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Theorem sgnpbi 26951
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 2449 . . . . 5  |-  ( (sgn
`  A )  =  0  ->  ( (sgn `  A )  =  1  <->  0  =  1 ) )
32imbi1d 317 . . . 4  |-  ( (sgn
`  A )  =  0  ->  ( (
(sgn `  A )  =  1  ->  0  <  A )  <->  ( 0  =  1  ->  0  <  A ) ) )
4 eqeq1 2449 . . . . 5  |-  ( (sgn
`  A )  =  1  ->  ( (sgn `  A )  =  1  <->  1  =  1 ) )
54imbi1d 317 . . . 4  |-  ( (sgn
`  A )  =  1  ->  ( (
(sgn `  A )  =  1  ->  0  <  A )  <->  ( 1  =  1  ->  0  <  A ) ) )
6 eqeq1 2449 . . . . 5  |-  ( (sgn
`  A )  = 
-u 1  ->  (
(sgn `  A )  =  1  <->  -u 1  =  1 ) )
76imbi1d 317 . . . 4  |-  ( (sgn
`  A )  = 
-u 1  ->  (
( (sgn `  A
)  =  1  -> 
0  <  A )  <->  (
-u 1  =  1  ->  0  <  A
) ) )
8 0ne1 10410 . . . . . . 7  |-  0  =/=  1
98neii 2624 . . . . . 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 989 . . . . 5  |-  ( ( A  e.  RR*  /\  0  <  A  /\  1  =  1 )  ->  0  <  A )
13123expia 1189 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
1  =  1  -> 
0  <  A )
)
14 neg1lt0 10449 . . . . . . . . 9  |-  -u 1  <  0
15 0lt1 9883 . . . . . . . . 9  |-  0  <  1
16 neg1rr 10447 . . . . . . . . . 10  |-  -u 1  e.  RR
17 0re 9407 . . . . . . . . . 10  |-  0  e.  RR
18 1re 9406 . . . . . . . . . 10  |-  1  e.  RR
1916, 17, 18lttri 9521 . . . . . . . . 9  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
2014, 15, 19mp2an 672 . . . . . . . 8  |-  -u 1  <  1
2116, 18ltnei 9519 . . . . . . . 8  |-  ( -u
1  <  1  ->  1  =/=  -u 1 )
2220, 21ax-mp 5 . . . . . . 7  |-  1  =/=  -u 1
2322nesymi 2672 . . . . . 6  |-  -.  -u 1  =  1
2423pm2.21i 131 . . . . 5  |-  ( -u
1  =  1  -> 
0  <  A )
2524a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( -u 1  =  1  -> 
0  <  A )
)
261, 3, 5, 7, 11, 13, 25sgn3da 26946 . . 3  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  =  1  ->  0  <  A ) )
2726imp 429 . 2  |-  ( ( A  e.  RR*  /\  (sgn `  A )  =  1 )  ->  0  <  A )
28 sgnp 12600 . 2  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (sgn `  A )  =  1 )
2927, 28impbida 828 1  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  =  1  <->  0  <  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   class class class wbr 4313   ` cfv 5439   0cc0 9303   1c1 9304   RR*cxr 9438    < clt 9439   -ucneg 9617  sgncsgn 12596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-sgn 12597
This theorem is referenced by:  sgnmulsgp  26955
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