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Theorem sgnneg 29411
Description: Negation of the signum. (Contributed by Thierry Arnoux, 1-Oct-2018.)
Assertion
Ref Expression
sgnneg  |-  ( A  e.  RR  ->  (sgn `  -u A )  =  -u (sgn `  A ) )

Proof of Theorem sgnneg
StepHypRef Expression
1 recn 9629 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
21negeq0d 9978 . . . 4  |-  ( A  e.  RR  ->  ( A  =  0  <->  -u A  =  0 ) )
32bicomd 205 . . 3  |-  ( A  e.  RR  ->  ( -u A  =  0  <->  A  =  0 ) )
4 eqidd 2452 . . 3  |-  ( ( A  e.  RR  /\  -u A  =  0 )  ->  0  =  0 )
53necon3bbid 2661 . . . . 5  |-  ( A  e.  RR  ->  ( -.  -u A  =  0  <-> 
A  =/=  0 ) )
65biimpa 487 . . . 4  |-  ( ( A  e.  RR  /\  -.  -u A  =  0 )  ->  A  =/=  0 )
7 lt0neg2 10121 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  A  <->  -u A  <  0 ) )
87adantr 467 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <  A  <->  -u A  <  0 ) )
9 id 22 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  RR )
10 0red 9644 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  0  e.  RR )
119, 10lttri2d 9774 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
1211biimpa 487 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  <  0  \/  0  <  A ) )
13 ltnsym2 9733 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  -.  ( A  <  0  /\  0  < 
A ) )
1410, 13mpdan 674 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -.  ( A  <  0  /\  0  <  A ) )
1514adantr 467 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  -.  ( A  <  0  /\  0  <  A ) )
1612, 15jca 535 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  <  0  \/  0  < 
A )  /\  -.  ( A  <  0  /\  0  <  A ) ) )
17 pm5.17 899 . . . . . . . . 9  |-  ( ( ( A  <  0  \/  0  <  A )  /\  -.  ( A  <  0  /\  0  <  A ) )  <->  ( A  <  0  <->  -.  0  <  A ) )
1816, 17sylib 200 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  <  0  <->  -.  0  <  A ) )
1918con2bid 331 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <  A  <->  -.  A  <  0 ) )
208, 19bitr3d 259 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( -u A  <  0  <->  -.  A  <  0 ) )
2120ifbid 3903 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  if ( -u A  <  0 ,  -u 1 ,  1 )  =  if ( -.  A  <  0 ,  -u 1 ,  1 ) )
22 ifnot 3926 . . . . 5  |-  if ( -.  A  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 ,  1 ,  -u 1
)
2321, 22syl6eq 2501 . . . 4  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  if ( -u A  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 ,  1 , 
-u 1 ) )
246, 23syldan 473 . . 3  |-  ( ( A  e.  RR  /\  -.  -u A  =  0 )  ->  if ( -u A  <  0 , 
-u 1 ,  1 )  =  if ( A  <  0 ,  1 ,  -u 1
) )
253, 4, 24ifbieq12d2 28159 . 2  |-  ( A  e.  RR  ->  if ( -u A  =  0 ,  0 ,  if ( -u A  <  0 ,  -u 1 ,  1 ) )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) ) )
26 renegcl 9937 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
27 rexr 9686 . . 3  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
28 sgnval 13151 . . 3  |-  ( -u A  e.  RR*  ->  (sgn `  -u A )  =  if ( -u A  =  0 ,  0 ,  if ( -u A  <  0 ,  -u 1 ,  1 ) ) )
2926, 27, 283syl 18 . 2  |-  ( A  e.  RR  ->  (sgn `  -u A )  =  if ( -u A  =  0 ,  0 ,  if ( -u A  <  0 ,  -u 1 ,  1 ) ) )
30 df-neg 9863 . . . 4  |-  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) )
3130a1i 11 . . 3  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) ) )
32 rexr 9686 . . . . 5  |-  ( A  e.  RR  ->  A  e.  RR* )
33 sgnval 13151 . . . . 5  |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
3432, 33syl 17 . . . 4  |-  ( A  e.  RR  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
3534oveq2d 6306 . . 3  |-  ( A  e.  RR  ->  (
0  -  (sgn `  A ) )  =  ( 0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
36 ovif2 6374 . . . . 5  |-  ( 0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u
1 ,  1 ) ) )  =  if ( A  =  0 ,  ( 0  -  0 ) ,  ( 0  -  if ( A  <  0 , 
-u 1 ,  1 ) ) )
37 biid 240 . . . . . 6  |-  ( A  =  0  <->  A  = 
0 )
38 0m0e0 10719 . . . . . 6  |-  ( 0  -  0 )  =  0
39 ovif2 6374 . . . . . . 7  |-  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) )  =  if ( A  <  0 ,  ( 0  -  -u 1
) ,  ( 0  -  1 ) )
40 biid 240 . . . . . . . 8  |-  ( A  <  0  <->  A  <  0 )
41 0cn 9635 . . . . . . . . . 10  |-  0  e.  CC
42 ax-1cn 9597 . . . . . . . . . 10  |-  1  e.  CC
4341, 42subnegi 9953 . . . . . . . . 9  |-  ( 0  -  -u 1 )  =  ( 0  +  1 )
44 0p1e1 10721 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4543, 44eqtr2i 2474 . . . . . . . 8  |-  1  =  ( 0  - 
-u 1 )
46 df-neg 9863 . . . . . . . 8  |-  -u 1  =  ( 0  -  1 )
4740, 45, 46ifbieq12i 3907 . . . . . . 7  |-  if ( A  <  0 ,  1 ,  -u 1
)  =  if ( A  <  0 ,  ( 0  -  -u 1
) ,  ( 0  -  1 ) )
4839, 47eqtr4i 2476 . . . . . 6  |-  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) )  =  if ( A  <  0 ,  1 ,  -u 1
)
4937, 38, 48ifbieq12i 3907 . . . . 5  |-  if ( A  =  0 ,  ( 0  -  0 ) ,  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) ) )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) )
5036, 49eqtri 2473 . . . 4  |-  ( 0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u
1 ,  1 ) ) )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) )
5150a1i 11 . . 3  |-  ( A  e.  RR  ->  (
0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 , 
-u 1 ,  1 ) ) )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 , 
-u 1 ) ) )
5231, 35, 513eqtrd 2489 . 2  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) ) )
5325, 29, 523eqtr4d 2495 1  |-  ( A  e.  RR  ->  (sgn `  -u A )  =  -u (sgn `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   ifcif 3881   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542   RR*cxr 9674    < clt 9675    - cmin 9860   -ucneg 9861  sgncsgn 13149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-sgn 13150
This theorem is referenced by: (None)
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