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Theorem sgnneg 26923
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 9372 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
21negeq0d 9711 . . . 4  |-  ( A  e.  RR  ->  ( A  =  0  <->  -u A  =  0 ) )
32bicomd 201 . . 3  |-  ( A  e.  RR  ->  ( -u A  =  0  <->  A  =  0 ) )
4 eqidd 2444 . . 3  |-  ( ( A  e.  RR  /\  -u A  =  0 )  ->  0  =  0 )
53necon3bbid 2642 . . . . 5  |-  ( A  e.  RR  ->  ( -.  -u A  =  0  <-> 
A  =/=  0 ) )
65biimpa 484 . . . 4  |-  ( ( A  e.  RR  /\  -.  -u A  =  0 )  ->  A  =/=  0 )
7 lt0neg2 9846 . . . . . . . 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 9386 . . . . . . . . . . . . 13  |-  0  e.  RR
1110a1i 11 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  0  e.  RR )
129, 11lttri2d 9513 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
1312biimpa 484 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  <  0  \/  0  <  A ) )
14 ltnsym2 9474 . . . . . . . . . . . 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 pm5.17 883 . . . . . . . . 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 3811 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  if ( -u A  <  0 ,  -u 1 ,  1 )  =  if ( -.  A  <  0 ,  -u 1 ,  1 ) )
23 ifnot 3834 . . . . 5  |-  if ( -.  A  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 ,  1 ,  -u 1
)
2422, 23syl6eq 2491 . . . 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 25903 . 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 9672 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
28 rexr 9429 . . 3  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
29 sgnval 12577 . . 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 9598 . . . 4  |-  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) )
3231a1i 11 . . 3  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) ) )
33 rexr 9429 . . . . 5  |-  ( A  e.  RR  ->  A  e.  RR* )
34 sgnval 12577 . . . . 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 6107 . . 3  |-  ( A  e.  RR  ->  (
0  -  (sgn `  A ) )  =  ( 0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
37 ovif2 6169 . . . . 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 10431 . . . . . 6  |-  ( 0  -  0 )  =  0
40 ovif2 6169 . . . . . . 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 9378 . . . . . . . . . 10  |-  0  e.  CC
43 ax-1cn 9340 . . . . . . . . . 10  |-  1  e.  CC
44 subneg 9658 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  1  e.  CC )  ->  ( 0  -  -u 1
)  =  ( 0  +  1 ) )
4542, 43, 44mp2an 672 . . . . . . . . 9  |-  ( 0  -  -u 1 )  =  ( 0  +  1 )
46 0p1e1 10433 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4745, 46eqtr2i 2464 . . . . . . . 8  |-  1  =  ( 0  - 
-u 1 )
48 df-neg 9598 . . . . . . . 8  |-  -u 1  =  ( 0  -  1 )
4941, 47, 48ifbieq12i 3815 . . . . . . 7  |-  if ( A  <  0 ,  1 ,  -u 1
)  =  if ( A  <  0 ,  ( 0  -  -u 1
) ,  ( 0  -  1 ) )
5040, 49eqtr4i 2466 . . . . . 6  |-  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) )  =  if ( A  <  0 ,  1 ,  -u 1
)
5138, 39, 50ifbieq12i 3815 . . . . 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 2463 . . . 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 2479 . 2  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) ) )
5526, 30, 543eqtr4d 2485 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 1369    e. wcel 1756    =/= wne 2606   ifcif 3791   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285   RR*cxr 9417    < clt 9418    - cmin 9595   -ucneg 9596  sgncsgn 12575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-sgn 12576
This theorem is referenced by: (None)
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