Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sgnsgn Structured version   Unicode version

Theorem sgnsgn 28670
Description: Signum is idempotent. (Contributed by Thierry Arnoux, 2-Oct-2018.)
Assertion
Ref Expression
sgnsgn  |-  ( A  e.  RR*  ->  (sgn `  (sgn `  A ) )  =  (sgn `  A
) )

Proof of Theorem sgnsgn
StepHypRef Expression
1 id 22 . 2  |-  ( A  e.  RR*  ->  A  e. 
RR* )
2 fveq2 5774 . . 3  |-  ( (sgn
`  A )  =  0  ->  (sgn `  (sgn `  A ) )  =  (sgn `  0 )
)
3 id 22 . . 3  |-  ( (sgn
`  A )  =  0  ->  (sgn `  A
)  =  0 )
42, 3eqeq12d 2404 . 2  |-  ( (sgn
`  A )  =  0  ->  ( (sgn `  (sgn `  A )
)  =  (sgn `  A )  <->  (sgn `  0
)  =  0 ) )
5 fveq2 5774 . . 3  |-  ( (sgn
`  A )  =  1  ->  (sgn `  (sgn `  A ) )  =  (sgn `  1 )
)
6 id 22 . . 3  |-  ( (sgn
`  A )  =  1  ->  (sgn `  A
)  =  1 )
75, 6eqeq12d 2404 . 2  |-  ( (sgn
`  A )  =  1  ->  ( (sgn `  (sgn `  A )
)  =  (sgn `  A )  <->  (sgn `  1
)  =  1 ) )
8 fveq2 5774 . . 3  |-  ( (sgn
`  A )  = 
-u 1  ->  (sgn `  (sgn `  A )
)  =  (sgn `  -u 1 ) )
9 id 22 . . 3  |-  ( (sgn
`  A )  = 
-u 1  ->  (sgn `  A )  =  -u
1 )
108, 9eqeq12d 2404 . 2  |-  ( (sgn
`  A )  = 
-u 1  ->  (
(sgn `  (sgn `  A
) )  =  (sgn
`  A )  <->  (sgn `  -u 1
)  =  -u 1
) )
11 sgn0 12924 . . 3  |-  (sgn ` 
0 )  =  0
1211a1i 11 . 2  |-  ( ( A  e.  RR*  /\  A  =  0 )  -> 
(sgn `  0 )  =  0 )
13 sgn1 12927 . . 3  |-  (sgn ` 
1 )  =  1
1413a1i 11 . 2  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  (sgn `  1 )  =  1 )
15 neg1rr 10557 . . . . 5  |-  -u 1  e.  RR
1615rexri 9557 . . . 4  |-  -u 1  e.  RR*
17 neg1lt0 10559 . . . 4  |-  -u 1  <  0
18 sgnn 12929 . . . 4  |-  ( (
-u 1  e.  RR*  /\  -u 1  <  0
)  ->  (sgn `  -u 1
)  =  -u 1
)
1916, 17, 18mp2an 670 . . 3  |-  (sgn `  -u 1 )  =  -u
1
2019a1i 11 . 2  |-  ( ( A  e.  RR*  /\  A  <  0 )  ->  (sgn `  -u 1 )  = 
-u 1 )
211, 4, 7, 10, 12, 14, 20sgn3da 28663 1  |-  ( A  e.  RR*  ->  (sgn `  (sgn `  A ) )  =  (sgn `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   class class class wbr 4367   ` cfv 5496   0cc0 9403   1c1 9404   RR*cxr 9538    < clt 9539   -ucneg 9719  sgncsgn 12921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-sgn 12922
This theorem is referenced by:  signsvfn  28722  signsvfpn  28725  signsvfnn  28726
  Copyright terms: Public domain W3C validator