Theorem sgn3da 30603

Description: A conditional containing a signum is true if it is true in all three possible cases. (Contributed by Thierry Arnoux, 1-Oct-2018.)

Hypotheses

Ref	Expression
sgn3da.0	|- ( ph -> A e. RR* )
sgn3da.1	|- ( (sgn ` A ) = 0 -> ( ps <-> ch ) )
sgn3da.2	|- ( (sgn ` A ) = 1 -> ( ps <-> th ) )
sgn3da.3	|- ( (sgn ` A ) = -u 1 -> ( ps <-> ta ) )
sgn3da.4	|- ( ( ph /\ A = 0 ) -> ch )
sgn3da.5	|- ( ( ph /\ 0 < A ) -> th )
sgn3da.6	|- ( ( ph /\ A < 0 ) -> ta )

Assertion

Ref	Expression
sgn3da	|- ( ph -> ps )

Proof of Theorem sgn3da

Step	Hyp	Ref	Expression
1		sgn3da.0	. . . . . . . . 9 |- ( ph -> A e. RR* )
2		sgnval 13828	. . . . . . . . 9 |- ( A e. RR* -> (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
3	1, 2	syl 17	. . . . . . . 8 |- ( ph -> (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
4	3	eqeq2d 2632	. . . . . . 7 |- ( ph -> ( 0 = (sgn ` A ) <-> 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) )
5	4	pm5.32i 669	. . . . . 6 |- ( ( ph /\ 0 = (sgn ` A ) ) <-> ( ph /\ 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) )
6		sgn3da.1	. . . . . . . . 9 |- ( (sgn ` A ) = 0 -> ( ps <-> ch ) )
7	6	eqcoms 2630	. . . . . . . 8 |- ( 0 = (sgn ` A ) -> ( ps <-> ch ) )
8	7	bicomd 213	. . . . . . 7 |- ( 0 = (sgn ` A ) -> ( ch <-> ps ) )
9	8	adantl 482	. . . . . 6 |- ( ( ph /\ 0 = (sgn ` A ) ) -> ( ch <-> ps ) )
10	5, 9	sylbir 225	. . . . 5 |- ( ( ph /\ 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) -> ( ch <-> ps ) )
11	10	expcom 451	. . . 4 |- ( 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) -> ( ph -> ( ch <-> ps ) ) )
12	11	pm5.74d 262	. . 3 |- ( 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) -> ( ( ph -> ch ) <-> ( ph -> ps ) ) )
13	3	eqeq2d 2632	. . . . . . 7 |- ( ph -> ( if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) <-> if ( A < 0 , -u 1 , 1 ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) )
14	13	pm5.32i 669	. . . . . 6 |- ( ( ph /\ if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) <-> ( ph /\ if ( A < 0 , -u 1 , 1 ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) )
15		eqeq1 2626	. . . . . . . . 9 |- ( -u 1 = if ( A < 0 , -u 1 , 1 ) -> ( -u 1 = (sgn ` A ) <-> if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) )
16	15	imbi1d 331	. . . . . . . 8 |- ( -u 1 = if ( A < 0 , -u 1 , 1 ) -> ( ( -u 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) <-> ( if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) ) )
17		eqeq1 2626	. . . . . . . . 9 |- ( 1 = if ( A < 0 , -u 1 , 1 ) -> ( 1 = (sgn ` A ) <-> if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) )
18	17	imbi1d 331	. . . . . . . 8 |- ( 1 = if ( A < 0 , -u 1 , 1 ) -> ( ( 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) <-> ( if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) ) )
19		sgn3da.6	. . . . . . . . . . . . . . 15 |- ( ( ph /\ A < 0 ) -> ta )
20	19	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A < 0 ) /\ ( A < 0 -> ta ) ) -> ta )
21		simp2 1062	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A < 0 ) /\ ta /\ A < 0 ) -> ta )
22	21	3expia 1267	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A < 0 ) /\ ta ) -> ( A < 0 -> ta ) )
23	20, 22	impbida 877	. . . . . . . . . . . . 13 |- ( ( ph /\ A < 0 ) -> ( ( A < 0 -> ta ) <-> ta ) )
24		pm3.24 926	. . . . . . . . . . . . . . . . 17 |- -. ( A < 0 /\ -. A < 0 )
25	24	pm2.21i 116	. . . . . . . . . . . . . . . 16 |- ( ( A < 0 /\ -. A < 0 ) -> th )
26	25	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( A < 0 /\ -. A < 0 ) ) -> th )
27	26	expr 643	. . . . . . . . . . . . . 14 |- ( ( ph /\ A < 0 ) -> ( -. A < 0 -> th ) )
28		tbtru 1494	. . . . . . . . . . . . . 14 |- ( ( -. A < 0 -> th ) <-> ( ( -. A < 0 -> th ) <-> T. ) )
29	27, 28	sylib 208	. . . . . . . . . . . . 13 |- ( ( ph /\ A < 0 ) -> ( ( -. A < 0 -> th ) <-> T. ) )
30	23, 29	anbi12d 747	. . . . . . . . . . . 12 |- ( ( ph /\ A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ( ta /\ T. ) ) )
31		ancom 466	. . . . . . . . . . . . 13 |- ( ( ta /\ T. ) <-> ( T. /\ ta ) )
32		truan 1501	. . . . . . . . . . . . 13 |- ( ( T. /\ ta ) <-> ta )
33	31, 32	bitri 264	. . . . . . . . . . . 12 |- ( ( ta /\ T. ) <-> ta )
34	30, 33	syl6bb 276	. . . . . . . . . . 11 |- ( ( ph /\ A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ta ) )
35	34	3adant3 1081	. . . . . . . . . 10 |- ( ( ph /\ A < 0 /\ -u 1 = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ta ) )
36		sgn3da.3	. . . . . . . . . . . 12 |- ( (sgn ` A ) = -u 1 -> ( ps <-> ta ) )
37	36	eqcoms 2630	. . . . . . . . . . 11 |- ( -u 1 = (sgn ` A ) -> ( ps <-> ta ) )
38	37	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( ph /\ A < 0 /\ -u 1 = (sgn ` A ) ) -> ( ps <-> ta ) )
39	35, 38	bitr4d 271	. . . . . . . . 9 |- ( ( ph /\ A < 0 /\ -u 1 = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
40	39	3expia 1267	. . . . . . . 8 |- ( ( ph /\ A < 0 ) -> ( -u 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
41	19	3adant2 1080	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. A < 0 /\ A < 0 ) -> ta )
42	41	3expia 1267	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. A < 0 ) -> ( A < 0 -> ta ) )
43		tbtru 1494	. . . . . . . . . . . . . 14 |- ( ( A < 0 -> ta ) <-> ( ( A < 0 -> ta ) <-> T. ) )
44	42, 43	sylib 208	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A < 0 ) -> ( ( A < 0 -> ta ) <-> T. ) )
45		pm3.35 611	. . . . . . . . . . . . . . 15 |- ( ( -. A < 0 /\ ( -. A < 0 -> th ) ) -> th )
46	45	adantll 750	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. A < 0 ) /\ ( -. A < 0 -> th ) ) -> th )
47		simp2 1062	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ -. A < 0 ) /\ th /\ -. A < 0 ) -> th )
48	47	3expia 1267	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. A < 0 ) /\ th ) -> ( -. A < 0 -> th ) )
49	46, 48	impbida 877	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A < 0 ) -> ( ( -. A < 0 -> th ) <-> th ) )
50	44, 49	anbi12d 747	. . . . . . . . . . . 12 |- ( ( ph /\ -. A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ( T. /\ th ) ) )
51		truan 1501	. . . . . . . . . . . 12 |- ( ( T. /\ th ) <-> th )
52	50, 51	syl6bb 276	. . . . . . . . . . 11 |- ( ( ph /\ -. A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> th ) )
53	52	3adant3 1081	. . . . . . . . . 10 |- ( ( ph /\ -. A < 0 /\ 1 = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> th ) )
54		sgn3da.2	. . . . . . . . . . . 12 |- ( (sgn ` A ) = 1 -> ( ps <-> th ) )
55	54	eqcoms 2630	. . . . . . . . . . 11 |- ( 1 = (sgn ` A ) -> ( ps <-> th ) )
56	55	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( ph /\ -. A < 0 /\ 1 = (sgn ` A ) ) -> ( ps <-> th ) )
57	53, 56	bitr4d 271	. . . . . . . . 9 |- ( ( ph /\ -. A < 0 /\ 1 = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
58	57	3expia 1267	. . . . . . . 8 |- ( ( ph /\ -. A < 0 ) -> ( 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
59	16, 18, 40, 58	ifbothda 4123	. . . . . . 7 |- ( ph -> ( if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
60	59	imp 445	. . . . . 6 |- ( ( ph /\ if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
61	14, 60	sylbir 225	. . . . 5 |- ( ( ph /\ if ( A < 0 , -u 1 , 1 ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
62	61	expcom 451	. . . 4 |- ( if ( A < 0 , -u 1 , 1 ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) -> ( ph -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
63	62	pm5.74d 262	. . 3 |- ( if ( A < 0 , -u 1 , 1 ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) -> ( ( ph -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) ) <-> ( ph -> ps ) ) )
64		sgn3da.4	. . . . 5 |- ( ( ph /\ A = 0 ) -> ch )
65	64	expcom 451	. . . 4 |- ( A = 0 -> ( ph -> ch ) )
66	65	adantl 482	. . 3 |- ( ( T. /\ A = 0 ) -> ( ph -> ch ) )
67	19	ex 450	. . . . . . 7 |- ( ph -> ( A < 0 -> ta ) )
68	67	adantr 481	. . . . . 6 |- ( ( ph /\ -. A = 0 ) -> ( A < 0 -> ta ) )
69		simp1 1061	. . . . . . . 8 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> ph )
70		df-ne 2795	. . . . . . . . . . . 12 |- ( A =/= 0 <-> -. A = 0 )
71		0xr 10086	. . . . . . . . . . . . 13 |- 0 e. RR*
72		xrlttri2 11975	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ 0 e. RR* ) -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
73	1, 71, 72	sylancl 694	. . . . . . . . . . . 12 |- ( ph -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
74	70, 73	syl5bbr 274	. . . . . . . . . . 11 |- ( ph -> ( -. A = 0 <-> ( A < 0 \/ 0 < A ) ) )
75	74	biimpa 501	. . . . . . . . . 10 |- ( ( ph /\ -. A = 0 ) -> ( A < 0 \/ 0 < A ) )
76	75	ord 392	. . . . . . . . 9 |- ( ( ph /\ -. A = 0 ) -> ( -. A < 0 -> 0 < A ) )
77	76	3impia 1261	. . . . . . . 8 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> 0 < A )
78		sgn3da.5	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> th )
79	69, 77, 78	syl2anc 693	. . . . . . 7 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> th )
80	79	3expia 1267	. . . . . 6 |- ( ( ph /\ -. A = 0 ) -> ( -. A < 0 -> th ) )
81	68, 80	jca 554	. . . . 5 |- ( ( ph /\ -. A = 0 ) -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) )
82	81	expcom 451	. . . 4 |- ( -. A = 0 -> ( ph -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) ) )
83	82	adantl 482	. . 3 |- ( ( T. /\ -. A = 0 ) -> ( ph -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) ) )
84	12, 63, 66, 83	ifbothda 4123	. 2 |- ( T. -> ( ph -> ps ) )
85	84	trud 1493	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   ifcif 4086   class class class wbr 4653   ` cfv 5888   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   -ucneg 10267  sgncsgn 13826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-neg 10269  df-sgn 13827

This theorem is referenced by:  sgnmul  30604  sgnsub  30606  sgnnbi  30607  sgnpbi  30608  sgn0bi  30609  sgnsgn  30610