Theorem srgbinomlem2

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

	Ref	Expression
Hypotheses	srgbinom.s	$$⊢S={\mathrm{Base}}_R$$
	srgbinom.m	$$⊢\underset{\dot{}}{\times }={\cdot }_R$$
	srgbinom.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	srgbinom.a	$$⊢\underset{\dot{}}{+}={+}_R$$
	srgbinom.g	$$⊢G={\mathrm{mulGrp}}_R$$
	srgbinom.e	$$⊢\underset{\dot{}}{\times }={\cdot }_G$$
	srgbinomlem.r	$$⊢\phi \to R\in \mathrm{SRing}$$
	srgbinomlem.a	$$⊢\phi \to A\in S$$
	srgbinomlem.b	$$⊢\phi \to B\in S$$
	srgbinomlem.c	$$⊢\phi \to A\underset{\dot{}}{\times }B=B\underset{\dot{}}{\times }A$$
	srgbinomlem.n	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	srgbinomlem2	$$⊢\left(\phi \wedge \left(C\in {\mathbb{N}}_0\wedge D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to C\underset{\dot{}}{·}\left(\left(D\underset{\dot{}}{\times }A\right)\underset{\dot{}}{\times }\left(E\underset{\dot{}}{\times }B\right)\right)\in S$$

Proof

Step	Hyp	Ref	Expression
1		srgbinom.s	$$⊢S={\mathrm{Base}}_R$$
2		srgbinom.m	$$⊢\underset{\dot{}}{\times }={\cdot }_R$$
3		srgbinom.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
4		srgbinom.a	$$⊢\underset{\dot{}}{+}={+}_R$$
5		srgbinom.g	$$⊢G={\mathrm{mulGrp}}_R$$
6		srgbinom.e	$$⊢\underset{\dot{}}{\times }={\cdot }_G$$
7		srgbinomlem.r	$$⊢\phi \to R\in \mathrm{SRing}$$
8		srgbinomlem.a	$$⊢\phi \to A\in S$$
9		srgbinomlem.b	$$⊢\phi \to B\in S$$
10		srgbinomlem.c	$$⊢\phi \to A\underset{\dot{}}{\times }B=B\underset{\dot{}}{\times }A$$
11		srgbinomlem.n	$$⊢\phi \to N\in {\mathbb{N}}_0$$
12		srgmnd	$$⊢R\in \mathrm{SRing}\to R\in \mathrm{Mnd}$$
13	7 12	syl	$$⊢\phi \to R\in \mathrm{Mnd}$$
14	13	adantr	$$⊢\left(\phi \wedge \left(C\in {\mathbb{N}}_0\wedge D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to R\in \mathrm{Mnd}$$
15		simpr1	$$⊢\left(\phi \wedge \left(C\in {\mathbb{N}}_0\wedge D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to C\in {\mathbb{N}}_0$$
16	1 2 3 4 5 6 7 8 9 10 11	srgbinomlem1	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to \left(D\underset{\dot{}}{\times }A\right)\underset{\dot{}}{\times }\left(E\underset{\dot{}}{\times }B\right)\in S$$
17	16	3adantr1	$$⊢\left(\phi \wedge \left(C\in {\mathbb{N}}_0\wedge D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to \left(D\underset{\dot{}}{\times }A\right)\underset{\dot{}}{\times }\left(E\underset{\dot{}}{\times }B\right)\in S$$
18	1 3	mulgnn0cl	$$⊢\left(R\in \mathrm{Mnd}\wedge C\in {\mathbb{N}}_0\wedge \left(D\underset{\dot{}}{\times }A\right)\underset{\dot{}}{\times }\left(E\underset{\dot{}}{\times }B\right)\in S\right)\to C\underset{\dot{}}{·}\left(\left(D\underset{\dot{}}{\times }A\right)\underset{\dot{}}{\times }\left(E\underset{\dot{}}{\times }B\right)\right)\in S$$
19	14 15 17 18	syl3anc	$$⊢\left(\phi \wedge \left(C\in {\mathbb{N}}_0\wedge D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to C\underset{\dot{}}{·}\left(\left(D\underset{\dot{}}{\times }A\right)\underset{\dot{}}{\times }\left(E\underset{\dot{}}{\times }B\right)\right)\in S$$