Theorem ressmplvsca

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

	Ref	Expression
Hypotheses	ressmpl.s	⊢ 𝑆 = ( 𝐼 mPoly 𝑅 )
	ressmpl.h	⊢ 𝐻 = ( 𝑅 ↾s 𝑇 )
	ressmpl.u	⊢ 𝑈 = ( 𝐼 mPoly 𝐻 )
	ressmpl.b	⊢ 𝐵 = ( Base ‘ 𝑈 )
	ressmpl.1	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
	ressmpl.2	⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
	ressmpl.p	⊢ 𝑃 = ( 𝑆 ↾s 𝐵 )
Assertion	ressmplvsca	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		ressmpl.s	⊢ 𝑆 = ( 𝐼 mPoly 𝑅 )
2		ressmpl.h	⊢ 𝐻 = ( 𝑅 ↾s 𝑇 )
3		ressmpl.u	⊢ 𝑈 = ( 𝐼 mPoly 𝐻 )
4		ressmpl.b	⊢ 𝐵 = ( Base ‘ 𝑈 )
5		ressmpl.1	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
6		ressmpl.2	⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
7		ressmpl.p	⊢ 𝑃 = ( 𝑆 ↾s 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	⊢ ( 𝑌 ∈ 𝐵 → 𝑌 ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) )
12		eqid	⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 )
13		eqid	⊢ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) = ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) )
14	12 2 8 9 13 6	resspsrvsca	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) → ( 𝑋 ( ·𝑠 ‘ ( 𝐼 mPwSer 𝐻 ) ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) 𝑌 ) )
15	11 14	sylanr2	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( ·𝑠 ‘ ( 𝐼 mPwSer 𝐻 ) ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) 𝑌 ) )
16	4	fvexi	⊢ 𝐵 ∈ V
17	3 8 4	mplval2	⊢ 𝑈 = ( ( 𝐼 mPwSer 𝐻 ) ↾s 𝐵 )
18		eqid	⊢ ( ·𝑠 ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( ·𝑠 ‘ ( 𝐼 mPwSer 𝐻 ) )
19	17 18	ressvsca	⊢ ( 𝐵 ∈ V → ( ·𝑠 ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( ·𝑠 ‘ 𝑈 ) )
20	16 19	ax-mp	⊢ ( ·𝑠 ‘ ( 𝐼 mPwSer 𝐻 ) ) = ( ·𝑠 ‘ 𝑈 )
21	20	oveqi	⊢ ( 𝑋 ( ·𝑠 ‘ ( 𝐼 mPwSer 𝐻 ) ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑌 )
22		fvex	⊢ ( Base ‘ 𝑆 ) ∈ V
23		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
24	1 12 23	mplval2	⊢ 𝑆 = ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ 𝑆 ) )
25		eqid	⊢ ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) )
26	24 25	ressvsca	⊢ ( ( Base ‘ 𝑆 ) ∈ V → ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( ·𝑠 ‘ 𝑆 ) )
27	22 26	ax-mp	⊢ ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( ·𝑠 ‘ 𝑆 )
28		fvex	⊢ ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∈ V
29	13 25	ressvsca	⊢ ( ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ∈ V → ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( ·𝑠 ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) )
30	28 29	ax-mp	⊢ ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( ·𝑠 ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) )
31		eqid	⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 )
32	7 31	ressvsca	⊢ ( 𝐵 ∈ V → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑃 ) )
33	16 32	ax-mp	⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑃 )
34	27 30 33	3eqtr3i	⊢ ( ·𝑠 ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) = ( ·𝑠 ‘ 𝑃 )
35	34	oveqi	⊢ ( 𝑋 ( ·𝑠 ‘ ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ ( 𝐼 mPwSer 𝐻 ) ) ) ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) 𝑌 )
36	15 21 35	3eqtr3g	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( ·𝑠 ‘ 𝑃 ) 𝑌 ) )