Theorem srgbinomlem2

Description: Lemma 2 for srgbinomlem . (Contributed by AV, 23-Aug-2019)

	Ref	Expression
Hypotheses	srgbinom.s	|- S = ( Base ` R )
	srgbinom.m	|- .X. = ( .r ` R )
	srgbinom.t	|- .x. = ( .g ` R )
	srgbinom.a	|- .+ = ( +g ` R )
	srgbinom.g	|- G = ( mulGrp ` R )
	srgbinom.e	|- .^ = ( .g ` G )
	srgbinomlem.r	|- ( ph -> R e. SRing )
	srgbinomlem.a	|- ( ph -> A e. S )
	srgbinomlem.b	|- ( ph -> B e. S )
	srgbinomlem.c	|- ( ph -> ( A .X. B ) = ( B .X. A ) )
	srgbinomlem.n	|- ( ph -> N e. NN0 )
Assertion	srgbinomlem2	|- ( ( ph /\ ( C e. NN0 /\ D e. NN0 /\ E e. NN0 ) ) -> ( C .x. ( ( D .^ A ) .X. ( E .^ B ) ) ) e. S )

Proof

Step	Hyp	Ref	Expression
1		srgbinom.s	|- S = ( Base ` R )
2		srgbinom.m	|- .X. = ( .r ` R )
3		srgbinom.t	|- .x. = ( .g ` R )
4		srgbinom.a	|- .+ = ( +g ` R )
5		srgbinom.g	|- G = ( mulGrp ` R )
6		srgbinom.e	|- .^ = ( .g ` G )
7		srgbinomlem.r	|- ( ph -> R e. SRing )
8		srgbinomlem.a	|- ( ph -> A e. S )
9		srgbinomlem.b	|- ( ph -> B e. S )
10		srgbinomlem.c	|- ( ph -> ( A .X. B ) = ( B .X. A ) )
11		srgbinomlem.n	|- ( ph -> N e. NN0 )
12		srgmnd	|- ( R e. SRing -> R e. Mnd )
13	7 12	syl	|- ( ph -> R e. Mnd )
14	13	adantr	|- ( ( ph /\ ( C e. NN0 /\ D e. NN0 /\ E e. NN0 ) ) -> R e. Mnd )
15		simpr1	|- ( ( ph /\ ( C e. NN0 /\ D e. NN0 /\ E e. NN0 ) ) -> C e. NN0 )
16	1 2 3 4 5 6 7 8 9 10 11	srgbinomlem1	|- ( ( ph /\ ( D e. NN0 /\ E e. NN0 ) ) -> ( ( D .^ A ) .X. ( E .^ B ) ) e. S )
17	16	3adantr1	|- ( ( ph /\ ( C e. NN0 /\ D e. NN0 /\ E e. NN0 ) ) -> ( ( D .^ A ) .X. ( E .^ B ) ) e. S )
18	1 3	mulgnn0cl	|- ( ( R e. Mnd /\ C e. NN0 /\ ( ( D .^ A ) .X. ( E .^ B ) ) e. S ) -> ( C .x. ( ( D .^ A ) .X. ( E .^ B ) ) ) e. S )
19	14 15 17 18	syl3anc	|- ( ( ph /\ ( C e. NN0 /\ D e. NN0 /\ E e. NN0 ) ) -> ( C .x. ( ( D .^ A ) .X. ( E .^ B ) ) ) e. S )