Theorem cnlmod4

Description: Lemma 4 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	cnlmod4	|- ( .s ` W ) = x.

Proof

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