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Metamath Proof Explorer

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 )