Theorem cnlmodlem2

Description: Lemma 2 for cnlmod . (Contributed by AV, 20-Sep-2021)

	Ref	Expression
Hypothesis	cnlmod.w	|- W = ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. } u. { <. ( Scalar ` ndx ) , CCfld >. , <. ( .s ` ndx ) , x. >. } )
Assertion	cnlmodlem2	|- ( +g ` W ) = +

Proof

Step	Hyp	Ref	Expression
1		cnlmod.w	|- W = ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. } u. { <. ( Scalar ` ndx ) , CCfld >. , <. ( .s ` ndx ) , x. >. } )
2		addex	|- + e. _V
3		qdass	|- ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. } u. { <. ( Scalar ` ndx ) , CCfld >. , <. ( .s ` ndx ) , x. >. } ) = ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( Scalar ` ndx ) , CCfld >. } u. { <. ( .s ` ndx ) , x. >. } )
4	1 3	eqtri	|- W = ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( Scalar ` ndx ) , CCfld >. } u. { <. ( .s ` ndx ) , x. >. } )
5	4	lmodplusg	|- ( + e. _V -> + = ( +g ` W ) )
6	5	eqcomd	|- ( + e. _V -> ( +g ` W ) = + )
7	2 6	ax-mp	|- ( +g ` W ) = +