Theorem sgn3da 29414

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 13145	. . . . . . . . 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 2437	. . . . . . 7 |- ( ph -> ( 0 = (sgn ` A ) <-> 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) )
5	4	pm5.32i 642	. . . . . 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 2435	. . . . . . . 8 |- ( 0 = (sgn ` A ) -> ( ps <-> ch ) )
8	7	bicomd 205	. . . . . . 7 |- ( 0 = (sgn ` A ) -> ( ch <-> ps ) )
9	8	adantl 468	. . . . . 6 |- ( ( ph /\ 0 = (sgn ` A ) ) -> ( ch <-> ps ) )
10	5, 9	sylbir 217	. . . . 5 |- ( ( ph /\ 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) -> ( ch <-> ps ) )
11	10	expcom 437	. . . 4 |- ( 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) -> ( ph -> ( ch <-> ps ) ) )
12	11	pm5.74d 251	. . 3 |- ( 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) -> ( ( ph -> ch ) <-> ( ph -> ps ) ) )
13	3	eqeq2d 2437	. . . . . . 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 642	. . . . . 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 2427	. . . . . . . . 9 |- ( -u 1 = if ( A < 0 , -u 1 , 1 ) -> ( -u 1 = (sgn ` A ) <-> if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) )
16	15	imbi1d 319	. . . . . . . 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 2427	. . . . . . . . 9 |- ( 1 = if ( A < 0 , -u 1 , 1 ) -> ( 1 = (sgn ` A ) <-> if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) )
18	17	imbi1d 319	. . . . . . . 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 467	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A < 0 ) /\ ( A < 0 -> ta ) ) -> ta )
21		simp2 1007	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A < 0 ) /\ ta /\ A < 0 ) -> ta )
22	21	3expia 1208	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A < 0 ) /\ ta ) -> ( A < 0 -> ta ) )
23	20, 22	impbida 841	. . . . . . . . . . . . 13 |- ( ( ph /\ A < 0 ) -> ( ( A < 0 -> ta ) <-> ta ) )
24		pm3.24 891	. . . . . . . . . . . . . . . . 17 |- -. ( A < 0 /\ -. A < 0 )
25	24	pm2.21i 135	. . . . . . . . . . . . . . . 16 |- ( ( A < 0 /\ -. A < 0 ) -> th )
26	25	adantl 468	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( A < 0 /\ -. A < 0 ) ) -> th )
27	26	expr 619	. . . . . . . . . . . . . 14 |- ( ( ph /\ A < 0 ) -> ( -. A < 0 -> th ) )
28		tbtru 1448	. . . . . . . . . . . . . 14 |- ( ( -. A < 0 -> th ) <-> ( ( -. A < 0 -> th ) <-> T. ) )
29	27, 28	sylib 200	. . . . . . . . . . . . 13 |- ( ( ph /\ A < 0 ) -> ( ( -. A < 0 -> th ) <-> T. ) )
30	23, 29	anbi12d 716	. . . . . . . . . . . 12 |- ( ( ph /\ A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ( ta /\ T. ) ) )
31		ancom 452	. . . . . . . . . . . . 13 |- ( ( ta /\ T. ) <-> ( T. /\ ta ) )
32		truan 1455	. . . . . . . . . . . . 13 |- ( ( T. /\ ta ) <-> ta )
33	31, 32	bitri 253	. . . . . . . . . . . 12 |- ( ( ta /\ T. ) <-> ta )
34	30, 33	syl6bb 265	. . . . . . . . . . 11 |- ( ( ph /\ A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ta ) )
35	34	3adant3 1026	. . . . . . . . . 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 2435	. . . . . . . . . . 11 |- ( -u 1 = (sgn ` A ) -> ( ps <-> ta ) )
38	37	3ad2ant3 1029	. . . . . . . . . 10 |- ( ( ph /\ A < 0 /\ -u 1 = (sgn ` A ) ) -> ( ps <-> ta ) )
39	35, 38	bitr4d 260	. . . . . . . . 9 |- ( ( ph /\ A < 0 /\ -u 1 = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
40	39	3expia 1208	. . . . . . . 8 |- ( ( ph /\ A < 0 ) -> ( -u 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
41	19	3adant2 1025	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. A < 0 /\ A < 0 ) -> ta )
42	41	3expia 1208	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. A < 0 ) -> ( A < 0 -> ta ) )
43		tbtru 1448	. . . . . . . . . . . . . 14 |- ( ( A < 0 -> ta ) <-> ( ( A < 0 -> ta ) <-> T. ) )
44	42, 43	sylib 200	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A < 0 ) -> ( ( A < 0 -> ta ) <-> T. ) )
45		pm3.35 590	. . . . . . . . . . . . . . 15 |- ( ( -. A < 0 /\ ( -. A < 0 -> th ) ) -> th )
46	45	adantll 719	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. A < 0 ) /\ ( -. A < 0 -> th ) ) -> th )
47		simp2 1007	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ -. A < 0 ) /\ th /\ -. A < 0 ) -> th )
48	47	3expia 1208	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. A < 0 ) /\ th ) -> ( -. A < 0 -> th ) )
49	46, 48	impbida 841	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A < 0 ) -> ( ( -. A < 0 -> th ) <-> th ) )
50	44, 49	anbi12d 716	. . . . . . . . . . . 12 |- ( ( ph /\ -. A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ( T. /\ th ) ) )
51		truan 1455	. . . . . . . . . . . 12 |- ( ( T. /\ th ) <-> th )
52	50, 51	syl6bb 265	. . . . . . . . . . 11 |- ( ( ph /\ -. A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> th ) )
53	52	3adant3 1026	. . . . . . . . . 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 2435	. . . . . . . . . . 11 |- ( 1 = (sgn ` A ) -> ( ps <-> th ) )
56	55	3ad2ant3 1029	. . . . . . . . . 10 |- ( ( ph /\ -. A < 0 /\ 1 = (sgn ` A ) ) -> ( ps <-> th ) )
57	53, 56	bitr4d 260	. . . . . . . . 9 |- ( ( ph /\ -. A < 0 /\ 1 = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
58	57	3expia 1208	. . . . . . . 8 |- ( ( ph /\ -. A < 0 ) -> ( 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
59	16, 18, 40, 58	ifbothda 3945	. . . . . . 7 |- ( ph -> ( if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
60	59	imp 431	. . . . . 6 |- ( ( ph /\ if ( A < 0 , -u 1 , 1 ) = (sgn ` A ) ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) )
61	14, 60	sylbir 217	. . . . 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 437	. . . 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 251	. . 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 437	. . . 4 |- ( A = 0 -> ( ph -> ch ) )
66	65	adantl 468	. . 3 |- ( ( T. /\ A = 0 ) -> ( ph -> ch ) )
67	19	ex 436	. . . . . . 7 |- ( ph -> ( A < 0 -> ta ) )
68	67	adantr 467	. . . . . 6 |- ( ( ph /\ -. A = 0 ) -> ( A < 0 -> ta ) )
69		simp1 1006	. . . . . . . 8 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> ph )
70		df-ne 2621	. . . . . . . . . . . 12 |- ( A =/= 0 <-> -. A = 0 )
71		0xr 9689	. . . . . . . . . . . . 13 |- 0 e. RR*
72		xrlttri2 11443	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ 0 e. RR* ) -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
73	1, 71, 72	sylancl 667	. . . . . . . . . . . 12 |- ( ph -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
74	70, 73	syl5bbr 263	. . . . . . . . . . 11 |- ( ph -> ( -. A = 0 <-> ( A < 0 \/ 0 < A ) ) )
75	74	biimpa 487	. . . . . . . . . 10 |- ( ( ph /\ -. A = 0 ) -> ( A < 0 \/ 0 < A ) )
76	75	ord 379	. . . . . . . . 9 |- ( ( ph /\ -. A = 0 ) -> ( -. A < 0 -> 0 < A ) )
77	76	3impia 1203	. . . . . . . 8 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> 0 < A )
78		sgn3da.5	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> th )
79	69, 77, 78	syl2anc 666	. . . . . . 7 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> th )
80	79	3expia 1208	. . . . . 6 |- ( ( ph /\ -. A = 0 ) -> ( -. A < 0 -> th ) )
81	68, 80	jca 535	. . . . 5 |- ( ( ph /\ -. A = 0 ) -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) )
82	81	expcom 437	. . . 4 |- ( -. A = 0 -> ( ph -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) ) )
83	82	adantl 468	. . 3 |- ( ( T. /\ -. A = 0 ) -> ( ph -> ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) ) )
84	12, 63, 66, 83	ifbothda 3945	. 2 |- ( T. -> ( ph -> ps ) )
85	84	trud 1447	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   /\ w3a 983   = wceq 1438   T. wtru 1439   e. wcel 1869   =/= wne 2619   ifcif 3910   class class class wbr 4421   ` cfv 5599   0cc0 9541   1c1 9542   RR*cxr 9676   < clt 9677   -ucneg 9863  sgncsgn 13143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-i2m1 9609  ax-1ne0 9610  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-po 4772  df-so 4773  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-neg 9865  df-sgn 13144

This theorem is referenced by:  sgnmul  29415  sgnsub  29417  sgnnbi  29418  sgnpbi  29419  sgn0bi  29420  sgnsgn  29421