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	|- ( ph -> R e. CRing )
	evl1varpw.n	|- ( ph -> N e. NN0 )
	evl1scvarpw.t1	|- .X. = ( .s ` W )
	evl1scvarpw.a	|- ( ph -> A e. B )
	evl1scvarpwval.c	|- ( ph -> C e. B )
	evl1scvarpwval.h	|- H = ( mulGrp ` R )
	evl1scvarpwval.e	|- E = ( .g ` H )
	evl1scvarpwval.t	|- .x. = ( .r ` R )
Assertion	evl1scvarpwval	|- ( ph -> ( ( Q ` ( A .X. ( N .^ X ) ) ) ` C ) = ( A .x. ( 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	|- ( ph -> R e. CRing )
8		evl1varpw.n	|- ( ph -> N e. NN0 )
9		evl1scvarpw.t1	|- .X. = ( .s ` W )
10		evl1scvarpw.a	|- ( ph -> A e. B )
11		evl1scvarpwval.c	|- ( ph -> C e. B )
12		evl1scvarpwval.h	|- H = ( mulGrp ` R )
13		evl1scvarpwval.e	|- E = ( .g ` H )
14		evl1scvarpwval.t	|- .x. = ( .r ` R )
15		eqid	|- ( Base ` W ) = ( Base ` W )
16		crngring	|- ( R e. CRing -> R e. Ring )
17	7 16	syl	|- ( ph -> R e. Ring )
18	2	ply1ring	|- ( R e. Ring -> W e. Ring )
19	17 18	syl	|- ( ph -> W e. Ring )
20	3	ringmgp	|- ( W e. Ring -> G e. Mnd )
21	19 20	syl	|- ( ph -> G e. Mnd )
22	4 2 15	vr1cl	|- ( R e. Ring -> X e. ( Base ` W ) )
23	17 22	syl	|- ( ph -> X e. ( Base ` W ) )
24	3 15	mgpbas	|- ( Base ` W ) = ( Base ` G )
25	24 6	mulgnn0cl	|- ( ( G e. Mnd /\ N e. NN0 /\ X e. ( Base ` W ) ) -> ( N .^ X ) e. ( Base ` W ) )
26	21 8 23 25	syl3anc	|- ( ph -> ( N .^ X ) e. ( Base ` W ) )
27	1 2 3 4 5 6 7 8 11 12 13	evl1varpwval	|- ( ph -> ( ( Q ` ( N .^ X ) ) ` C ) = ( N E C ) )
28	26 27	jca	|- ( ph -> ( ( N .^ X ) e. ( Base ` W ) /\ ( ( Q ` ( N .^ X ) ) ` C ) = ( N E C ) ) )
29	1 2 5 15 7 11 28 10 9 14	evl1vsd	|- ( ph -> ( ( A .X. ( N .^ X ) ) e. ( Base ` W ) /\ ( ( Q ` ( A .X. ( N .^ X ) ) ) ` C ) = ( A .x. ( N E C ) ) ) )
30	29	simprd	|- ( ph -> ( ( Q ` ( A .X. ( N .^ X ) ) ) ` C ) = ( A .x. ( N E C ) ) )