Theorem evl1scvarpw

Description: Univariate polynomial evaluation maps a multiple of an exponentiation of a variable to the multiple of an 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 )
	evl1scvarpw.s	⊢ 𝑆 = ( 𝑅 ↑s 𝐵 )
	evl1scvarpw.t2	⊢ ∙ = ( .r ‘ 𝑆 )
	evl1scvarpw.m	⊢ 𝑀 = ( mulGrp ‘ 𝑆 )
	evl1scvarpw.f	⊢ 𝐹 = ( .g ‘ 𝑀 )
Assertion	evl1scvarpw	⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) = ( ( 𝐵 × { 𝐴 } ) ∙ ( 𝑁 𝐹 ( 𝑄 ‘ 𝑋 ) ) ) )

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		evl1scvarpw.s	⊢ 𝑆 = ( 𝑅 ↑s 𝐵 )
12		evl1scvarpw.t2	⊢ ∙ = ( .r ‘ 𝑆 )
13		evl1scvarpw.m	⊢ 𝑀 = ( mulGrp ‘ 𝑆 )
14		evl1scvarpw.f	⊢ 𝐹 = ( .g ‘ 𝑀 )
15	2	ply1assa	⊢ ( 𝑅 ∈ CRing → 𝑊 ∈ AssAlg )
16	7 15	syl	⊢ ( 𝜑 → 𝑊 ∈ AssAlg )
17	10 5	eleqtrdi	⊢ ( 𝜑 → 𝐴 ∈ ( Base ‘ 𝑅 ) )
18	2	ply1sca	⊢ ( 𝑅 ∈ CRing → 𝑅 = ( Scalar ‘ 𝑊 ) )
19	18	eqcomd	⊢ ( 𝑅 ∈ CRing → ( Scalar ‘ 𝑊 ) = 𝑅 )
20	7 19	syl	⊢ ( 𝜑 → ( Scalar ‘ 𝑊 ) = 𝑅 )
21	20	fveq2d	⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ 𝑅 ) )
22	17 21	eleqtrrd	⊢ ( 𝜑 → 𝐴 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
23		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
24	3 23	mgpbas	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝐺 )
25		crngring	⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring )
26	7 25	syl	⊢ ( 𝜑 → 𝑅 ∈ Ring )
27	2	ply1ring	⊢ ( 𝑅 ∈ Ring → 𝑊 ∈ Ring )
28	26 27	syl	⊢ ( 𝜑 → 𝑊 ∈ Ring )
29	3	ringmgp	⊢ ( 𝑊 ∈ Ring → 𝐺 ∈ Mnd )
30	28 29	syl	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
31	4 2 23	vr1cl	⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ ( Base ‘ 𝑊 ) )
32	26 31	syl	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑊 ) )
33	24 6 30 8 32	mulgnn0cld	⊢ ( 𝜑 → ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) )
34		eqid	⊢ ( algSc ‘ 𝑊 ) = ( algSc ‘ 𝑊 )
35		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
36		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
37		eqid	⊢ ( .r ‘ 𝑊 ) = ( .r ‘ 𝑊 )
38	34 35 36 23 37 9	asclmul1	⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝐴 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) → ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) = ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) )
39	16 22 33 38	syl3anc	⊢ ( 𝜑 → ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) = ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) )
40	39	eqcomd	⊢ ( 𝜑 → ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) = ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) )
41	40	fveq2d	⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) = ( 𝑄 ‘ ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) ) )
42	1 2 11 5	evl1rhm	⊢ ( 𝑅 ∈ CRing → 𝑄 ∈ ( 𝑊 RingHom 𝑆 ) )
43	7 42	syl	⊢ ( 𝜑 → 𝑄 ∈ ( 𝑊 RingHom 𝑆 ) )
44	2	ply1lmod	⊢ ( 𝑅 ∈ Ring → 𝑊 ∈ LMod )
45	26 44	syl	⊢ ( 𝜑 → 𝑊 ∈ LMod )
46	34 35 28 45 36 23	asclf	⊢ ( 𝜑 → ( algSc ‘ 𝑊 ) : ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⟶ ( Base ‘ 𝑊 ) )
47	46 22	ffvelcdmd	⊢ ( 𝜑 → ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ∈ ( Base ‘ 𝑊 ) )
48	23 37 12	rhmmul	⊢ ( ( 𝑄 ∈ ( 𝑊 RingHom 𝑆 ) ∧ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) → ( 𝑄 ‘ ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) ) = ( ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) ∙ ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ) )
49	43 47 33 48	syl3anc	⊢ ( 𝜑 → ( 𝑄 ‘ ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) ) = ( ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) ∙ ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ) )
50	1 2 5 34	evl1sca	⊢ ( ( 𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐵 ) → ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) = ( 𝐵 × { 𝐴 } ) )
51	7 10 50	syl2anc	⊢ ( 𝜑 → ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) = ( 𝐵 × { 𝐴 } ) )
52	1 2 3 4 5 6 7 8	evl1varpw	⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) = ( 𝑁 ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) ( 𝑄 ‘ 𝑋 ) ) )
53	11	fveq2i	⊢ ( mulGrp ‘ 𝑆 ) = ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) )
54	13 53	eqtri	⊢ 𝑀 = ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) )
55	54	fveq2i	⊢ ( .g ‘ 𝑀 ) = ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) )
56	14 55	eqtri	⊢ 𝐹 = ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) )
57	56	a1i	⊢ ( 𝜑 → 𝐹 = ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) )
58	57	eqcomd	⊢ ( 𝜑 → ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) = 𝐹 )
59	58	oveqd	⊢ ( 𝜑 → ( 𝑁 ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) ( 𝑄 ‘ 𝑋 ) ) = ( 𝑁 𝐹 ( 𝑄 ‘ 𝑋 ) ) )
60	52 59	eqtrd	⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) = ( 𝑁 𝐹 ( 𝑄 ‘ 𝑋 ) ) )
61	51 60	oveq12d	⊢ ( 𝜑 → ( ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) ∙ ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ) = ( ( 𝐵 × { 𝐴 } ) ∙ ( 𝑁 𝐹 ( 𝑄 ‘ 𝑋 ) ) ) )
62	41 49 61	3eqtrd	⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) = ( ( 𝐵 × { 𝐴 } ) ∙ ( 𝑁 𝐹 ( 𝑄 ‘ 𝑋 ) ) ) )