Theorem subm0

Description: Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015)

	Ref	Expression
Hypotheses	submmnd.h	|- H = ( M |`s S )
	subm0.z	|- .0. = ( 0g ` M )
Assertion	subm0	|- ( S e. ( SubMnd ` M ) -> .0. = ( 0g ` H ) )

Proof

Step	Hyp	Ref	Expression
1		submmnd.h	|- H = ( M |`s S )
2		subm0.z	|- .0. = ( 0g ` M )
3		submrcl	|- ( S e. ( SubMnd ` M ) -> M e. Mnd )
4	1	submmnd	|- ( S e. ( SubMnd ` M ) -> H e. Mnd )
5		eqid	|- ( Base ` M ) = ( Base ` M )
6	5	submss	|- ( S e. ( SubMnd ` M ) -> S C_ ( Base ` M ) )
7	2	subm0cl	|- ( S e. ( SubMnd ` M ) -> .0. e. S )
8	5 2 1	submnd0	|- ( ( ( M e. Mnd /\ H e. Mnd ) /\ ( S C_ ( Base ` M ) /\ .0. e. S ) ) -> .0. = ( 0g ` H ) )
9	3 4 6 7 8	syl22anc	|- ( S e. ( SubMnd ` M ) -> .0. = ( 0g ` H ) )