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Theorem sgnneg 29484
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 9647 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
21negeq0d 9997 . . . 4  |-  ( A  e.  RR  ->  ( A  =  0  <->  -u A  =  0 ) )
32bicomd 206 . . 3  |-  ( A  e.  RR  ->  ( -u A  =  0  <->  A  =  0 ) )
4 eqidd 2472 . . 3  |-  ( ( A  e.  RR  /\  -u A  =  0 )  ->  0  =  0 )
53necon3bbid 2680 . . . . 5  |-  ( A  e.  RR  ->  ( -.  -u A  =  0  <-> 
A  =/=  0 ) )
65biimpa 492 . . . 4  |-  ( ( A  e.  RR  /\  -.  -u A  =  0 )  ->  A  =/=  0 )
7 lt0neg2 10142 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  A  <->  -u A  <  0 ) )
87adantr 472 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <  A  <->  -u A  <  0 ) )
9 id 22 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  RR )
10 0red 9662 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  0  e.  RR )
119, 10lttri2d 9791 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
1211biimpa 492 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  <  0  \/  0  <  A ) )
13 ltnsym2 9751 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  -.  ( A  <  0  /\  0  < 
A ) )
1410, 13mpdan 681 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -.  ( A  <  0  /\  0  <  A ) )
1514adantr 472 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  -.  ( A  <  0  /\  0  <  A ) )
1612, 15jca 541 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  <  0  \/  0  < 
A )  /\  -.  ( A  <  0  /\  0  <  A ) ) )
17 pm5.17 905 . . . . . . . . 9  |-  ( ( ( A  <  0  \/  0  <  A )  /\  -.  ( A  <  0  /\  0  <  A ) )  <->  ( A  <  0  <->  -.  0  <  A ) )
1816, 17sylib 201 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  <  0  <->  -.  0  <  A ) )
1918con2bid 336 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <  A  <->  -.  A  <  0 ) )
208, 19bitr3d 263 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( -u A  <  0  <->  -.  A  <  0 ) )
2120ifbid 3894 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  if ( -u A  <  0 ,  -u 1 ,  1 )  =  if ( -.  A  <  0 ,  -u 1 ,  1 ) )
22 ifnot 3917 . . . . 5  |-  if ( -.  A  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 ,  1 ,  -u 1
)
2321, 22syl6eq 2521 . . . 4  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  if ( -u A  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 ,  1 , 
-u 1 ) )
246, 23syldan 478 . . 3  |-  ( ( A  e.  RR  /\  -.  -u A  =  0 )  ->  if ( -u A  <  0 , 
-u 1 ,  1 )  =  if ( A  <  0 ,  1 ,  -u 1
) )
253, 4, 24ifbieq12d2 28237 . 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 9957 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
27 rexr 9704 . . 3  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
28 sgnval 13228 . . 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 9883 . . . 4  |-  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) )
3130a1i 11 . . 3  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) ) )
32 rexr 9704 . . . . 5  |-  ( A  e.  RR  ->  A  e.  RR* )
33 sgnval 13228 . . . . 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 6324 . . 3  |-  ( A  e.  RR  ->  (
0  -  (sgn `  A ) )  =  ( 0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
36 ovif2 6393 . . . . 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 244 . . . . . 6  |-  ( A  =  0  <->  A  = 
0 )
38 0m0e0 10741 . . . . . 6  |-  ( 0  -  0 )  =  0
39 ovif2 6393 . . . . . . 7  |-  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) )  =  if ( A  <  0 ,  ( 0  -  -u 1
) ,  ( 0  -  1 ) )
40 biid 244 . . . . . . . 8  |-  ( A  <  0  <->  A  <  0 )
41 0cn 9653 . . . . . . . . . 10  |-  0  e.  CC
42 ax-1cn 9615 . . . . . . . . . 10  |-  1  e.  CC
4341, 42subnegi 9973 . . . . . . . . 9  |-  ( 0  -  -u 1 )  =  ( 0  +  1 )
44 0p1e1 10743 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4543, 44eqtr2i 2494 . . . . . . . 8  |-  1  =  ( 0  - 
-u 1 )
46 df-neg 9883 . . . . . . . 8  |-  -u 1  =  ( 0  -  1 )
4740, 45, 46ifbieq12i 3898 . . . . . . 7  |-  if ( A  <  0 ,  1 ,  -u 1
)  =  if ( A  <  0 ,  ( 0  -  -u 1
) ,  ( 0  -  1 ) )
4839, 47eqtr4i 2496 . . . . . 6  |-  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) )  =  if ( A  <  0 ,  1 ,  -u 1
)
4937, 38, 48ifbieq12i 3898 . . . . 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 2493 . . . 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 2509 . 2  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) ) )
5325, 29, 523eqtr4d 2515 1  |-  ( A  e.  RR  ->  (sgn `  -u A )  =  -u (sgn `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   ifcif 3872   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560   RR*cxr 9692    < clt 9693    - cmin 9880   -ucneg 9881  sgncsgn 13226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-sgn 13227
This theorem is referenced by: (None)
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