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

Step	Hyp	Ref	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 )
4	2, 3	eqeq12d 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 )
7	5, 6	eqeq12d 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 )
10	8, 9	eqeq12d 2404	. 2 |- ( (sgn ` A ) = -u 1 -> ( (sgn ` (sgn ` A ) ) = (sgn ` A ) <-> (sgn ` -u 1 ) = -u 1 ) )
11		sgn0 12924	. . 3 |- (sgn ` 0 ) = 0
12	11	a1i 11	. 2 |- ( ( A e. RR* /\ A = 0 ) -> (sgn ` 0 ) = 0 )
13		sgn1 12927	. . 3 |- (sgn ` 1 ) = 1
14	13	a1i 11	. 2 |- ( ( A e. RR* /\ 0 < A ) -> (sgn ` 1 ) = 1 )
15		neg1rr 10557	. . . . 5 |- -u 1 e. RR
16	15	rexri 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 )
19	16, 17, 18	mp2an 670	. . 3 |- (sgn ` -u 1 ) = -u 1
20	19	a1i 11	. 2 |- ( ( A e. RR* /\ A < 0 ) -> (sgn ` -u 1 ) = -u 1 )
21	1, 4, 7, 10, 12, 14, 20	sgn3da 28663	1 |- ( A e. RR* -> (sgn ` (sgn ` A ) ) = (sgn ` A ) )

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