Theorem sgn3da 31909

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 14439	. . . . . . . . 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 2809	. . . . . . 7 ⊢ (𝜑 → (0 = (sgn‘𝐴) ↔ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
5	4	pm5.32i 578	. . . . . 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 2806	. . . . . . . 8 ⊢ (0 = (sgn‘𝐴) → (𝜓 ↔ 𝜒))
8	7	bicomd 226	. . . . . . 7 ⊢ (0 = (sgn‘𝐴) → (𝜒 ↔ 𝜓))
9	8	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 0 = (sgn‘𝐴)) → (𝜒 ↔ 𝜓))
10	5, 9	sylbir 238	. . . . 5 ⊢ ((𝜑 ∧ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (𝜒 ↔ 𝜓))
11	10	expcom 417	. . . 4 ⊢ (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → (𝜑 → (𝜒 ↔ 𝜓)))
12	11	pm5.74d 276	. . 3 ⊢ (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → ((𝜑 → 𝜒) ↔ (𝜑 → 𝜓)))
13	3	eqeq2d 2809	. . . . . . 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 578	. . . . . 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 2802	. . . . . . . . 9 ⊢ (-1 = if(𝐴 < 0, -1, 1) → (-1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
16	15	imbi1d 345	. . . . . . . 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 2802	. . . . . . . . 9 ⊢ (1 = if(𝐴 < 0, -1, 1) → (1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
18	17	imbi1d 345	. . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ (𝐴 < 0 → 𝜏)) → 𝜏)
21		simp2 1134	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ 𝜏 ∧ 𝐴 < 0) → 𝜏)
22	21	3expia 1118	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 < 0) ∧ 𝜏) → (𝐴 < 0 → 𝜏))
23	20, 22	impbida 800	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ 𝜏))
24		pm3.24 406	. . . . . . . . . . . . . . . . 17 ⊢ ¬ (𝐴 < 0 ∧ ¬ 𝐴 < 0)
25	24	pm2.21i 119	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 < 0 ∧ ¬ 𝐴 < 0) → 𝜃)
26	25	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐴 < 0 ∧ ¬ 𝐴 < 0)) → 𝜃)
27	26	expr 460	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 < 0) → (¬ 𝐴 < 0 → 𝜃))
28		tbtru 1546	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝐴 < 0 → 𝜃) ↔ ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
29	27, 28	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
30	23, 29	anbi12d 633	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (𝜏 ∧ ⊤)))
31		ancom 464	. . . . . . . . . . . . 13 ⊢ ((𝜏 ∧ ⊤) ↔ (⊤ ∧ 𝜏))
32		truan 1549	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝜏) ↔ 𝜏)
33	31, 32	bitri 278	. . . . . . . . . . . 12 ⊢ ((𝜏 ∧ ⊤) ↔ 𝜏)
34	30, 33	syl6bb 290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
35	34	3adant3 1129	. . . . . . . . . 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 2806	. . . . . . . . . . 11 ⊢ (-1 = (sgn‘𝐴) → (𝜓 ↔ 𝜏))
38	37	3ad2ant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (𝜓 ↔ 𝜏))
39	35, 38	bitr4d 285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
40	39	3expia 1118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 0) → (-1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
41	19	3adant2 1128	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 𝐴 < 0) → 𝜏)
42	41	3expia 1118	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → (𝐴 < 0 → 𝜏))
43		tbtru 1546	. . . . . . . . . . . . . 14 ⊢ ((𝐴 < 0 → 𝜏) ↔ ((𝐴 < 0 → 𝜏) ↔ ⊤))
44	42, 43	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ ⊤))
45		pm3.35 802	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝐴 < 0 ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
46	45	adantll 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐴 < 0) ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
47		simp2 1134	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃 ∧ ¬ 𝐴 < 0) → 𝜃)
48	47	3expia 1118	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃) → (¬ 𝐴 < 0 → 𝜃))
49	46, 48	impbida 800	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ 𝜃))
50	44, 49	anbi12d 633	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (⊤ ∧ 𝜃)))
51		truan 1549	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝜃) ↔ 𝜃)
52	50, 51	syl6bb 290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
53	52	3adant3 1129	. . . . . . . . . 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 2806	. . . . . . . . . . 11 ⊢ (1 = (sgn‘𝐴) → (𝜓 ↔ 𝜃))
56	55	3ad2ant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (𝜓 ↔ 𝜃))
57	53, 56	bitr4d 285	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
58	57	3expia 1118	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 < 0) → (1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
59	16, 18, 40, 58	ifbothda 4462	. . . . . . 7 ⊢ (𝜑 → (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
60	59	imp 410	. . . . . 6 ⊢ ((𝜑 ∧ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
61	14, 60	sylbir 238	. . . . 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 417	. . . 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 276	. . 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 417	. . . 4 ⊢ (𝐴 = 0 → (𝜑 → 𝜒))
66	65	adantl 485	. . 3 ⊢ ((⊤ ∧ 𝐴 = 0) → (𝜑 → 𝜒))
67	19	ex 416	. . . . . . 7 ⊢ (𝜑 → (𝐴 < 0 → 𝜏))
68	67	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 → 𝜏))
69		simp1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜑)
70		df-ne 2988	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
71		0xr 10677	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ*
72		xrlttri2 12523	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
73	1, 71, 72	sylancl 589	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
74	70, 73	bitr3id 288	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝐴 = 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
75	74	biimpa 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 ∨ 0 < 𝐴))
76	75	ord 861	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 0 < 𝐴))
77	76	3impia 1114	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 0 < 𝐴)
78		sgn3da.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝜃)
79	69, 77, 78	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜃)
80	79	3expia 1118	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 𝜃))
81	68, 80	jca 515	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = 0) → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)))
82	81	expcom 417	. . . 4 ⊢ (¬ 𝐴 = 0 → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
83	82	adantl 485	. . 3 ⊢ ((⊤ ∧ ¬ 𝐴 = 0) → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
84	12, 63, 66, 83	ifbothda 4462	. 2 ⊢ (⊤ → (𝜑 → 𝜓))
85	84	mptru 1545	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111   ≠ wne 2987  ifcif 4425   class class class wbr 5030  ‘cfv 6324  0cc0 10526  1c1 10527  ℝ^*cxr 10663   < clt 10664  -cneg 10860  sgncsgn 14437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-i2m1 10594  ax-rnegex 10597  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-neg 10862  df-sgn 14438

This theorem is referenced by:  sgnmul  31910  sgnsub  31912  sgnnbi  31913  sgnpbi  31914  sgn0bi  31915  sgnsgn  31916