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Theorem sgnnbi 28681
Description: Negative signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
Assertion
Ref Expression
sgnnbi  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  = 
-u 1  <->  A  <  0 ) )

Proof of Theorem sgnnbi
StepHypRef Expression
1 id 22 . . . 4  |-  ( A  e.  RR*  ->  A  e. 
RR* )
2 eqeq1 2461 . . . . 5  |-  ( (sgn
`  A )  =  0  ->  ( (sgn `  A )  =  -u
1  <->  0  =  -u
1 ) )
32imbi1d 317 . . . 4  |-  ( (sgn
`  A )  =  0  ->  ( (
(sgn `  A )  =  -u 1  ->  A  <  0 )  <->  ( 0  =  -u 1  ->  A  <  0 ) ) )
4 eqeq1 2461 . . . . 5  |-  ( (sgn
`  A )  =  1  ->  ( (sgn `  A )  =  -u
1  <->  1  =  -u
1 ) )
54imbi1d 317 . . . 4  |-  ( (sgn
`  A )  =  1  ->  ( (
(sgn `  A )  =  -u 1  ->  A  <  0 )  <->  ( 1  =  -u 1  ->  A  <  0 ) ) )
6 eqeq1 2461 . . . . 5  |-  ( (sgn
`  A )  = 
-u 1  ->  (
(sgn `  A )  =  -u 1  <->  -u 1  = 
-u 1 ) )
76imbi1d 317 . . . 4  |-  ( (sgn
`  A )  = 
-u 1  ->  (
( (sgn `  A
)  =  -u 1  ->  A  <  0 )  <-> 
( -u 1  =  -u
1  ->  A  <  0 ) ) )
8 neg1ne0 10662 . . . . . . 7  |-  -u 1  =/=  0
98nesymi 2730 . . . . . 6  |-  -.  0  =  -u 1
109pm2.21i 131 . . . . 5  |-  ( 0  =  -u 1  ->  A  <  0 )
1110a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
( 0  =  -u
1  ->  A  <  0 ) )
12 neg1rr 10661 . . . . . . . 8  |-  -u 1  e.  RR
13 neg1lt0 10663 . . . . . . . . 9  |-  -u 1  <  0
14 0lt1 10096 . . . . . . . . 9  |-  0  <  1
15 0re 9613 . . . . . . . . . 10  |-  0  e.  RR
16 1re 9612 . . . . . . . . . 10  |-  1  e.  RR
1712, 15, 16lttri 9727 . . . . . . . . 9  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
1813, 14, 17mp2an 672 . . . . . . . 8  |-  -u 1  <  1
1912, 18gtneii 9713 . . . . . . 7  |-  1  =/=  -u 1
2019neii 2656 . . . . . 6  |-  -.  1  =  -u 1
2120pm2.21i 131 . . . . 5  |-  ( 1  =  -u 1  ->  A  <  0 )
2221a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
1  =  -u 1  ->  A  <  0 ) )
23 simp2 997 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0  /\  -u 1  =  -u 1 )  ->  A  <  0 )
24233expia 1198 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( -u 1  =  -u 1  ->  A  <  0 ) )
251, 3, 5, 7, 11, 22, 24sgn3da 28677 . . 3  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  = 
-u 1  ->  A  <  0 ) )
2625imp 429 . 2  |-  ( ( A  e.  RR*  /\  (sgn `  A )  =  -u
1 )  ->  A  <  0 )
27 sgnn 12939 . 2  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  A )  =  -u
1 )
2826, 27impbida 832 1  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  = 
-u 1  <->  A  <  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594   0cc0 9509   1c1 9510   RR*cxr 9644    < clt 9645   -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-cnex 9565  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  ax-pre-mulgt0 9586
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:  sgnmulsgn  28685
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