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Theorem sgncl 27057
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 461 . . . . 5  |-  ( ( A  e.  RR*  /\  A  =  0 )  ->  A  =  0 )
21fveq2d 5795 . . . 4  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  A )  =  (sgn `  0 )
)
3 sgn0 12682 . . . 4  |-  (sgn ` 
0 )  =  0
42, 3syl6eq 2508 . . 3  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  A )  =  0 )
5 c0ex 9483 . . . 4  |-  0  e.  _V
65tpid2 4089 . . 3  |-  0  e.  { -u 1 ,  0 ,  1 }
74, 6syl6eqel 2547 . 2  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
8 simpl 457 . . . 4  |-  ( ( A  e.  RR*  /\  A  =/=  0 )  ->  A  e.  RR* )
9 sgnn 12687 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  A )  =  -u
1 )
10 negex 9711 . . . . . 6  |-  -u 1  e.  _V
1110tpid1 4088 . . . . 5  |-  -u 1  e.  { -u 1 ,  0 ,  1 }
129, 11syl6eqel 2547 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
138, 12sylan 471 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/=  0 )  /\  A  <  0
)  ->  (sgn `  A
)  e.  { -u
1 ,  0 ,  1 } )
14 sgnp 12683 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (sgn `  A )  =  1 )
15 1ex 9484 . . . . . 6  |-  1  e.  _V
1615tpid3 4091 . . . . 5  |-  1  e.  { -u 1 ,  0 ,  1 }
1714, 16syl6eqel 2547 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
188, 17sylan 471 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/=  0 )  /\  0  <  A
)  ->  (sgn `  A
)  e.  { -u
1 ,  0 ,  1 } )
19 0xr 9533 . . . . 5  |-  0  e.  RR*
2019a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  A  =/=  0 )  ->  0  e.  RR* )
21 simpr 461 . . . 4  |-  ( ( A  e.  RR*  /\  A  =/=  0 )  ->  A  =/=  0 )
22 xrlttri2 11222 . . . . 5  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
2322biimpa 484 . . . 4  |-  ( ( ( A  e.  RR*  /\  0  e.  RR* )  /\  A  =/=  0
)  ->  ( A  <  0  \/  0  < 
A ) )
248, 20, 21, 23syl21anc 1218 . . 3  |-  ( ( A  e.  RR*  /\  A  =/=  0 )  ->  ( A  <  0  \/  0  <  A ) )
2513, 18, 24mpjaodan 784 . 2  |-  ( ( A  e.  RR*  /\  A  =/=  0 )  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
267, 25pm2.61dane 2766 1  |-  ( A  e.  RR*  ->  (sgn `  A )  e.  { -u 1 ,  0 ,  1 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   {ctp 3981   class class class wbr 4392   ` cfv 5518   0cc0 9385   1c1 9386   RR*cxr 9520    < clt 9521   -ucneg 9699  sgncsgn 12679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-i2m1 9453  ax-1ne0 9454  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-neg 9701  df-sgn 12680
This theorem is referenced by:  sgnclre  27058  sgnmulsgn  27068  sgnmulsgp  27069  signstcl  27102  signstf  27103  signstf0  27105  signstfvn  27106  signsvtn0  27107  signstfvneq0  27109  signsvfn  27119
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