Theorem quseccl

Description: Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015) (Proof shortened by AV, 9-Mar-2025)

	Ref	Expression
Hypotheses	qusgrp.h	|- H = ( G /s ( G ~QG S ) )
	qusadd.v	|- V = ( Base ` G )
	quseccl.b	|- B = ( Base ` H )
Assertion	quseccl	|- ( ( S e. ( NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. B )

Proof

Step	Hyp	Ref	Expression
1		qusgrp.h	|- H = ( G /s ( G ~QG S ) )
2		qusadd.v	|- V = ( Base ` G )
3		quseccl.b	|- B = ( Base ` H )
4		nsgsubg	|- ( S e. ( NrmSGrp ` G ) -> S e. ( SubGrp ` G ) )
5		subgrcl	|- ( S e. ( SubGrp ` G ) -> G e. Grp )
6	4 5	syl	|- ( S e. ( NrmSGrp ` G ) -> G e. Grp )
7		eqid	|- ( G ~QG S ) = ( G ~QG S )
8	7 1 2 3	quseccl0	|- ( ( G e. Grp /\ X e. V ) -> [ X ] ( G ~QG S ) e. B )
9	6 8	sylan	|- ( ( S e. ( NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. B )