Theorem sgn3da 28231

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 12887	. . . . . . . . 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 2481	. . . . . . 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 2479	. . . . . . . 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 2481	. . . . . . 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 2471	. . . . . . . . 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 2471	. . . . . . . . 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 997	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A < 0 ) /\ ta /\ A < 0 ) -> ta )
22	21	3expia 1198	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A < 0 ) /\ ta ) -> ( A < 0 -> ta ) )
23	20, 22	impbida 830	. . . . . . . . . . . . 13 |- ( ( ph /\ A < 0 ) -> ( ( A < 0 -> ta ) <-> ta ) )
24		pm3.24 880	. . . . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . 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 1016	. . . . . . . . . 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 2479	. . . . . . . . . . 11 |- ( -u 1 = (sgn ` A ) -> ( ps <-> ta ) )
38	37	3ad2ant3 1019	. . . . . . . . . 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 1198	. . . . . . . 8 |- ( ( ph /\ A < 0 ) -> ( -u 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
41	19	3adant2 1015	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. A < 0 /\ A < 0 ) -> ta )
42	41	3expia 1198	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. A < 0 ) -> ( A < 0 -> ta ) )
43		tbtru 1389	. . . . . . . . . . . . . 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 997	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ -. A < 0 ) /\ th /\ -. A < 0 ) -> th )
48	47	3expia 1198	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. A < 0 ) /\ th ) -> ( -. A < 0 -> th ) )
49	46, 48	impbida 830	. . . . . . . . . . . . 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 1396	. . . . . . . . . . . 12 |- ( ( T. /\ th ) <-> th )
52	50, 51	syl6bb 261	. . . . . . . . . . 11 |- ( ( ph /\ -. A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> th ) )
53	52	3adant3 1016	. . . . . . . . . 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 2479	. . . . . . . . . . 11 |- ( 1 = (sgn ` A ) -> ( ps <-> th ) )
56	55	3ad2ant3 1019	. . . . . . . . . 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 1198	. . . . . . . 8 |- ( ( ph /\ -. A < 0 ) -> ( 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
59	16, 18, 40, 58	ifbothda 3974	. . . . . . 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 996	. . . . . . . 8 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> ph )
70		df-ne 2664	. . . . . . . . . . . 12 |- ( A =/= 0 <-> -. A = 0 )
71		0xr 9641	. . . . . . . . . . . . 13 |- 0 e. RR*
72		xrlttri2 11349	. . . . . . . . . . . . 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 1193	. . . . . . . 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 1198	. . . . . 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 3974	. 2 |- ( T. -> ( ph -> ps ) )
86	85	trud 1388	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   T. wtru 1380   e. wcel 1767   =/= wne 2662   ifcif 3939   class class class wbr 4447   ` cfv 5588   0cc0 9493   1c1 9494   RR*cxr 9628   < clt 9629   -ucneg 9807  sgncsgn 12885

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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-i2m1 9561  ax-1ne0 9562  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568

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-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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-neg 9809  df-sgn 12886

This theorem is referenced by:  sgnmul  28232  sgnsub  28234  sgnnbi  28235  sgnpbi  28236  sgn0bi  28237  sgnsgn  28238