Theorem isrnghm2d

Description: Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020)

	Ref	Expression
Hypotheses	isrnghmd.b	$$⊢B={\mathrm{Base}}_R$$
	isrnghmd.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	isrnghmd.u	$$⊢\underset{\dot{}}{\times }={\cdot }_S$$
	isrnghmd.r	$$⊢\phi \to R\in \mathrm{Rng}$$
	isrnghmd.s	$$⊢\phi \to S\in \mathrm{Rng}$$
	isrnghmd.ht	$$⊢\left(\phi \wedge \left(x\in B\wedge y\in B\right)\right)\to F\left(x\underset{\dot{}}{·}y\right)=F\left(x\right)\underset{\dot{}}{\times }F\left(y\right)$$
	isrnghm2d.f	$$⊢\phi \to F\in \left(R\mathrm{GrpHom}S\right)$$
Assertion	isrnghm2d	$$⊢\phi \to F\in \left(R\mathrm{RngHomo}S\right)$$

Proof

Step	Hyp	Ref	Expression
1		isrnghmd.b	$$⊢B={\mathrm{Base}}_R$$
2		isrnghmd.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		isrnghmd.u	$$⊢\underset{\dot{}}{\times }={\cdot }_S$$
4		isrnghmd.r	$$⊢\phi \to R\in \mathrm{Rng}$$
5		isrnghmd.s	$$⊢\phi \to S\in \mathrm{Rng}$$
6		isrnghmd.ht	$$⊢\left(\phi \wedge \left(x\in B\wedge y\in B\right)\right)\to F\left(x\underset{\dot{}}{·}y\right)=F\left(x\right)\underset{\dot{}}{\times }F\left(y\right)$$
7		isrnghm2d.f	$$⊢\phi \to F\in \left(R\mathrm{GrpHom}S\right)$$
8	4 5	jca	$$⊢\phi \to \left(R\in \mathrm{Rng}\wedge S\in \mathrm{Rng}\right)$$
9	6	ralrimivva	$$⊢\phi \to \forall x\in B\forall y\in BF\left(x\underset{\dot{}}{·}y\right)=F\left(x\right)\underset{\dot{}}{\times }F\left(y\right)$$
10	7 9	jca	$$⊢\phi \to \left(F\in \left(R\mathrm{GrpHom}S\right)\wedge \forall x\in B\forall y\in BF\left(x\underset{\dot{}}{·}y\right)=F\left(x\right)\underset{\dot{}}{\times }F\left(y\right)\right)$$
11	1 2 3	isrnghm	$$⊢F\in \left(R\mathrm{RngHomo}S\right)\leftrightarrow \left(\left(R\in \mathrm{Rng}\wedge S\in \mathrm{Rng}\right)\wedge \left(F\in \left(R\mathrm{GrpHom}S\right)\wedge \forall x\in B\forall y\in BF\left(x\underset{\dot{}}{·}y\right)=F\left(x\right)\underset{\dot{}}{\times }F\left(y\right)\right)\right)$$
12	8 10 11	sylanbrc	$$⊢\phi \to F\in \left(R\mathrm{RngHomo}S\right)$$