Theorem sgn3da 31799

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	⊢ (𝜑 → 𝐴 ∈ ℝ*)
sgn3da.1	⊢ ((sgn‘𝐴) = 0 → (𝜓 ↔ 𝜒))
sgn3da.2	⊢ ((sgn‘𝐴) = 1 → (𝜓 ↔ 𝜃))
sgn3da.3	⊢ ((sgn‘𝐴) = -1 → (𝜓 ↔ 𝜏))
sgn3da.4	⊢ ((𝜑 ∧ 𝐴 = 0) → 𝜒)
sgn3da.5	⊢ ((𝜑 ∧ 0 < 𝐴) → 𝜃)
sgn3da.6	⊢ ((𝜑 ∧ 𝐴 < 0) → 𝜏)

Assertion

Ref	Expression
sgn3da	⊢ (𝜑 → 𝜓)

Proof of Theorem sgn3da

Step	Hyp	Ref	Expression
1		sgn3da.0	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
2		sgnval 14446	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
4	3	eqeq2d 2832	. . . . . . 7 ⊢ (𝜑 → (0 = (sgn‘𝐴) ↔ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
5	4	pm5.32i 577	. . . . . 6 ⊢ ((𝜑 ∧ 0 = (sgn‘𝐴)) ↔ (𝜑 ∧ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
6		sgn3da.1	. . . . . . . . 9 ⊢ ((sgn‘𝐴) = 0 → (𝜓 ↔ 𝜒))
7	6	eqcoms 2829	. . . . . . . 8 ⊢ (0 = (sgn‘𝐴) → (𝜓 ↔ 𝜒))
8	7	bicomd 225	. . . . . . 7 ⊢ (0 = (sgn‘𝐴) → (𝜒 ↔ 𝜓))
9	8	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 0 = (sgn‘𝐴)) → (𝜒 ↔ 𝜓))
10	5, 9	sylbir 237	. . . . 5 ⊢ ((𝜑 ∧ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (𝜒 ↔ 𝜓))
11	10	expcom 416	. . . 4 ⊢ (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → (𝜑 → (𝜒 ↔ 𝜓)))
12	11	pm5.74d 275	. . 3 ⊢ (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → ((𝜑 → 𝜒) ↔ (𝜑 → 𝜓)))
13	3	eqeq2d 2832	. . . . . . 7 ⊢ (𝜑 → (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
14	13	pm5.32i 577	. . . . . 6 ⊢ ((𝜑 ∧ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)) ↔ (𝜑 ∧ if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
15		eqeq1 2825	. . . . . . . . 9 ⊢ (-1 = if(𝐴 < 0, -1, 1) → (-1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
16	15	imbi1d 344	. . . . . . . 8 ⊢ (-1 = if(𝐴 < 0, -1, 1) → ((-1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)) ↔ (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))))
17		eqeq1 2825	. . . . . . . . 9 ⊢ (1 = if(𝐴 < 0, -1, 1) → (1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
18	17	imbi1d 344	. . . . . . . 8 ⊢ (1 = if(𝐴 < 0, -1, 1) → ((1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)) ↔ (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))))
19		sgn3da.6	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝜏)
20	19	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝐴 < 0 → 𝜏)) → 𝜏)
21		simp2 1133	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ 𝜏 ∧ 𝐴 < 0) → 𝜏)
22	21	3expia 1117	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ 𝜏) → (𝐴 < 0 → 𝜏))
23	20, 22	impbida 799	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ 𝜏))
24		pm3.24 405	. . . . . . . . . . . . . . . . 17 ⊢ ¬ (𝐴 < 0 ∧ ¬ 𝐴 < 0)
25	24	pm2.21i 119	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 < 0 ∧ ¬ 𝐴 < 0) → 𝜃)
26	25	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐴 < 0 ∧ ¬ 𝐴 < 0)) → 𝜃)
27	26	expr 459	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 < 0) → (¬ 𝐴 < 0 → 𝜃))
28		tbtru 1541	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝐴 < 0 → 𝜃) ↔ ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
29	27, 28	sylib 220	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
30	23, 29	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (𝜏 ∧ ⊤)))
31		ancom 463	. . . . . . . . . . . . 13 ⊢ ((𝜏 ∧ ⊤) ↔ (⊤ ∧ 𝜏))
32		truan 1544	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝜏) ↔ 𝜏)
33	31, 32	bitri 277	. . . . . . . . . . . 12 ⊢ ((𝜏 ∧ ⊤) ↔ 𝜏)
34	30, 33	syl6bb 289	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
35	34	3adant3 1128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
36		sgn3da.3	. . . . . . . . . . . 12 ⊢ ((sgn‘𝐴) = -1 → (𝜓 ↔ 𝜏))
37	36	eqcoms 2829	. . . . . . . . . . 11 ⊢ (-1 = (sgn‘𝐴) → (𝜓 ↔ 𝜏))
38	37	3ad2ant3 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (𝜓 ↔ 𝜏))
39	35, 38	bitr4d 284	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
40	39	3expia 1117	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 0) → (-1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
41	19	3adant2 1127	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 𝐴 < 0) → 𝜏)
42	41	3expia 1117	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → (𝐴 < 0 → 𝜏))
43		tbtru 1541	. . . . . . . . . . . . . 14 ⊢ ((𝐴 < 0 → 𝜏) ↔ ((𝐴 < 0 → 𝜏) ↔ ⊤))
44	42, 43	sylib 220	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ ⊤))
45		pm3.35 801	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝐴 < 0 ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
46	45	adantll 712	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐴 < 0) ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
47		simp2 1133	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃 ∧ ¬ 𝐴 < 0) → 𝜃)
48	47	3expia 1117	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃) → (¬ 𝐴 < 0 → 𝜃))
49	46, 48	impbida 799	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ 𝜃))
50	44, 49	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (⊤ ∧ 𝜃)))
51		truan 1544	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝜃) ↔ 𝜃)
52	50, 51	syl6bb 289	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
53	52	3adant3 1128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
54		sgn3da.2	. . . . . . . . . . . 12 ⊢ ((sgn‘𝐴) = 1 → (𝜓 ↔ 𝜃))
55	54	eqcoms 2829	. . . . . . . . . . 11 ⊢ (1 = (sgn‘𝐴) → (𝜓 ↔ 𝜃))
56	55	3ad2ant3 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (𝜓 ↔ 𝜃))
57	53, 56	bitr4d 284	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
58	57	3expia 1117	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → (1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
59	16, 18, 40, 58	ifbothda 4503	. . . . . . 7 ⊢ (𝜑 → (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
60	59	imp 409	. . . . . 6 ⊢ ((𝜑 ∧ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
61	14, 60	sylbir 237	. . . . 5 ⊢ ((𝜑 ∧ if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
62	61	expcom 416	. . . 4 ⊢ (if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → (𝜑 → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
63	62	pm5.74d 275	. . 3 ⊢ (if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → ((𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))) ↔ (𝜑 → 𝜓)))
64		sgn3da.4	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 0) → 𝜒)
65	64	expcom 416	. . . 4 ⊢ (𝐴 = 0 → (𝜑 → 𝜒))
66	65	adantl 484	. . 3 ⊢ ((⊤ ∧ 𝐴 = 0) → (𝜑 → 𝜒))
67	19	ex 415	. . . . . . 7 ⊢ (𝜑 → (𝐴 < 0 → 𝜏))
68	67	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 → 𝜏))
69		simp1 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜑)
70		df-ne 3017	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
71		0xr 10687	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ*
72		xrlttri2 12534	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
73	1, 71, 72	sylancl 588	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
74	70, 73	syl5bbr 287	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝐴 = 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
75	74	biimpa 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 ∨ 0 < 𝐴))
76	75	ord 860	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 0 < 𝐴))
77	76	3impia 1113	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 0 < 𝐴)
78		sgn3da.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝜃)
79	69, 77, 78	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜃)
80	79	3expia 1117	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 𝜃))
81	68, 80	jca 514	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)))
82	81	expcom 416	. . . 4 ⊢ (¬ 𝐴 = 0 → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
83	82	adantl 484	. . 3 ⊢ ((⊤ ∧ ¬ 𝐴 = 0) → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
84	12, 63, 66, 83	ifbothda 4503	. 2 ⊢ (⊤ → (𝜑 → 𝜓))
85	84	mptru 1540	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1533  ⊤wtru 1534   ∈ wcel 2110   ≠ wne 3016  ifcif 4466   class class class wbr 5065  ‘cfv 6354  0cc0 10536  1c1 10537  ℝ^*cxr 10673   < clt 10674  -cneg 10870  sgncsgn 14444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-i2m1 10604  ax-rnegex 10607  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-neg 10872  df-sgn 14445

This theorem is referenced by:  sgnmul  31800  sgnsub  31802  sgnnbi  31803  sgnpbi  31804  sgn0bi  31805  sgnsgn  31806