Theorem ressply1mul

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

	Ref	Expression
Hypotheses	ressply1.s	⊢ 𝑆 = ( Poly1 ‘ 𝑅 )
	ressply1.h	⊢ 𝐻 = ( 𝑅 ↾s 𝑇 )
	ressply1.u	⊢ 𝑈 = ( Poly1 ‘ 𝐻 )
	ressply1.b	⊢ 𝐵 = ( Base ‘ 𝑈 )
	ressply1.2	⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
	ressply1.p	⊢ 𝑃 = ( 𝑆 ↾s 𝐵 )
Assertion	ressply1mul	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( .r ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( .r ‘ 𝑃 ) 𝑌 ) )

Proof

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	ressmplmul	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( .r ‘ ( 1o mPoly 𝐻 ) ) 𝑌 ) = ( 𝑋 ( .r ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) 𝑌 ) )
14		eqid	⊢ ( .r ‘ 𝑈 ) = ( .r ‘ 𝑈 )
15	3 8 14	ply1mulr	⊢ ( .r ‘ 𝑈 ) = ( .r ‘ ( 1o mPoly 𝐻 ) )
16	15	oveqi	⊢ ( 𝑋 ( .r ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( .r ‘ ( 1o mPoly 𝐻 ) ) 𝑌 )
17		eqid	⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
18	1 7 17	ply1mulr	⊢ ( .r ‘ 𝑆 ) = ( .r ‘ ( 1o mPoly 𝑅 ) )
19	4	fvexi	⊢ 𝐵 ∈ V
20	6 17	ressmulr	⊢ ( 𝐵 ∈ V → ( .r ‘ 𝑆 ) = ( .r ‘ 𝑃 ) )
21	19 20	ax-mp	⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑃 )
22		eqid	⊢ ( .r ‘ ( 1o mPoly 𝑅 ) ) = ( .r ‘ ( 1o mPoly 𝑅 ) )
23	12 22	ressmulr	⊢ ( 𝐵 ∈ V → ( .r ‘ ( 1o mPoly 𝑅 ) ) = ( .r ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) )
24	19 23	ax-mp	⊢ ( .r ‘ ( 1o mPoly 𝑅 ) ) = ( .r ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) )
25	18 21 24	3eqtr3i	⊢ ( .r ‘ 𝑃 ) = ( .r ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) )
26	25	oveqi	⊢ ( 𝑋 ( .r ‘ 𝑃 ) 𝑌 ) = ( 𝑋 ( .r ‘ ( ( 1o mPoly 𝑅 ) ↾s 𝐵 ) ) 𝑌 )
27	13 16 26	3eqtr4g	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( .r ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( .r ‘ 𝑃 ) 𝑌 ) )