Theorem gicqusker

Description: The image H of a group homomorphism F is isomorphic with the quotient group Q over F 's kernel K . Together with ghmker and ghmima , this is sometimes called the first isomorphism theorem for groups. (Contributed by Thierry Arnoux, 10-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		gicqusker.1	$$⊢\underset{\dot{}}{0}=0_H$$
2		gicqusker.f	$$⊢\phi \to F\in \left(G\mathrm{GrpHom}H\right)$$
3		gicqusker.k	$$⊢K=F^{-1}\left[\left\{\underset{\dot{}}{0}\right\}\right]$$
4		gicqusker.q	$$⊢Q=G{/}_{𝑠}\left(G{~}_{\mathit{QG}}K\right)$$
5		gicqusker.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 8 5	ghmqusker	$$⊢\phi \to \left(p\in {\mathrm{Base}}_Q⟼\bigcup F\left[p\right]\right)\in \left(Q\mathrm{GrpIso}H\right)$$
10		brgici	$$⊢\left(p\in {\mathrm{Base}}_Q⟼\bigcup F\left[p\right]\right)\in \left(Q\mathrm{GrpIso}H\right)\to Q{\simeq }_{𝑔}H$$
11	9 10	syl	$$⊢\phi \to Q{\simeq }_{𝑔}H$$