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Theorem sgnnbi 26947
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 2449 . . . . 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 2449 . . . . 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 2449 . . . . 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 10446 . . . . . . . 8  |-  -u 1  =/=  0
98necomi 2635 . . . . . . 7  |-  0  =/=  -u 1
109neii 2624 . . . . . 6  |-  -.  0  =  -u 1
1110pm2.21i 131 . . . . 5  |-  ( 0  =  -u 1  ->  A  <  0 )
1211a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
( 0  =  -u
1  ->  A  <  0 ) )
13 neg1lt0 10447 . . . . . . . . 9  |-  -u 1  <  0
14 0lt1 9881 . . . . . . . . 9  |-  0  <  1
15 neg1rr 10445 . . . . . . . . . 10  |-  -u 1  e.  RR
16 0re 9405 . . . . . . . . . 10  |-  0  e.  RR
17 1re 9404 . . . . . . . . . 10  |-  1  e.  RR
1815, 16, 17lttri 9519 . . . . . . . . 9  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
1913, 14, 18mp2an 672 . . . . . . . 8  |-  -u 1  <  1
2015, 17ltnei 9517 . . . . . . . 8  |-  ( -u
1  <  1  ->  1  =/=  -u 1 )
2119, 20ax-mp 5 . . . . . . 7  |-  1  =/=  -u 1
2221neii 2624 . . . . . 6  |-  -.  1  =  -u 1
2322pm2.21i 131 . . . . 5  |-  ( 1  =  -u 1  ->  A  <  0 )
2423a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (
1  =  -u 1  ->  A  <  0 ) )
25 simp2 989 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0  /\  -u 1  =  -u 1 )  ->  A  <  0 )
26253expia 1189 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( -u 1  =  -u 1  ->  A  <  0 ) )
271, 3, 5, 7, 12, 24, 26sgn3da 26943 . . 3  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  = 
-u 1  ->  A  <  0 ) )
2827imp 429 . 2  |-  ( ( A  e.  RR*  /\  (sgn `  A )  =  -u
1 )  ->  A  <  0 )
29 sgnn 12602 . 2  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  A )  =  -u
1 )
3028, 29impbida 828 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 1369    e. wcel 1756    =/= wne 2620   class class class wbr 4311   ` cfv 5437   0cc0 9301   1c1 9302   RR*cxr 9436    < clt 9437   -ucneg 9615  sgncsgn 12594
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 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
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 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-po 4660  df-so 4661  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-sgn 12595
This theorem is referenced by:  sgnmulsgn  26951
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