Theorem sgnsv 29727

Description: The sign mapping. (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
sgnsv	|- ( R e. V -> S = ( x e. B |-> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) ) )

Distinct variable groups:   x, .0.   x, .<   x, B   x, R   x, V

Allowed substitution hint:   S( x)

Proof of Theorem sgnsv

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sgnsval.s	. 2 |- S = (sgns ` R )
2		elex 3212	. . 3 |- ( R e. V -> R e. _V )
3		fveq2 6191	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
4		sgnsval.b	. . . . . 6 |- B = ( Base ` R )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( r = R -> ( Base ` r ) = B )
6		fveq2 6191	. . . . . . . . 9 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
7		sgnsval.0	. . . . . . . . 9 |- .0. = ( 0g ` R )
8	6, 7	syl6eqr 2674	. . . . . . . 8 |- ( r = R -> ( 0g ` r ) = .0. )
9	8	adantr 481	. . . . . . 7 |- ( ( r = R /\ x e. ( Base ` r ) ) -> ( 0g ` r ) = .0. )
10	9	eqeq2d 2632	. . . . . 6 |- ( ( r = R /\ x e. ( Base ` r ) ) -> ( x = ( 0g ` r ) <-> x = .0. ) )
11		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> ( lt ` r ) = ( lt ` R ) )
12		sgnsval.l	. . . . . . . . . 10 |- .< = ( lt ` R )
13	11, 12	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> ( lt ` r ) = .< )
14	13	adantr 481	. . . . . . . 8 |- ( ( r = R /\ x e. ( Base ` r ) ) -> ( lt ` r ) = .< )
15		eqidd 2623	. . . . . . . 8 |- ( ( r = R /\ x e. ( Base ` r ) ) -> x = x )
16	9, 14, 15	breq123d 4667	. . . . . . 7 |- ( ( r = R /\ x e. ( Base ` r ) ) -> ( ( 0g ` r ) ( lt ` r ) x <-> .0. .< x ) )
17	16	ifbid 4108	. . . . . 6 |- ( ( r = R /\ x e. ( Base ` r ) ) -> if ( ( 0g ` r ) ( lt ` r ) x , 1 , -u 1 ) = if ( .0. .< x , 1 , -u 1 ) )
18	10, 17	ifbieq2d 4111	. . . . 5 |- ( ( r = R /\ x e. ( Base ` r ) ) -> if ( x = ( 0g ` r ) , 0 , if ( ( 0g ` r ) ( lt ` r ) x , 1 , -u 1 ) ) = if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) )
19	5, 18	mpteq12dva 4732	. . . 4 |- ( r = R -> ( x e. ( Base ` r ) |-> if ( x = ( 0g ` r ) , 0 , if ( ( 0g ` r ) ( lt ` r ) x , 1 , -u 1 ) ) ) = ( x e. B |-> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) ) )
20		df-sgns 29726	. . . 4 |- sgns = ( r e. _V |-> ( x e. ( Base ` r ) |-> if ( x = ( 0g ` r ) , 0 , if ( ( 0g ` r ) ( lt ` r ) x , 1 , -u 1 ) ) ) )
21		fvex 6201	. . . . 5 |- ( Base ` r ) e. _V
22	21	mptex 6486	. . . 4 |- ( x e. ( Base ` r ) |-> if ( x = ( 0g ` r ) , 0 , if ( ( 0g ` r ) ( lt ` r ) x , 1 , -u 1 ) ) ) e. _V
23	19, 20, 22	fvmpt3i 6287	. . 3 |- ( R e. _V -> (sgns ` R ) = ( x e. B |-> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) ) )
24	2, 23	syl 17	. 2 |- ( R e. V -> (sgns ` R ) = ( x e. B |-> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) ) )
25	1, 24	syl5eq 2668	1 |- ( R e. V -> S = ( x e. B |-> if ( x = .0. , 0 , if ( .0. .< x , 1 , -u 1 ) ) ) )

Syntax hints:   -> 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

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-sgns 29726

This theorem is referenced by:  sgnsval  29728  sgnsf  29729