Theorem sgn3da 26946

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 12598	. . . . . . . . 9 |- ( A e. RR* -> (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
3	1, 2	syl 16	. . . . . . . 8 |- ( ph -> (sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
4	3	eqeq2d 2454	. . . . . . 7 |- ( ph -> ( 0 = (sgn ` A ) <-> 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) )
5	4	pm5.32i 637	. . . . . 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 2446	. . . . . . . 8 |- ( 0 = (sgn ` A ) -> ( ps <-> ch ) )
8	7	bicomd 201	. . . . . . 7 |- ( 0 = (sgn ` A ) -> ( ch <-> ps ) )
9	8	adantl 466	. . . . . 6 |- ( ( ph /\ 0 = (sgn ` A ) ) -> ( ch <-> ps ) )
10	5, 9	sylbir 213	. . . . 5 |- ( ( ph /\ 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) -> ( ch <-> ps ) )
11	10	expcom 435	. . . 4 |- ( 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) -> ( ph -> ( ch <-> ps ) ) )
12	11	pm5.74d 247	. . 3 |- ( 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) -> ( ( ph -> ch ) <-> ( ph -> ps ) ) )
13	3	eqeq2d 2454	. . . . . . 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 637	. . . . . 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 2449	. . . . . . . . 9 |- ( -u 1 = if ( A < 0 , -u 1 , 1 ) -> ( -u 1 = (sgn ` A ) <-> if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) )
16	15	imbi1d 317	. . . . . . . 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 2449	. . . . . . . . 9 |- ( 1 = if ( A < 0 , -u 1 , 1 ) -> ( 1 = (sgn ` A ) <-> if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) )
18	17	imbi1d 317	. . . . . . . 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 465	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A < 0 ) /\ ( A < 0 -> ta ) ) -> ta )
21		simp2 989	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A < 0 ) /\ ta /\ A < 0 ) -> ta )
22	21	3expia 1189	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A < 0 ) /\ ta ) -> ( A < 0 -> ta ) )
23	20, 22	impbida 828	. . . . . . . . . . . . 13 |- ( ( ph /\ A < 0 ) -> ( ( A < 0 -> ta ) <-> ta ) )
24		pm3.24 877	. . . . . . . . . . . . . . . . 17 |- -. ( A < 0 /\ -. A < 0 )
25	24	pm2.21i 131	. . . . . . . . . . . . . . . 16 |- ( ( A < 0 /\ -. A < 0 ) -> th )
26	25	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( A < 0 /\ -. A < 0 ) ) -> th )
27	26	expr 615	. . . . . . . . . . . . . 14 |- ( ( ph /\ A < 0 ) -> ( -. A < 0 -> th ) )
28		tbtru 1379	. . . . . . . . . . . . . 14 |- ( ( -. A < 0 -> th ) <-> ( ( -. A < 0 -> th ) <-> T. ) )
29	27, 28	sylib 196	. . . . . . . . . . . . 13 |- ( ( ph /\ A < 0 ) -> ( ( -. A < 0 -> th ) <-> T. ) )
30	23, 29	anbi12d 710	. . . . . . . . . . . 12 |- ( ( ph /\ A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ( ta /\ T. ) ) )
31		ancom 450	. . . . . . . . . . . . 13 |- ( ( ta /\ T. ) <-> ( T. /\ ta ) )
32		truan 1386	. . . . . . . . . . . . 13 |- ( ( T. /\ ta ) <-> ta )
33	31, 32	bitri 249	. . . . . . . . . . . 12 |- ( ( ta /\ T. ) <-> ta )
34	30, 33	syl6bb 261	. . . . . . . . . . 11 |- ( ( ph /\ A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ta ) )
35	34	3adant3 1008	. . . . . . . . . 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 2446	. . . . . . . . . . 11 |- ( -u 1 = (sgn ` A ) -> ( ps <-> ta ) )
38	37	3ad2ant3 1011	. . . . . . . . . 10 |- ( ( ph /\ A < 0 /\ -u 1 = (sgn ` A ) ) -> ( ps <-> ta ) )
39	35, 38	bitr4d 256	. . . . . . . . 9 |- ( ( ph /\ A < 0 /\ -u 1 = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
40	39	3expia 1189	. . . . . . . 8 |- ( ( ph /\ A < 0 ) -> ( -u 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
41	19	3adant2 1007	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. A < 0 /\ A < 0 ) -> ta )
42	41	3expia 1189	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. A < 0 ) -> ( A < 0 -> ta ) )
43		tbtru 1379	. . . . . . . . . . . . . 14 |- ( ( A < 0 -> ta ) <-> ( ( A < 0 -> ta ) <-> T. ) )
44	42, 43	sylib 196	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A < 0 ) -> ( ( A < 0 -> ta ) <-> T. ) )
45		pm3.35 587	. . . . . . . . . . . . . . 15 |- ( ( -. A < 0 /\ ( -. A < 0 -> th ) ) -> th )
46	45	adantll 713	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. A < 0 ) /\ ( -. A < 0 -> th ) ) -> th )
47		simp2 989	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ -. A < 0 ) /\ th /\ -. A < 0 ) -> th )
48	47	3expia 1189	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. A < 0 ) /\ th ) -> ( -. A < 0 -> th ) )
49	46, 48	impbida 828	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A < 0 ) -> ( ( -. A < 0 -> th ) <-> th ) )
50	44, 49	anbi12d 710	. . . . . . . . . . . 12 |- ( ( ph /\ -. A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ( T. /\ th ) ) )
51		truan 1386	. . . . . . . . . . . 12 |- ( ( T. /\ th ) <-> th )
52	50, 51	syl6bb 261	. . . . . . . . . . 11 |- ( ( ph /\ -. A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> th ) )
53	52	3adant3 1008	. . . . . . . . . 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 2446	. . . . . . . . . . 11 |- ( 1 = (sgn ` A ) -> ( ps <-> th ) )
56	55	3ad2ant3 1011	. . . . . . . . . 10 |- ( ( ph /\ -. A < 0 /\ 1 = (sgn ` A ) ) -> ( ps <-> th ) )
57	53, 56	bitr4d 256	. . . . . . . . 9 |- ( ( ph /\ -. A < 0 /\ 1 = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
58	57	3expia 1189	. . . . . . . 8 |- ( ( ph /\ -. A < 0 ) -> ( 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
59	16, 18, 40, 58	ifbothda 3845	. . . . . . 7 |- ( ph -> ( if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
60	59	imp 429	. . . . . 6 |- ( ( ph /\ if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
61	14, 60	sylbir 213	. . . . 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 435	. . . 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 247	. . 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 435	. . . 4 |- ( A = 0 -> ( ph -> ch ) )
66	65	adantl 466	. . 3 |- ( ( T. /\ A = 0 ) -> ( ph -> ch ) )
67	19	ex 434	. . . . . . 7 |- ( ph -> ( A < 0 -> ta ) )
68	67	adantr 465	. . . . . 6 |- ( ( ph /\ -. A = 0 ) -> ( A < 0 -> ta ) )
69		simp1 988	. . . . . . . 8 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> ph )
70		df-ne 2622	. . . . . . . . . . . 12 |- ( A =/= 0 <-> -. A = 0 )
71		0xr 9451	. . . . . . . . . . . . 13 |- 0 e. RR*
72		xrlttri2 11140	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ 0 e. RR* ) -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
73	1, 71, 72	sylancl 662	. . . . . . . . . . . 12 |- ( ph -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
74	70, 73	syl5bbr 259	. . . . . . . . . . 11 |- ( ph -> ( -. A = 0 <-> ( A < 0 \/ 0 < A ) ) )
75	74	biimpa 484	. . . . . . . . . 10 |- ( ( ph /\ -. A = 0 ) -> ( A < 0 \/ 0 < A ) )
76		df-or 370	. . . . . . . . . 10 |- ( ( A < 0 \/ 0 < A ) <-> ( -. A < 0 -> 0 < A ) )
77	75, 76	sylib 196	. . . . . . . . 9 |- ( ( ph /\ -. A = 0 ) -> ( -. A < 0 -> 0 < A ) )
78	77	3impia 1184	. . . . . . . 8 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> 0 < A )
79		sgn3da.5	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> th )
80	69, 78, 79	syl2anc 661	. . . . . . 7 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> th )
81	80	3expia 1189	. . . . . 6 |- ( ( ph /\ -. A = 0 ) -> ( -. A < 0 -> th ) )
82	68, 81	jca 532	. . . . 5 |- ( ( ph /\ -. A = 0 ) -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) )
83	82	expcom 435	. . . 4 |- ( -. A = 0 -> ( ph -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) ) )
84	83	adantl 466	. . 3 |- ( ( T. /\ -. A = 0 ) -> ( ph -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) ) )
85	12, 63, 66, 84	ifbothda 3845	. 2 |- ( T. -> ( ph -> ps ) )
86	85	trud 1378	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1369   T. wtru 1370   e. wcel 1756   =/= wne 2620   ifcif 3812   class class class wbr 4313   ` cfv 5439   0cc0 9303   1c1 9304   RR*cxr 9438   < clt 9439   -ucneg 9617  sgncsgn 12596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-i2m1 9371  ax-1ne0 9372  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-neg 9619  df-sgn 12597

This theorem is referenced by:  sgnmul  26947  sgnsub  26949  sgnnbi  26950  sgnpbi  26951  sgn0bi  26952  sgnsgn  26953