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Theorem sgncl 29482
Description: Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.)
Assertion
Ref Expression
sgncl  |-  ( A  e.  RR*  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )

Proof of Theorem sgncl
StepHypRef Expression
1 simpr 468 . . . . 5  |-  ( ( A  e.  RR*  /\  A  =  0 )  ->  A  =  0 )
21fveq2d 5883 . . . 4  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  A )  =  (sgn `  0 )
)
3 sgn0 13229 . . . 4  |-  (sgn ` 
0 )  =  0
42, 3syl6eq 2521 . . 3  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  A )  =  0 )
5 c0ex 9655 . . . 4  |-  0  e.  _V
65tpid2 4077 . . 3  |-  0  e.  { -u 1 ,  0 ,  1 }
74, 6syl6eqel 2557 . 2  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
8 sgnn 13234 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  A )  =  -u
1 )
9 negex 9893 . . . . . 6  |-  -u 1  e.  _V
109tpid1 4076 . . . . 5  |-  -u 1  e.  { -u 1 ,  0 ,  1 }
118, 10syl6eqel 2557 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
1211adantlr 729 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/=  0 )  /\  A  <  0
)  ->  (sgn `  A
)  e.  { -u
1 ,  0 ,  1 } )
13 sgnp 13230 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (sgn `  A )  =  1 )
14 1ex 9656 . . . . . 6  |-  1  e.  _V
1514tpid3 4079 . . . . 5  |-  1  e.  { -u 1 ,  0 ,  1 }
1613, 15syl6eqel 2557 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
1716adantlr 729 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/=  0 )  /\  0  <  A
)  ->  (sgn `  A
)  e.  { -u
1 ,  0 ,  1 } )
18 0xr 9705 . . . 4  |-  0  e.  RR*
19 xrlttri2 11464 . . . . 5  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
2019biimpa 492 . . . 4  |-  ( ( ( A  e.  RR*  /\  0  e.  RR* )  /\  A  =/=  0
)  ->  ( A  <  0  \/  0  < 
A ) )
2118, 20mpanl2 695 . . 3  |-  ( ( A  e.  RR*  /\  A  =/=  0 )  ->  ( A  <  0  \/  0  <  A ) )
2212, 17, 21mpjaodan 803 . 2  |-  ( ( A  e.  RR*  /\  A  =/=  0 )  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
237, 22pm2.61dane 2730 1  |-  ( A  e.  RR*  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {ctp 3963   class class class wbr 4395   ` cfv 5589   0cc0 9557   1c1 9558   RR*cxr 9692    < clt 9693   -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-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-i2m1 9625  ax-1ne0 9626  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632
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-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-tp 3964  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-ov 6311  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-neg 9883  df-sgn 13227
This theorem is referenced by:  sgnclre  29483  sgnmulsgn  29493  sgnmulsgp  29494  signstcl  29526  signstf  29527  signstf0  29529  signstfvn  29530  signsvtn0  29531  signstfvneq0  29533  signsvfn  29543
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