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Theorem sgnpbi 28991
Description: Positive signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
Assertion
Ref Expression
sgnpbi  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  =  1  <->  0  <  A
) )

Proof of Theorem sgnpbi
StepHypRef Expression
1 id 22 . . . 4  |-  ( A  e.  RR*  ->  A  e. 
RR* )
2 eqeq1 2406 . . . . 5  |-  ( (sgn
`  A )  =  0  ->  ( (sgn `  A )  =  1  <->  0  =  1 ) )
32imbi1d 315 . . . 4  |-  ( (sgn
`  A )  =  0  ->  ( (
(sgn `  A )  =  1  ->  0  <  A )  <->  ( 0  =  1  ->  0  <  A ) ) )
4 eqeq1 2406 . . . . 5  |-  ( (sgn
`  A )  =  1  ->  ( (sgn `  A )  =  1  <->  1  =  1 ) )
54imbi1d 315 . . . 4  |-  ( (sgn
`  A )  =  1  ->  ( (
(sgn `  A )  =  1  ->  0  <  A )  <->  ( 1  =  1  ->  0  <  A ) ) )
6 eqeq1 2406 . . . . 5  |-  ( (sgn
`  A )  = 
-u 1  ->  (
(sgn `  A )  =  1  <->  -u 1  =  1 ) )
76imbi1d 315 . . . 4  |-  ( (sgn
`  A )  = 
-u 1  ->  (
( (sgn `  A
)  =  1  -> 
0  <  A )  <->  (
-u 1  =  1  ->  0  <  A
) ) )
8 0ne1 10644 . . . . . . 7  |-  0  =/=  1
98neii 2602 . . . . . 6  |-  -.  0  =  1
109pm2.21i 131 . . . . 5  |-  ( 0  =  1  ->  0  <  A )
1110a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
( 0  =  1  ->  0  <  A
) )
12 simp2 998 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A  /\  1  =  1 )  ->  0  <  A )
13123expia 1199 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
1  =  1  -> 
0  <  A )
)
14 neg1rr 10681 . . . . . . . 8  |-  -u 1  e.  RR
15 neg1lt0 10683 . . . . . . . . 9  |-  -u 1  <  0
16 0lt1 10115 . . . . . . . . 9  |-  0  <  1
17 0re 9626 . . . . . . . . . 10  |-  0  e.  RR
18 1re 9625 . . . . . . . . . 10  |-  1  e.  RR
1914, 17, 18lttri 9742 . . . . . . . . 9  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
2015, 16, 19mp2an 670 . . . . . . . 8  |-  -u 1  <  1
2114, 20gtneii 9728 . . . . . . 7  |-  1  =/=  -u 1
2221nesymi 2676 . . . . . 6  |-  -.  -u 1  =  1
2322pm2.21i 131 . . . . 5  |-  ( -u
1  =  1  -> 
0  <  A )
2423a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( -u 1  =  1  -> 
0  <  A )
)
251, 3, 5, 7, 11, 13, 24sgn3da 28986 . . 3  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  =  1  ->  0  <  A ) )
2625imp 427 . 2  |-  ( ( A  e.  RR*  /\  (sgn `  A )  =  1 )  ->  0  <  A )
27 sgnp 13072 . 2  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (sgn `  A )  =  1 )
2826, 27impbida 833 1  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  =  1  <->  0  <  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   class class class wbr 4395   ` cfv 5569   0cc0 9522   1c1 9523   RR*cxr 9657    < clt 9658   -ucneg 9842  sgncsgn 13068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-sgn 13069
This theorem is referenced by:  sgnmulsgp  28995
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