Theorem sgnneg 29411

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 9629	. . . . 5 |- ( A e. RR -> A e. CC )
2	1	negeq0d 9978	. . . 4 |- ( A e. RR -> ( A = 0 <-> -u A = 0 ) )
3	2	bicomd 205	. . 3 |- ( A e. RR -> ( -u A = 0 <-> A = 0 ) )
4		eqidd 2452	. . 3 |- ( ( A e. RR /\ -u A = 0 ) -> 0 = 0 )
5	3	necon3bbid 2661	. . . . 5 |- ( A e. RR -> ( -. -u A = 0 <-> A =/= 0 ) )
6	5	biimpa 487	. . . 4 |- ( ( A e. RR /\ -. -u A = 0 ) -> A =/= 0 )
7		lt0neg2 10121	. . . . . . . 8 |- ( A e. RR -> ( 0 < A <-> -u A < 0 ) )
8	7	adantr 467	. . . . . . 7 |- ( ( A e. RR /\ A =/= 0 ) -> ( 0 < A <-> -u A < 0 ) )
9		id 22	. . . . . . . . . . . 12 |- ( A e. RR -> A e. RR )
10		0red 9644	. . . . . . . . . . . 12 |- ( A e. RR -> 0 e. RR )
11	9, 10	lttri2d 9774	. . . . . . . . . . 11 |- ( A e. RR -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
12	11	biimpa 487	. . . . . . . . . 10 |- ( ( A e. RR /\ A =/= 0 ) -> ( A < 0 \/ 0 < A ) )
13		ltnsym2 9733	. . . . . . . . . . . 12 |- ( ( A e. RR /\ 0 e. RR ) -> -. ( A < 0 /\ 0 < A ) )
14	10, 13	mpdan 674	. . . . . . . . . . 11 |- ( A e. RR -> -. ( A < 0 /\ 0 < A ) )
15	14	adantr 467	. . . . . . . . . 10 |- ( ( A e. RR /\ A =/= 0 ) -> -. ( A < 0 /\ 0 < A ) )
16	12, 15	jca 535	. . . . . . . . 9 |- ( ( A e. RR /\ A =/= 0 ) -> ( ( A < 0 \/ 0 < A ) /\ -. ( A < 0 /\ 0 < A ) ) )
17		pm5.17 899	. . . . . . . . 9 |- ( ( ( A < 0 \/ 0 < A ) /\ -. ( A < 0 /\ 0 < A ) ) <-> ( A < 0 <-> -. 0 < A ) )
18	16, 17	sylib 200	. . . . . . . 8 |- ( ( A e. RR /\ A =/= 0 ) -> ( A < 0 <-> -. 0 < A ) )
19	18	con2bid 331	. . . . . . 7 |- ( ( A e. RR /\ A =/= 0 ) -> ( 0 < A <-> -. A < 0 ) )
20	8, 19	bitr3d 259	. . . . . 6 |- ( ( A e. RR /\ A =/= 0 ) -> ( -u A < 0 <-> -. A < 0 ) )
21	20	ifbid 3903	. . . . 5 |- ( ( A e. RR /\ A =/= 0 ) -> if ( -u A < 0 , -u 1 , 1 ) = if ( -. A < 0 , -u 1 , 1 ) )
22		ifnot 3926	. . . . 5 |- if ( -. A < 0 , -u 1 , 1 ) = if ( A < 0 , 1 , -u 1 )
23	21, 22	syl6eq 2501	. . . 4 |- ( ( A e. RR /\ A =/= 0 ) -> if ( -u A < 0 , -u 1 , 1 ) = if ( A < 0 , 1 , -u 1 ) )
24	6, 23	syldan 473	. . 3 |- ( ( A e. RR /\ -. -u A = 0 ) -> if ( -u A < 0 , -u 1 , 1 ) = if ( A < 0 , 1 , -u 1 ) )
25	3, 4, 24	ifbieq12d2 28159	. 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 ) ) )
26		renegcl 9937	. . 3 |- ( A e. RR -> -u A e. RR )
27		rexr 9686	. . 3 |- ( -u A e. RR -> -u A e. RR* )
28		sgnval 13151	. . 3 |- ( -u A e. RR* -> (sgn ` -u A ) = if ( -u A = 0 , 0 , if ( -u A < 0 , -u 1 , 1 ) ) )
29	26, 27, 28	3syl 18	. 2 |- ( A e. RR -> (sgn ` -u A ) = if ( -u A = 0 , 0 , if ( -u A < 0 , -u 1 , 1 ) ) )
30		df-neg 9863	. . . 4 |- -u (sgn ` A ) = ( 0 - (sgn ` A ) )
31	30	a1i 11	. . 3 |- ( A e. RR -> -u (sgn ` A ) = ( 0 - (sgn ` A ) ) )
32		rexr 9686	. . . . 5 |- ( A e. RR -> A e. RR* )
33		sgnval 13151	. . . . 5 |- ( A e. RR* -> (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
34	32, 33	syl 17	. . . 4 |- ( A e. RR -> (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
35	34	oveq2d 6306	. . 3 |- ( A e. RR -> ( 0 - (sgn ` A ) ) = ( 0 - if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) )
36		ovif2 6374	. . . . 5 |- ( 0 - if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) = if ( A = 0 , ( 0 - 0 ) , ( 0 - if ( A < 0 , -u 1 , 1 ) ) )
37		biid 240	. . . . . 6 |- ( A = 0 <-> A = 0 )
38		0m0e0 10719	. . . . . 6 |- ( 0 - 0 ) = 0
39		ovif2 6374	. . . . . . 7 |- ( 0 - if ( A < 0 , -u 1 , 1 ) ) = if ( A < 0 , ( 0 - -u 1 ) , ( 0 - 1 ) )
40		biid 240	. . . . . . . 8 |- ( A < 0 <-> A < 0 )
41		0cn 9635	. . . . . . . . . 10 |- 0 e. CC
42		ax-1cn 9597	. . . . . . . . . 10 |- 1 e. CC
43	41, 42	subnegi 9953	. . . . . . . . 9 |- ( 0 - -u 1 ) = ( 0 + 1 )
44		0p1e1 10721	. . . . . . . . 9 |- ( 0 + 1 ) = 1
45	43, 44	eqtr2i 2474	. . . . . . . 8 |- 1 = ( 0 - -u 1 )
46		df-neg 9863	. . . . . . . 8 |- -u 1 = ( 0 - 1 )
47	40, 45, 46	ifbieq12i 3907	. . . . . . 7 |- if ( A < 0 , 1 , -u 1 ) = if ( A < 0 , ( 0 - -u 1 ) , ( 0 - 1 ) )
48	39, 47	eqtr4i 2476	. . . . . 6 |- ( 0 - if ( A < 0 , -u 1 , 1 ) ) = if ( A < 0 , 1 , -u 1 )
49	37, 38, 48	ifbieq12i 3907	. . . . 5 |- if ( A = 0 , ( 0 - 0 ) , ( 0 - if ( A < 0 , -u 1 , 1 ) ) ) = if ( A = 0 , 0 , if ( A < 0 , 1 , -u 1 ) )
50	36, 49	eqtri 2473	. . . 4 |- ( 0 - if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) = if ( A = 0 , 0 , if ( A < 0 , 1 , -u 1 ) )
51	50	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 ) ) )
52	31, 35, 51	3eqtrd 2489	. 2 |- ( A e. RR -> -u (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , 1 , -u 1 ) ) )
53	25, 29, 52	3eqtr4d 2495	1 |- ( A e. RR -> (sgn ` -u A ) = -u (sgn ` A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   ifcif 3881   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   RR*cxr 9674   < clt 9675   - cmin 9860   -ucneg 9861  sgncsgn 13149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-sgn 13150

This theorem is referenced by: (None)