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Theorem sgnneg 28116
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 9578 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
21negeq0d 9918 . . . 4  |-  ( A  e.  RR  ->  ( A  =  0  <->  -u A  =  0 ) )
32bicomd 201 . . 3  |-  ( A  e.  RR  ->  ( -u A  =  0  <->  A  =  0 ) )
4 eqidd 2468 . . 3  |-  ( ( A  e.  RR  /\  -u A  =  0 )  ->  0  =  0 )
53necon3bbid 2714 . . . . 5  |-  ( A  e.  RR  ->  ( -.  -u A  =  0  <-> 
A  =/=  0 ) )
65biimpa 484 . . . 4  |-  ( ( A  e.  RR  /\  -.  -u A  =  0 )  ->  A  =/=  0 )
7 lt0neg2 10055 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  A  <->  -u A  <  0 ) )
87adantr 465 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <  A  <->  -u A  <  0 ) )
9 id 22 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  RR )
10 0re 9592 . . . . . . . . . . . . 13  |-  0  e.  RR
1110a1i 11 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  0  e.  RR )
129, 11lttri2d 9719 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
1312biimpa 484 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  <  0  \/  0  <  A ) )
14 ltnsym2 9680 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  -.  ( A  <  0  /\  0  < 
A ) )
159, 11, 14syl2anc 661 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -.  ( A  <  0  /\  0  <  A ) )
1615adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  -.  ( A  <  0  /\  0  <  A ) )
1713, 16jca 532 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  <  0  \/  0  < 
A )  /\  -.  ( A  <  0  /\  0  <  A ) ) )
18 pm5.17 886 . . . . . . . . 9  |-  ( ( ( A  <  0  \/  0  <  A )  /\  -.  ( A  <  0  /\  0  <  A ) )  <->  ( A  <  0  <->  -.  0  <  A ) )
1917, 18sylib 196 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  <  0  <->  -.  0  <  A ) )
2019con2bid 329 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <  A  <->  -.  A  <  0 ) )
218, 20bitr3d 255 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( -u A  <  0  <->  -.  A  <  0 ) )
2221ifbid 3961 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  if ( -u A  <  0 ,  -u 1 ,  1 )  =  if ( -.  A  <  0 ,  -u 1 ,  1 ) )
23 ifnot 3984 . . . . 5  |-  if ( -.  A  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 ,  1 ,  -u 1
)
2422, 23syl6eq 2524 . . . 4  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  if ( -u A  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 ,  1 , 
-u 1 ) )
256, 24syldan 470 . . 3  |-  ( ( A  e.  RR  /\  -.  -u A  =  0 )  ->  if ( -u A  <  0 , 
-u 1 ,  1 )  =  if ( A  <  0 ,  1 ,  -u 1
) )
263, 4, 25ifbieq12d2 27091 . 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 ) ) )
27 renegcl 9878 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
28 rexr 9635 . . 3  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
29 sgnval 12878 . . 3  |-  ( -u A  e.  RR*  ->  (sgn `  -u A )  =  if ( -u A  =  0 ,  0 ,  if ( -u A  <  0 ,  -u 1 ,  1 ) ) )
3027, 28, 293syl 20 . 2  |-  ( A  e.  RR  ->  (sgn `  -u A )  =  if ( -u A  =  0 ,  0 ,  if ( -u A  <  0 ,  -u 1 ,  1 ) ) )
31 df-neg 9804 . . . 4  |-  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) )
3231a1i 11 . . 3  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) ) )
33 rexr 9635 . . . . 5  |-  ( A  e.  RR  ->  A  e.  RR* )
34 sgnval 12878 . . . . 5  |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
3533, 34syl 16 . . . 4  |-  ( A  e.  RR  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
3635oveq2d 6298 . . 3  |-  ( A  e.  RR  ->  (
0  -  (sgn `  A ) )  =  ( 0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
37 ovif2 6362 . . . . 5  |-  ( 0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u
1 ,  1 ) ) )  =  if ( A  =  0 ,  ( 0  -  0 ) ,  ( 0  -  if ( A  <  0 , 
-u 1 ,  1 ) ) )
38 biid 236 . . . . . 6  |-  ( A  =  0  <->  A  = 
0 )
39 0m0e0 10641 . . . . . 6  |-  ( 0  -  0 )  =  0
40 ovif2 6362 . . . . . . 7  |-  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) )  =  if ( A  <  0 ,  ( 0  -  -u 1
) ,  ( 0  -  1 ) )
41 biid 236 . . . . . . . 8  |-  ( A  <  0  <->  A  <  0 )
42 0cn 9584 . . . . . . . . . 10  |-  0  e.  CC
43 ax-1cn 9546 . . . . . . . . . 10  |-  1  e.  CC
44 subneg 9864 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  1  e.  CC )  ->  ( 0  -  -u 1
)  =  ( 0  +  1 ) )
4542, 43, 44mp2an 672 . . . . . . . . 9  |-  ( 0  -  -u 1 )  =  ( 0  +  1 )
46 0p1e1 10643 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4745, 46eqtr2i 2497 . . . . . . . 8  |-  1  =  ( 0  - 
-u 1 )
48 df-neg 9804 . . . . . . . 8  |-  -u 1  =  ( 0  -  1 )
4941, 47, 48ifbieq12i 3965 . . . . . . 7  |-  if ( A  <  0 ,  1 ,  -u 1
)  =  if ( A  <  0 ,  ( 0  -  -u 1
) ,  ( 0  -  1 ) )
5040, 49eqtr4i 2499 . . . . . 6  |-  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) )  =  if ( A  <  0 ,  1 ,  -u 1
)
5138, 39, 50ifbieq12i 3965 . . . . 5  |-  if ( A  =  0 ,  ( 0  -  0 ) ,  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) ) )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) )
5237, 51eqtri 2496 . . . 4  |-  ( 0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u
1 ,  1 ) ) )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) )
5352a1i 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 ) ) )
5432, 36, 533eqtrd 2512 . 2  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) ) )
5526, 30, 543eqtr4d 2518 1  |-  ( A  e.  RR  ->  (sgn `  -u A )  =  -u (sgn `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3939   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491   RR*cxr 9623    < clt 9624    - cmin 9801   -ucneg 9802  sgncsgn 12876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-sgn 12877
This theorem is referenced by: (None)
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