Theorem srgbinomlem1

Description: Lemma 1 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	srgbinomlem1	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to \left(D\underset{\dot{}}{\times }A\right)\underset{\dot{}}{\times }\left(E\underset{\dot{}}{\times }B\right)\in S$$

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	7	adantr	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to R\in \mathrm{SRing}$$
13	5 1	mgpbas	$$⊢S={\mathrm{Base}}_G$$
14	5	srgmgp	$$⊢R\in \mathrm{SRing}\to G\in \mathrm{Mnd}$$
15	7 14	syl	$$⊢\phi \to G\in \mathrm{Mnd}$$
16	15	adantr	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to G\in \mathrm{Mnd}$$
17		simprl	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to D\in {\mathbb{N}}_0$$
18	8	adantr	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to A\in S$$
19	13 6 16 17 18	mulgnn0cld	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to D\underset{\dot{}}{\times }A\in S$$
20		simprr	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to E\in {\mathbb{N}}_0$$
21	9	adantr	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to B\in S$$
22	13 6 16 20 21	mulgnn0cld	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to E\underset{\dot{}}{\times }B\in S$$
23	1 2	srgcl	$$⊢\left(R\in \mathrm{SRing}\wedge D\underset{\dot{}}{\times }A\in S\wedge E\underset{\dot{}}{\times }B\in S\right)\to \left(D\underset{\dot{}}{\times }A\right)\underset{\dot{}}{\times }\left(E\underset{\dot{}}{\times }B\right)\in S$$
24	12 19 22 23	syl3anc	$$⊢\left(\phi \wedge \left(D\in {\mathbb{N}}_0\wedge E\in {\mathbb{N}}_0\right)\right)\to \left(D\underset{\dot{}}{\times }A\right)\underset{\dot{}}{\times }\left(E\underset{\dot{}}{\times }B\right)\in S$$