Theorem sgnsval 29728

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

Hypotheses

Ref	Expression
sgnsval.b	|- B = ( Base ` R )
sgnsval.0	|- .0. = ( 0g ` R )
sgnsval.l	|- .< = ( lt ` R )
sgnsval.s	|- S = (sgns ` R )

Assertion

Ref	Expression
sgnsval	|- ( ( R e. V /\ X e. B ) -> ( S ` X ) = if ( X = .0. , 0 , if ( .0. .< X , 1 , -u 1 ) ) )

Proof of Theorem sgnsval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sgnsval.b	. . . 4 |- B = ( Base ` R )
2		sgnsval.0	. . . 4 |- .0. = ( 0g ` R )
3		sgnsval.l	. . . 4 |- .< = ( lt ` R )
4		sgnsval.s	. . . 4 |- S = (sgns ` R )
5	1, 2, 3, 4	sgnsv 29727	. . 3 |- ( R e. V -> S = ( x e. B |-> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) ) )
6	5	adantr 481	. 2 |- ( ( R e. V /\ X e. B ) -> S = ( x e. B |-> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) ) )
7		eqeq1 2626	. . . 4 |- ( x = X -> ( x = .0. <-> X = .0. ) )
8		breq2 4657	. . . . 5 |- ( x = X -> ( .0. .< x <-> .0. .< X ) )
9	8	ifbid 4108	. . . 4 |- ( x = X -> if ( .0. .< x , 1 , -u 1 ) = if ( .0. .< X , 1 , -u 1 ) )
10	7, 9	ifbieq2d 4111	. . 3 |- ( 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 482	. 2 |- ( ( ( 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 477	. 2 |- ( ( R e. V /\ X e. B ) -> X e. B )
13		c0ex 10034	. . . 4 |- 0 e. _V
14	13	a1i 11	. . 3 |- ( ( ( R e. V /\ X e. B ) /\ X = .0. ) -> 0 e. _V )
15		1ex 10035	. . . . 5 |- 1 e. _V
16	15	a1i 11	. . . 4 |- ( ( ( ( R e. V /\ X e. B ) /\ -. X = .0. ) /\ .0. .< X ) -> 1 e. _V )
17		negex 10279	. . . . 5 |- -u 1 e. _V
18	17	a1i 11	. . . 4 |- ( ( ( ( R e. V /\ X e. B ) /\ -. X = .0. ) /\ -. .0. .< X ) -> -u 1 e. _V )
19	16, 18	ifclda 4120	. . 3 |- ( ( ( R e. V /\ X e. B ) /\ -. X = .0. ) -> if ( .0. .< X , 1 , -u 1 ) e. _V )
20	14, 19	ifclda 4120	. 2 |- ( ( R e. V /\ X e. B ) -> if ( X = .0. , 0 , if ( .0. .< X , 1 , -u 1 ) ) e. _V )
21	6, 11, 12, 20	fvmptd 6288	1 |- ( ( R e. V /\ X e. B ) -> ( S ` X ) = if ( X = .0. , 0 , if ( .0. .< X , 1 , -u 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   0cc0 9936   1c1 9937   -ucneg 10267   Basecbs 15857   0gc0g 16100   ltcplt 16941  sgn_scsgns 29725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-neg 10269  df-sgns 29726

This theorem is referenced by: (None)