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	|- .0. = ( 0g ` H )
	lmhmqusker.f	|- ( ph -> F e. ( G LMHom H ) )
	lmhmqusker.k	|- K = ( `' F " { .0. } )
	lmhmqusker.q	|- Q = ( G /s ( G ~QG K ) )
	lmhmqusker.s	|- ( ph -> ran F = ( Base ` H ) )
Assertion	lmicqusker	|- ( ph -> Q ~=m H )

Proof

Step	Hyp	Ref	Expression
1		lmhmqusker.1	|- .0. = ( 0g ` H )
2		lmhmqusker.f	|- ( ph -> F e. ( G LMHom H ) )
3		lmhmqusker.k	|- K = ( `' F " { .0. } )
4		lmhmqusker.q	|- Q = ( G /s ( G ~QG K ) )
5		lmhmqusker.s	|- ( ph -> ran F = ( Base ` H ) )
6		imaeq2	|- ( p = q -> ( F " p ) = ( F " q ) )
7	6	unieqd	|- ( p = q -> U. ( F " p ) = U. ( F " q ) )
8	7	cbvmptv	|- ( p e. ( Base ` Q ) |-> U. ( F " p ) ) = ( q e. ( Base ` Q ) |-> U. ( F " q ) )
9	1 2 3 4 5 8	lmhmqusker	|- ( ph -> ( p e. ( Base ` Q ) |-> U. ( F " p ) ) e. ( Q LMIso H ) )
10		brlmici	|- ( ( p e. ( Base ` Q ) |-> U. ( F " p ) ) e. ( Q LMIso H ) -> Q ~=m H )
11	9 10	syl	|- ( ph -> Q ~=m H )