Theorem evl1varpwval

Description: Value of a univariate polynomial evaluation mapping the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019)

	Ref	Expression
Hypotheses	evl1varpw.q	⊢ 𝑄 = ( eval1 ‘ 𝑅 )
	evl1varpw.w	⊢ 𝑊 = ( Poly1 ‘ 𝑅 )
	evl1varpw.g	⊢ 𝐺 = ( mulGrp ‘ 𝑊 )
	evl1varpw.x	⊢ 𝑋 = ( var1 ‘ 𝑅 )
	evl1varpw.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	evl1varpw.e	⊢ ↑ = ( .g ‘ 𝐺 )
	evl1varpw.r	⊢ ( 𝜑 → 𝑅 ∈ CRing )
	evl1varpw.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	evl1varpwval.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )
	evl1varpwval.h	⊢ 𝐻 = ( mulGrp ‘ 𝑅 )
	evl1varpwval.e	⊢ 𝐸 = ( .g ‘ 𝐻 )
Assertion	evl1varpwval	⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ‘ 𝐶 ) = ( 𝑁 𝐸 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		evl1varpw.q	⊢ 𝑄 = ( eval1 ‘ 𝑅 )
2		evl1varpw.w	⊢ 𝑊 = ( Poly1 ‘ 𝑅 )
3		evl1varpw.g	⊢ 𝐺 = ( mulGrp ‘ 𝑊 )
4		evl1varpw.x	⊢ 𝑋 = ( var1 ‘ 𝑅 )
5		evl1varpw.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
6		evl1varpw.e	⊢ ↑ = ( .g ‘ 𝐺 )
7		evl1varpw.r	⊢ ( 𝜑 → 𝑅 ∈ CRing )
8		evl1varpw.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
9		evl1varpwval.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )
10		evl1varpwval.h	⊢ 𝐻 = ( mulGrp ‘ 𝑅 )
11		evl1varpwval.e	⊢ 𝐸 = ( .g ‘ 𝐻 )
12		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
13	1 4 5 2 12 7 9	evl1vard	⊢ ( 𝜑 → ( 𝑋 ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝑄 ‘ 𝑋 ) ‘ 𝐶 ) = 𝐶 ) )
14	3	fveq2i	⊢ ( .g ‘ 𝐺 ) = ( .g ‘ ( mulGrp ‘ 𝑊 ) )
15	6 14	eqtri	⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑊 ) )
16	10	fveq2i	⊢ ( .g ‘ 𝐻 ) = ( .g ‘ ( mulGrp ‘ 𝑅 ) )
17	11 16	eqtri	⊢ 𝐸 = ( .g ‘ ( mulGrp ‘ 𝑅 ) )
18	1 2 5 12 7 9 13 15 17 8	evl1expd	⊢ ( 𝜑 → ( ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ‘ 𝐶 ) = ( 𝑁 𝐸 𝐶 ) ) )
19	18	simprd	⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ‘ 𝐶 ) = ( 𝑁 𝐸 𝐶 ) )