Theorem sgnneg 28116

Description: Negation of the signum. (Contributed by Thierry Arnoux, 1-Oct-2018.)

Assertion

Ref	Expression
sgnneg	|- ( A e. RR -> (sgn ` -u A ) = -u (sgn ` A ) )

Proof of Theorem sgnneg

Step	Hyp	Ref	Expression
1		recn 9578	. . . . 5 |- ( A e. RR -> A e. CC )
2	1	negeq0d 9918	. . . 4 |- ( A e. RR -> ( A = 0 <-> -u A = 0 ) )
3	2	bicomd 201	. . 3 |- ( A e. RR -> ( -u A = 0 <-> A = 0 ) )
4		eqidd 2468	. . 3 |- ( ( A e. RR /\ -u A = 0 ) -> 0 = 0 )
5	3	necon3bbid 2714	. . . . 5 |- ( A e. RR -> ( -. -u A = 0 <-> A =/= 0 ) )
6	5	biimpa 484	. . . 4 |- ( ( A e. RR /\ -. -u A = 0 ) -> A =/= 0 )
7		lt0neg2 10055	. . . . . . . 8 |- ( A e. RR -> ( 0 < A <-> -u A < 0 ) )
8	7	adantr 465	. . . . . . 7 |- ( ( A e. RR /\ A =/= 0 ) -> ( 0 < A <-> -u A < 0 ) )
9		id 22	. . . . . . . . . . . 12 |- ( A e. RR -> A e. RR )
10		0re 9592	. . . . . . . . . . . . 13 |- 0 e. RR
11	10	a1i 11	. . . . . . . . . . . 12 |- ( A e. RR -> 0 e. RR )
12	9, 11	lttri2d 9719	. . . . . . . . . . 11 |- ( A e. RR -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
13	12	biimpa 484	. . . . . . . . . 10 |- ( ( A e. RR /\ A =/= 0 ) -> ( A < 0 \/ 0 < A ) )
14		ltnsym2 9680	. . . . . . . . . . . 12 |- ( ( A e. RR /\ 0 e. RR ) -> -. ( A < 0 /\ 0 < A ) )
15	9, 11, 14	syl2anc 661	. . . . . . . . . . 11 |- ( A e. RR -> -. ( A < 0 /\ 0 < A ) )
16	15	adantr 465	. . . . . . . . . 10 |- ( ( A e. RR /\ A =/= 0 ) -> -. ( A < 0 /\ 0 < A ) )
17	13, 16	jca 532	. . . . . . . . 9 |- ( ( A e. RR /\ A =/= 0 ) -> ( ( A < 0 \/ 0 < A ) /\ -. ( A < 0 /\ 0 < A ) ) )
18		pm5.17 886	. . . . . . . . 9 |- ( ( ( A < 0 \/ 0 < A ) /\ -. ( A < 0 /\ 0 < A ) ) <-> ( A < 0 <-> -. 0 < A ) )
19	17, 18	sylib 196	. . . . . . . 8 |- ( ( A e. RR /\ A =/= 0 ) -> ( A < 0 <-> -. 0 < A ) )
20	19	con2bid 329	. . . . . . 7 |- ( ( A e. RR /\ A =/= 0 ) -> ( 0 < A <-> -. A < 0 ) )
21	8, 20	bitr3d 255	. . . . . 6 |- ( ( A e. RR /\ A =/= 0 ) -> ( -u A < 0 <-> -. A < 0 ) )
22	21	ifbid 3961	. . . . 5 |- ( ( A e. RR /\ A =/= 0 ) -> if ( -u A < 0 , -u 1 , 1 ) = if ( -. A < 0 , -u 1 , 1 ) )
23		ifnot 3984	. . . . 5 |- if ( -. A < 0 , -u 1 , 1 ) = if ( A < 0 , 1 , -u 1 )
24	22, 23	syl6eq 2524	. . . 4 |- ( ( A e. RR /\ A =/= 0 ) -> if ( -u A < 0 , -u 1 , 1 ) = if ( A < 0 , 1 , -u 1 ) )
25	6, 24	syldan 470	. . 3 |- ( ( A e. RR /\ -. -u A = 0 ) -> if ( -u A < 0 , -u 1 , 1 ) = if ( A < 0 , 1 , -u 1 ) )
26	3, 4, 25	ifbieq12d2 27091	. 2 |- ( A e. RR -> if ( -u A = 0 , 0 , if ( -u A < 0 , -u 1 , 1 ) ) = if ( A = 0 , 0 , if ( A < 0 , 1 , -u 1 ) ) )
27		renegcl 9878	. . 3 |- ( A e. RR -> -u A e. RR )
28		rexr 9635	. . 3 |- ( -u A e. RR -> -u A e. RR* )
29		sgnval 12878	. . 3 |- ( -u A e. RR* -> (sgn ` -u A ) = if ( -u A = 0 , 0 , if ( -u A < 0 , -u 1 , 1 ) ) )
30	27, 28, 29	3syl 20	. 2 |- ( A e. RR -> (sgn ` -u A ) = if ( -u A = 0 , 0 , if ( -u A < 0 , -u 1 , 1 ) ) )
31		df-neg 9804	. . . 4 |- -u (sgn ` A ) = ( 0 - (sgn ` A ) )
32	31	a1i 11	. . 3 |- ( A e. RR -> -u (sgn ` A ) = ( 0 - (sgn ` A ) ) )
33		rexr 9635	. . . . 5 |- ( A e. RR -> A e. RR* )
34		sgnval 12878	. . . . 5 |- ( A e. RR* -> (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
35	33, 34	syl 16	. . . 4 |- ( A e. RR -> (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
36	35	oveq2d 6298	. . 3 |- ( A e. RR -> ( 0 - (sgn ` A ) ) = ( 0 - if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) )
37		ovif2 6362	. . . . 5 |- ( 0 - if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) = if ( A = 0 , ( 0 - 0 ) , ( 0 - if ( A < 0 , -u 1 , 1 ) ) )
38		biid 236	. . . . . 6 |- ( A = 0 <-> A = 0 )
39		0m0e0 10641	. . . . . 6 |- ( 0 - 0 ) = 0
40		ovif2 6362	. . . . . . 7 |- ( 0 - if ( A < 0 , -u 1 , 1 ) ) = if ( A < 0 , ( 0 - -u 1 ) , ( 0 - 1 ) )
41		biid 236	. . . . . . . 8 |- ( A < 0 <-> A < 0 )
42		0cn 9584	. . . . . . . . . 10 |- 0 e. CC
43		ax-1cn 9546	. . . . . . . . . 10 |- 1 e. CC
44		subneg 9864	. . . . . . . . . 10 |- ( ( 0 e. CC /\ 1 e. CC ) -> ( 0 - -u 1 ) = ( 0 + 1 ) )
45	42, 43, 44	mp2an 672	. . . . . . . . 9 |- ( 0 - -u 1 ) = ( 0 + 1 )
46		0p1e1 10643	. . . . . . . . 9 |- ( 0 + 1 ) = 1
47	45, 46	eqtr2i 2497	. . . . . . . 8 |- 1 = ( 0 - -u 1 )
48		df-neg 9804	. . . . . . . 8 |- -u 1 = ( 0 - 1 )
49	41, 47, 48	ifbieq12i 3965	. . . . . . 7 |- if ( A < 0 , 1 , -u 1 ) = if ( A < 0 , ( 0 - -u 1 ) , ( 0 - 1 ) )
50	40, 49	eqtr4i 2499	. . . . . 6 |- ( 0 - if ( A < 0 , -u 1 , 1 ) ) = if ( A < 0 , 1 , -u 1 )
51	38, 39, 50	ifbieq12i 3965	. . . . 5 |- if ( A = 0 , ( 0 - 0 ) , ( 0 - if ( A < 0 , -u 1 , 1 ) ) ) = if ( A = 0 , 0 , if ( A < 0 , 1 , -u 1 ) )
52	37, 51	eqtri 2496	. . . 4 |- ( 0 - if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) = if ( A = 0 , 0 , if ( A < 0 , 1 , -u 1 ) )
53	52	a1i 11	. . 3 |- ( A e. RR -> ( 0 - if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) = if ( A = 0 , 0 , if ( A < 0 , 1 , -u 1 ) ) )
54	32, 36, 53	3eqtrd 2512	. 2 |- ( A e. RR -> -u (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , 1 , -u 1 ) ) )
55	26, 30, 54	3eqtr4d 2518	1 |- ( A e. RR -> (sgn ` -u A ) = -u (sgn ` A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   ifcif 3939   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   + caddc 9491   RR*cxr 9623   < clt 9624   - cmin 9801   -ucneg 9802  sgncsgn 12876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-sgn 12877

This theorem is referenced by: (None)