Theorem sgn3da 29412

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 13151	. . . . . . . . 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 2461	. . . . . . 7 |- ( ph -> ( 0 = (sgn ` A ) <-> 0 = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) )
5	4	pm5.32i 643	. . . . . 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 2459	. . . . . . . 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 2461	. . . . . . 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 643	. . . . . 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 2455	. . . . . . . . 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 2455	. . . . . . . . 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 1009	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A < 0 ) /\ ta /\ A < 0 ) -> ta )
22	21	3expia 1210	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A < 0 ) /\ ta ) -> ( A < 0 -> ta ) )
23	20, 22	impbida 843	. . . . . . . . . . . . 13 |- ( ( ph /\ A < 0 ) -> ( ( A < 0 -> ta ) <-> ta ) )
24		pm3.24 893	. . . . . . . . . . . . . . . . 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 620	. . . . . . . . . . . . . 14 |- ( ( ph /\ A < 0 ) -> ( -. A < 0 -> th ) )
28		tbtru 1454	. . . . . . . . . . . . . 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 717	. . . . . . . . . . . 12 |- ( ( ph /\ A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ( ta /\ T. ) ) )
31		ancom 452	. . . . . . . . . . . . 13 |- ( ( ta /\ T. ) <-> ( T. /\ ta ) )
32		truan 1461	. . . . . . . . . . . . 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 1028	. . . . . . . . . 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 2459	. . . . . . . . . . 11 |- ( -u 1 = (sgn ` A ) -> ( ps <-> ta ) )
38	37	3ad2ant3 1031	. . . . . . . . . 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 1210	. . . . . . . 8 |- ( ( ph /\ A < 0 ) -> ( -u 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
41	19	3adant2 1027	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. A < 0 /\ A < 0 ) -> ta )
42	41	3expia 1210	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. A < 0 ) -> ( A < 0 -> ta ) )
43		tbtru 1454	. . . . . . . . . . . . . 14 |- ( ( A < 0 -> ta ) <-> ( ( A < 0 -> ta ) <-> T. ) )
44	42, 43	sylib 200	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A < 0 ) -> ( ( A < 0 -> ta ) <-> T. ) )
45		pm3.35 591	. . . . . . . . . . . . . . 15 |- ( ( -. A < 0 /\ ( -. A < 0 -> th ) ) -> th )
46	45	adantll 720	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. A < 0 ) /\ ( -. A < 0 -> th ) ) -> th )
47		simp2 1009	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ -. A < 0 ) /\ th /\ -. A < 0 ) -> th )
48	47	3expia 1210	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. A < 0 ) /\ th ) -> ( -. A < 0 -> th ) )
49	46, 48	impbida 843	. . . . . . . . . . . . 13 |- ( ( ph /\ -. A < 0 ) -> ( ( -. A < 0 -> th ) <-> th ) )
50	44, 49	anbi12d 717	. . . . . . . . . . . 12 |- ( ( ph /\ -. A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ( T. /\ th ) ) )
51		truan 1461	. . . . . . . . . . . 12 |- ( ( T. /\ th ) <-> th )
52	50, 51	syl6bb 265	. . . . . . . . . . 11 |- ( ( ph /\ -. A < 0 ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> th ) )
53	52	3adant3 1028	. . . . . . . . . 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 2459	. . . . . . . . . . 11 |- ( 1 = (sgn ` A ) -> ( ps <-> th ) )
56	55	3ad2ant3 1031	. . . . . . . . . 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 1210	. . . . . . . 8 |- ( ( ph /\ -. A < 0 ) -> ( 1 = (sgn ` A ) -> ( ( ( A < 0 -> ta ) /\ ( -. A < 0 -> th ) ) <-> ps ) ) )
59	16, 18, 40, 58	ifbothda 3916	. . . . . . 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 1008	. . . . . . . 8 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> ph )
70		df-ne 2624	. . . . . . . . . . . 12 |- ( A =/= 0 <-> -. A = 0 )
71		0xr 9687	. . . . . . . . . . . . 13 |- 0 e. RR*
72		xrlttri2 11441	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ 0 e. RR* ) -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
73	1, 71, 72	sylancl 668	. . . . . . . . . . . 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 1205	. . . . . . . 8 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> 0 < A )
78		sgn3da.5	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> th )
79	69, 77, 78	syl2anc 667	. . . . . . 7 |- ( ( ph /\ -. A = 0 /\ -. A < 0 ) -> th )
80	79	3expia 1210	. . . . . 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 3916	. 2 |- ( T. -> ( ph -> ps ) )
85	84	trud 1453	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   /\ w3a 985   = wceq 1444   T. wtru 1445   e. wcel 1887   =/= wne 2622   ifcif 3881   class class class wbr 4402   ` cfv 5582   0cc0 9539   1c1 9540   RR*cxr 9674   < clt 9675   -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-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-i2m1 9607  ax-1ne0 9608  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614

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-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-ov 6293  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-neg 9863  df-sgn 13150

This theorem is referenced by:  sgnmul  29413  sgnsub  29415  sgnnbi  29416  sgnpbi  29417  sgn0bi  29418  sgnsgn  29419