Theorem sgnsgn 27051

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 5775	. . 3 |- ( (sgn ` A ) = 0 -> (sgn ` (sgn ` A ) ) = (sgn ` 0 ) )
3		id 22	. . 3 |- ( (sgn ` A ) = 0 -> (sgn ` A ) = 0 )
4	2, 3	eqeq12d 2471	. 2 |- ( (sgn ` A ) = 0 -> ( (sgn ` (sgn ` A ) ) = (sgn ` A ) <-> (sgn ` 0 ) = 0 ) )
5		fveq2 5775	. . 3 |- ( (sgn ` A ) = 1 -> (sgn ` (sgn ` A ) ) = (sgn ` 1 ) )
6		id 22	. . 3 |- ( (sgn ` A ) = 1 -> (sgn ` A ) = 1 )
7	5, 6	eqeq12d 2471	. 2 |- ( (sgn ` A ) = 1 -> ( (sgn ` (sgn ` A ) ) = (sgn ` A ) <-> (sgn ` 1 ) = 1 ) )
8		fveq2 5775	. . 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 2471	. 2 |- ( (sgn ` A ) = -u 1 -> ( (sgn ` (sgn ` A ) ) = (sgn ` A ) <-> (sgn ` -u 1 ) = -u 1 ) )
11		sgn0 12666	. . 3 |- (sgn ` 0 ) = 0
12	11	a1i 11	. 2 |- ( ( A e. RR* /\ A = 0 ) -> (sgn ` 0 ) = 0 )
13		1re 9472	. . . . 5 |- 1 e. RR
14	13	rexri 9523	. . . 4 |- 1 e. RR*
15		0lt1 9949	. . . 4 |- 0 < 1
16		sgnp 12667	. . . 4 |- ( ( 1 e. RR* /\ 0 < 1 ) -> (sgn ` 1 ) = 1 )
17	14, 15, 16	mp2an 672	. . 3 |- (sgn ` 1 ) = 1
18	17	a1i 11	. 2 |- ( ( A e. RR* /\ 0 < A ) -> (sgn ` 1 ) = 1 )
19		neg1rr 10513	. . . . 5 |- -u 1 e. RR
20	19	rexri 9523	. . . 4 |- -u 1 e. RR*
21		neg1lt0 10515	. . . 4 |- -u 1 < 0
22		sgnn 12671	. . . 4 |- ( ( -u 1 e. RR* /\ -u 1 < 0 ) -> (sgn ` -u 1 ) = -u 1 )
23	20, 21, 22	mp2an 672	. . 3 |- (sgn ` -u 1 ) = -u 1
24	23	a1i 11	. 2 |- ( ( A e. RR* /\ A < 0 ) -> (sgn ` -u 1 ) = -u 1 )
25	1, 4, 7, 10, 12, 18, 24	sgn3da 27044	1 |- ( A e. RR* -> (sgn ` (sgn ` A ) ) = (sgn ` A ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1757   class class class wbr 4376   ` cfv 5502   0cc0 9369   1c1 9370   RR*cxr 9504   < clt 9505   -ucneg 9683  sgncsgn 12663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-po 4725  df-so 4726  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-sgn 12664

This theorem is referenced by:  signsvfn  27103  signsvfpn  27106  signsvfnn  27107