Theorem sgnsval

Description: The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018)

	Ref	Expression
Hypotheses	sgnsval.b	|- B = ( Base ` R )
	sgnsval.0	|- .0. = ( 0g ` R )
	sgnsval.l	|- .< = ( lt ` R )
	sgnsval.s	|- S = ( sgns ` R )
Assertion	sgnsval	|- ( ( R e. V /\ X e. B ) -> ( S ` X ) = if ( X = .0. , 0 , if ( .0. .< X , 1 , -u 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sgnsval.b	|- B = ( Base ` R )
2		sgnsval.0	|- .0. = ( 0g ` R )
3		sgnsval.l	|- .< = ( lt ` R )
4		sgnsval.s	|- S = ( sgns ` R )
5	1 2 3 4	sgnsv	|- ( R e. V -> S = ( x e. B |-> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) ) )
6	5	adantr	|- ( ( R e. V /\ X e. B ) -> S = ( x e. B |-> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) ) )
7		eqeq1	|- ( x = X -> ( x = .0. <-> X = .0. ) )
8		breq2	|- ( x = X -> ( .0. .< x <-> .0. .< X ) )
9	8	ifbid	|- ( x = X -> if ( .0. .< x , 1 , -u 1 ) = if ( .0. .< X , 1 , -u 1 ) )
10	7 9	ifbieq2d	|- ( x = X -> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) = if ( X = .0. , 0 , if ( .0. .< X , 1 , -u 1 ) ) )
11	10	adantl	|- ( ( ( R e. V /\ X e. B ) /\ x = X ) -> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) = if ( X = .0. , 0 , if ( .0. .< X , 1 , -u 1 ) ) )
12		simpr	|- ( ( R e. V /\ X e. B ) -> X e. B )
13		c0ex	|- 0 e. _V
14	13	a1i	|- ( ( ( R e. V /\ X e. B ) /\ X = .0. ) -> 0 e. _V )
15		1ex	|- 1 e. _V
16	15	a1i	|- ( ( ( ( R e. V /\ X e. B ) /\ -. X = .0. ) /\ .0. .< X ) -> 1 e. _V )
17		negex	|- -u 1 e. _V
18	17	a1i	|- ( ( ( ( R e. V /\ X e. B ) /\ -. X = .0. ) /\ -. .0. .< X ) -> -u 1 e. _V )
19	16 18	ifclda	|- ( ( ( R e. V /\ X e. B ) /\ -. X = .0. ) -> if ( .0. .< X , 1 , -u 1 ) e. _V )
20	14 19	ifclda	|- ( ( R e. V /\ X e. B ) -> if ( X = .0. , 0 , if ( .0. .< X , 1 , -u 1 ) ) e. _V )
21	6 11 12 20	fvmptd	|- ( ( R e. V /\ X e. B ) -> ( S ` X ) = if ( X = .0. , 0 , if ( .0. .< X , 1 , -u 1 ) ) )