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Theorem sgncl 28741
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 459 . . . . 5  |-  ( ( A  e.  RR*  /\  A  =  0 )  ->  A  =  0 )
21fveq2d 5852 . . . 4  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  A )  =  (sgn `  0 )
)
3 sgn0 13004 . . . 4  |-  (sgn ` 
0 )  =  0
42, 3syl6eq 2511 . . 3  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  A )  =  0 )
5 c0ex 9579 . . . 4  |-  0  e.  _V
65tpid2 4130 . . 3  |-  0  e.  { -u 1 ,  0 ,  1 }
74, 6syl6eqel 2550 . 2  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
8 sgnn 13009 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  A )  =  -u
1 )
9 negex 9809 . . . . . 6  |-  -u 1  e.  _V
109tpid1 4129 . . . . 5  |-  -u 1  e.  { -u 1 ,  0 ,  1 }
118, 10syl6eqel 2550 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
1211adantlr 712 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/=  0 )  /\  A  <  0
)  ->  (sgn `  A
)  e.  { -u
1 ,  0 ,  1 } )
13 sgnp 13005 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (sgn `  A )  =  1 )
14 1ex 9580 . . . . . 6  |-  1  e.  _V
1514tpid3 4132 . . . . 5  |-  1  e.  { -u 1 ,  0 ,  1 }
1613, 15syl6eqel 2550 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
1716adantlr 712 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/=  0 )  /\  0  <  A
)  ->  (sgn `  A
)  e.  { -u
1 ,  0 ,  1 } )
18 0xr 9629 . . . 4  |-  0  e.  RR*
19 xrlttri2 11351 . . . . 5  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
2019biimpa 482 . . . 4  |-  ( ( ( A  e.  RR*  /\  0  e.  RR* )  /\  A  =/=  0
)  ->  ( A  <  0  \/  0  < 
A ) )
2118, 20mpanl2 679 . . 3  |-  ( ( A  e.  RR*  /\  A  =/=  0 )  ->  ( A  <  0  \/  0  <  A ) )
2212, 17, 21mpjaodan 784 . 2  |-  ( ( A  e.  RR*  /\  A  =/=  0 )  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
237, 22pm2.61dane 2772 1  |-  ( A  e.  RR*  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {ctp 4020   class class class wbr 4439   ` cfv 5570   0cc0 9481   1c1 9482   RR*cxr 9616    < clt 9617   -ucneg 9797  sgncsgn 13001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-neg 9799  df-sgn 13002
This theorem is referenced by:  sgnclre  28742  sgnmulsgn  28752  sgnmulsgp  28753  signstcl  28786  signstf  28787  signstf0  28789  signstfvn  28790  signsvtn0  28791  signstfvneq0  28793  signsvfn  28803
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