼 mPwSer 𝐻 ) ) 𝑌 ) = ( 𝑋 ( .r ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) 𝑌 ) )
17 13 16 sylan2 ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 ( .r ‘ ( 𝐼 mPwSer 𝐻 ) ) 𝑌 ) = ( 𝑋 ( .r ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) 𝑌 ) )
18 4 fvexi 𝐵 ∈ V
19 3 8 4 mplval2 𝑈 = ( ( 𝐼 mPwSer 𝐻 ) ↾s 𝐵 )
20 eqid ( .r ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( .r ‘ ( 𝐼 mPwSer 𝐻 ) )
21 19 20 ressmulr ( 𝐵 ∈ V → ( .r ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( .r𝑈 ) )
22 18 21 ax-mp ( .r ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( .r𝑈 )
23 22 oveqi ( 𝑋 ( .r ‘ ( 𝐼 mPwSer 𝐻 ) ) 𝑌 ) = ( 𝑋 ( .r𝑈 ) 𝑌 )
24 fvex ( Base ‘ 𝑆 ) ∈ V
25 eqid ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
26 1 14 25 mplval2 𝑆 = ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ 𝑆 ) )
27 eqid ( .r ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( .r ‘ ( 𝐼 mPwSer 𝑅 ) )
28 26 27 ressmulr ( ( Base ‘ 𝑆 ) ∈ V → ( .r ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( .r𝑆 ) )
29 24 28 ax-mp ( .r ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( .r𝑆 )
30 fvex ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∈ V
31 15 27 ressmulr ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∈ V → ( .r ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( .r ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) )
32 30 31 ax-mp ( .r ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( .r ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) )
33 eqid ( .r𝑆 ) = ( .r𝑆 )
34 7 33 ressmulr ( 𝐵 ∈ V → ( .r𝑆 ) = ( .r𝑃 ) )
35 18 34 ax-mp ( .r𝑆 ) = ( .r𝑃 )
36 29 32 35 3eqtr3i ( .r ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) = ( .r𝑃 )
37 36 oveqi ( 𝑋 ( .r ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) 𝑌 ) = ( 𝑋 ( .r𝑃 ) 𝑌 )
38 17 23 37 3eqtr3g ( ( 𝜑 ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 ( .r𝑈 ) 𝑌 ) = ( 𝑋 ( .r𝑃 ) 𝑌 ) )