Theorem kerlidl

Description: The kernel of a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 1-Jul-2024)

	Ref	Expression
Hypotheses	kerlidl.i	$$⊢I=\mathrm{LIdeal}\left(R\right)$$
	kerlidl.0	$$⊢\underset{\dot{}}{0}=0_S$$
Assertion	kerlidl	$$⊢F\in \left(R\mathrm{RingHom}S\right)\to F^{-1}\left[\left\{\underset{\dot{}}{0}\right\}\right]\in I$$

Proof

Step	Hyp	Ref	Expression
1		kerlidl.i	$$⊢I=\mathrm{LIdeal}\left(R\right)$$
2		kerlidl.0	$$⊢\underset{\dot{}}{0}=0_S$$
3		rhmrcl2	$$⊢F\in \left(R\mathrm{RingHom}S\right)\to S\in \mathrm{Ring}$$
4		eqid	$$⊢\mathrm{LIdeal}\left(S\right)=\mathrm{LIdeal}\left(S\right)$$
5	4 2	lidl0	$$⊢S\in \mathrm{Ring}\to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{LIdeal}\left(S\right)$$
6	3 5	syl	$$⊢F\in \left(R\mathrm{RingHom}S\right)\to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{LIdeal}\left(S\right)$$
7	1	rhmpreimaidl	$$⊢\left(F\in \left(R\mathrm{RingHom}S\right)\wedge \left\{\underset{\dot{}}{0}\right\}\in \mathrm{LIdeal}\left(S\right)\right)\to F^{-1}\left[\left\{\underset{\dot{}}{0}\right\}\right]\in I$$
8	6 7	mpdan	$$⊢F\in \left(R\mathrm{RingHom}S\right)\to F^{-1}\left[\left\{\underset{\dot{}}{0}\right\}\right]\in I$$