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Theorem sgnnbi 28110
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 2464 . . . . 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 2464 . . . . 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 2464 . . . . 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 10630 . . . . . . . 8  |-  -u 1  =/=  0
98necomi 2730 . . . . . . 7  |-  0  =/=  -u 1
109neii 2659 . . . . . 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 10631 . . . . . . . . 9  |-  -u 1  <  0
14 0lt1 10064 . . . . . . . . 9  |-  0  <  1
15 neg1rr 10629 . . . . . . . . . 10  |-  -u 1  e.  RR
16 0re 9585 . . . . . . . . . 10  |-  0  e.  RR
17 1re 9584 . . . . . . . . . 10  |-  1  e.  RR
1815, 16, 17lttri 9699 . . . . . . . . 9  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
1913, 14, 18mp2an 672 . . . . . . . 8  |-  -u 1  <  1
2015, 17ltnei 9697 . . . . . . . 8  |-  ( -u
1  <  1  ->  1  =/=  -u 1 )
2119, 20ax-mp 5 . . . . . . 7  |-  1  =/=  -u 1
2221neii 2659 . . . . . 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 992 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <  0  /\  -u 1  =  -u 1 )  ->  A  <  0 )
26253expia 1193 . . . 4  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  ( -u 1  =  -u 1  ->  A  <  0 ) )
271, 3, 5, 7, 12, 24, 26sgn3da 28106 . . 3  |-  ( A  e.  RR*  ->  ( (sgn
`  A )  = 
-u 1  ->  A  <  0 ) )
2827imp 429 . 2  |-  ( ( A  e.  RR*  /\  (sgn `  A )  =  -u
1 )  ->  A  <  0 )
29 sgnn 12877 . 2  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  A )  =  -u
1 )
3028, 29impbida 829 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 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579   0cc0 9481   1c1 9482   RR*cxr 9616    < clt 9617   -ucneg 9795  sgncsgn 12869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-sgn 12870
This theorem is referenced by:  sgnmulsgn  28114
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