Theorem signswplusg

Description: The operation of W . (Contributed by Thierry Arnoux, 9-Sep-2018)

	Ref	Expression
Hypotheses	signsw.p	|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
	signsw.w	|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
Assertion	signswplusg	|- .+^ = ( +g ` W )

Proof

Step	Hyp	Ref	Expression
1		signsw.p	|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
2		signsw.w	|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
3		tpex	|- { -u 1 , 0 , 1 } e. _V
4	3 3	mpoex	|- ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) ) e. _V
5	1 4	eqeltri	|- .+^ e. _V
6	2	grpplusg	|- ( .+^ e. _V -> .+^ = ( +g ` W ) )
7	5 6	ax-mp	|- .+^ = ( +g ` W )