Theorem lmicqusker

Description: The image H of a module homomorphism F is isomorphic with the quotient module Q over F 's kernel K . This is part of what is sometimes called the first isomorphism theorem for modules. (Contributed by Thierry Arnoux, 10-Mar-2025)

	Ref	Expression
Hypotheses	lmhmqusker.1	$$⊢\underset{\dot{}}{0}=0_H$$
	lmhmqusker.f	$$⊢\phi \to F\in \left(G\mathrm{LMHom}H\right)$$
	lmhmqusker.k	$$⊢K=F^{-1}\left[\left\{\underset{\dot{}}{0}\right\}\right]$$
	lmhmqusker.q	$$⊢Q=G{/}_{𝑠}\left(G{~}_{\mathit{QG}}K\right)$$
	lmhmqusker.s	$$⊢\phi \to \mathrm{ran}F={\mathrm{Base}}_H$$
Assertion	lmicqusker	$$⊢\phi \to Q{\simeq }_{𝑚}H$$

Proof

Step	Hyp	Ref	Expression
1		lmhmqusker.1	$$⊢\underset{\dot{}}{0}=0_H$$
2		lmhmqusker.f	$$⊢\phi \to F\in \left(G\mathrm{LMHom}H\right)$$
3		lmhmqusker.k	$$⊢K=F^{-1}\left[\left\{\underset{\dot{}}{0}\right\}\right]$$
4		lmhmqusker.q	$$⊢Q=G{/}_{𝑠}\left(G{~}_{\mathit{QG}}K\right)$$
5		lmhmqusker.s	$$⊢\phi \to \mathrm{ran}F={\mathrm{Base}}_H$$
6		imaeq2	$$⊢p=q\to F\left[p\right]=F\left[q\right]$$
7	6	unieqd	$$⊢p=q\to \bigcup F\left[p\right]=\bigcup F\left[q\right]$$
8	7	cbvmptv	$$⊢\left(p\in {\mathrm{Base}}_Q⟼\bigcup F\left[p\right]\right)=\left(q\in {\mathrm{Base}}_Q⟼\bigcup F\left[q\right]\right)$$
9	1 2 3 4 5 8	lmhmqusker	$$⊢\phi \to \left(p\in {\mathrm{Base}}_Q⟼\bigcup F\left[p\right]\right)\in \left(Q\mathrm{LMIso}H\right)$$
10		brlmici	$$⊢\left(p\in {\mathrm{Base}}_Q⟼\bigcup F\left[p\right]\right)\in \left(Q\mathrm{LMIso}H\right)\to Q{\simeq }_{𝑚}H$$
11	9 10	syl	$$⊢\phi \to Q{\simeq }_{𝑚}H$$