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Theorem sgnneg 28676
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 9599 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
21negeq0d 9942 . . . 4  |-  ( A  e.  RR  ->  ( A  =  0  <->  -u A  =  0 ) )
32bicomd 201 . . 3  |-  ( A  e.  RR  ->  ( -u A  =  0  <->  A  =  0 ) )
4 eqidd 2458 . . 3  |-  ( ( A  e.  RR  /\  -u A  =  0 )  ->  0  =  0 )
53necon3bbid 2704 . . . . 5  |-  ( A  e.  RR  ->  ( -.  -u A  =  0  <-> 
A  =/=  0 ) )
65biimpa 484 . . . 4  |-  ( ( A  e.  RR  /\  -.  -u A  =  0 )  ->  A  =/=  0 )
7 lt0neg2 10080 . . . . . . . 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 0red 9614 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  0  e.  RR )
119, 10lttri2d 9741 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
1211biimpa 484 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  <  0  \/  0  <  A ) )
13 ltnsym2 9701 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  -.  ( A  <  0  /\  0  < 
A ) )
1410, 13mpdan 668 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -.  ( A  <  0  /\  0  <  A ) )
1514adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  -.  ( A  <  0  /\  0  <  A ) )
1612, 15jca 532 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( ( A  <  0  \/  0  < 
A )  /\  -.  ( A  <  0  /\  0  <  A ) ) )
17 pm5.17 888 . . . . . . . . 9  |-  ( ( ( A  <  0  \/  0  <  A )  /\  -.  ( A  <  0  /\  0  <  A ) )  <->  ( A  <  0  <->  -.  0  <  A ) )
1816, 17sylib 196 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( A  <  0  <->  -.  0  <  A ) )
1918con2bid 329 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 0  <  A  <->  -.  A  <  0 ) )
208, 19bitr3d 255 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( -u A  <  0  <->  -.  A  <  0 ) )
2120ifbid 3966 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  if ( -u A  <  0 ,  -u 1 ,  1 )  =  if ( -.  A  <  0 ,  -u 1 ,  1 ) )
22 ifnot 3989 . . . . 5  |-  if ( -.  A  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 ,  1 ,  -u 1
)
2321, 22syl6eq 2514 . . . 4  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  if ( -u A  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 ,  1 , 
-u 1 ) )
246, 23syldan 470 . . 3  |-  ( ( A  e.  RR  /\  -.  -u A  =  0 )  ->  if ( -u A  <  0 , 
-u 1 ,  1 )  =  if ( A  <  0 ,  1 ,  -u 1
) )
253, 4, 24ifbieq12d2 27555 . 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 9901 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
27 rexr 9656 . . 3  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
28 sgnval 12933 . . 3  |-  ( -u A  e.  RR*  ->  (sgn `  -u A )  =  if ( -u A  =  0 ,  0 ,  if ( -u A  <  0 ,  -u 1 ,  1 ) ) )
2926, 27, 283syl 20 . 2  |-  ( A  e.  RR  ->  (sgn `  -u A )  =  if ( -u A  =  0 ,  0 ,  if ( -u A  <  0 ,  -u 1 ,  1 ) ) )
30 df-neg 9827 . . . 4  |-  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) )
3130a1i 11 . . 3  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  ( 0  -  (sgn `  A ) ) )
32 rexr 9656 . . . . 5  |-  ( A  e.  RR  ->  A  e.  RR* )
33 sgnval 12933 . . . . 5  |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
3432, 33syl 16 . . . 4  |-  ( A  e.  RR  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
3534oveq2d 6312 . . 3  |-  ( A  e.  RR  ->  (
0  -  (sgn `  A ) )  =  ( 0  -  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
36 ovif2 6379 . . . . 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 236 . . . . . 6  |-  ( A  =  0  <->  A  = 
0 )
38 0m0e0 10666 . . . . . 6  |-  ( 0  -  0 )  =  0
39 ovif2 6379 . . . . . . 7  |-  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) )  =  if ( A  <  0 ,  ( 0  -  -u 1
) ,  ( 0  -  1 ) )
40 biid 236 . . . . . . . 8  |-  ( A  <  0  <->  A  <  0 )
41 0cn 9605 . . . . . . . . . 10  |-  0  e.  CC
42 ax-1cn 9567 . . . . . . . . . 10  |-  1  e.  CC
4341, 42subnegi 9917 . . . . . . . . 9  |-  ( 0  -  -u 1 )  =  ( 0  +  1 )
44 0p1e1 10668 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4543, 44eqtr2i 2487 . . . . . . . 8  |-  1  =  ( 0  - 
-u 1 )
46 df-neg 9827 . . . . . . . 8  |-  -u 1  =  ( 0  -  1 )
4740, 45, 46ifbieq12i 3970 . . . . . . 7  |-  if ( A  <  0 ,  1 ,  -u 1
)  =  if ( A  <  0 ,  ( 0  -  -u 1
) ,  ( 0  -  1 ) )
4839, 47eqtr4i 2489 . . . . . 6  |-  ( 0  -  if ( A  <  0 ,  -u
1 ,  1 ) )  =  if ( A  <  0 ,  1 ,  -u 1
)
4937, 38, 48ifbieq12i 3970 . . . . 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 2486 . . . 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 2502 . 2  |-  ( A  e.  RR  ->  -u (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  1 ,  -u
1 ) ) )
5325, 29, 523eqtr4d 2508 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 1395    e. wcel 1819    =/= wne 2652   ifcif 3944   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512   RR*cxr 9644    < clt 9645    - cmin 9824   -ucneg 9825  sgncsgn 12931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-sgn 12932
This theorem is referenced by: (None)
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