Theorem evl1scvarpwval

Description: Value of a univariate polynomial evaluation mapping a multiple of an exponentiation of a variable to the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-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 )
	evl1scvarpw.t1	⊢ × = ( ·𝑠 ‘ 𝑊 )
	evl1scvarpw.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	evl1scvarpwval.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )
	evl1scvarpwval.h	⊢ 𝐻 = ( mulGrp ‘ 𝑅 )
	evl1scvarpwval.e	⊢ 𝐸 = ( .g ‘ 𝐻 )
	evl1scvarpwval.t	⊢ · = ( .r ‘ 𝑅 )
Assertion	evl1scvarpwval	⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) ‘ 𝐶 ) = ( 𝐴 · ( 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		evl1scvarpw.t1	⊢ × = ( ·𝑠 ‘ 𝑊 )
10		evl1scvarpw.a	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
11		evl1scvarpwval.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )
12		evl1scvarpwval.h	⊢ 𝐻 = ( mulGrp ‘ 𝑅 )
13		evl1scvarpwval.e	⊢ 𝐸 = ( .g ‘ 𝐻 )
14		evl1scvarpwval.t	⊢ · = ( .r ‘ 𝑅 )
15		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
16		crngring	⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring )
17	7 16	syl	⊢ ( 𝜑 → 𝑅 ∈ Ring )
18	2	ply1ring	⊢ ( 𝑅 ∈ Ring → 𝑊 ∈ Ring )
19	17 18	syl	⊢ ( 𝜑 → 𝑊 ∈ Ring )
20	3	ringmgp	⊢ ( 𝑊 ∈ Ring → 𝐺 ∈ Mnd )
21	19 20	syl	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
22	4 2 15	vr1cl	⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ ( Base ‘ 𝑊 ) )
23	17 22	syl	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑊 ) )
24	3 15	mgpbas	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝐺 )
25	24 6	mulgnn0cl	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) )
26	21 8 23 25	syl3anc	⊢ ( 𝜑 → ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) )
27	1 2 3 4 5 6 7 8 11 12 13	evl1varpwval	⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ‘ 𝐶 ) = ( 𝑁 𝐸 𝐶 ) )
28	26 27	jca	⊢ ( 𝜑 → ( ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ‘ 𝐶 ) = ( 𝑁 𝐸 𝐶 ) ) )
29	1 2 5 15 7 11 28 10 9 14	evl1vsd	⊢ ( 𝜑 → ( ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) ‘ 𝐶 ) = ( 𝐴 · ( 𝑁 𝐸 𝐶 ) ) ) )
30	29	simprd	⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) ‘ 𝐶 ) = ( 𝐴 · ( 𝑁 𝐸 𝐶 ) ) )