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Theorem sgn3da 30376
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 (𝜑𝐴 ∈ ℝ*)
sgn3da.1 ((sgn‘𝐴) = 0 → (𝜓𝜒))
sgn3da.2 ((sgn‘𝐴) = 1 → (𝜓𝜃))
sgn3da.3 ((sgn‘𝐴) = -1 → (𝜓𝜏))
sgn3da.4 ((𝜑𝐴 = 0) → 𝜒)
sgn3da.5 ((𝜑 ∧ 0 < 𝐴) → 𝜃)
sgn3da.6 ((𝜑𝐴 < 0) → 𝜏)
Assertion
Ref Expression
sgn3da (𝜑𝜓)

Proof of Theorem sgn3da
StepHypRef Expression
1 sgn3da.0 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ*)
2 sgnval 13757 . . . . . . . . 9 (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
31, 2syl 17 . . . . . . . 8 (𝜑 → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
43eqeq2d 2636 . . . . . . 7 (𝜑 → (0 = (sgn‘𝐴) ↔ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
54pm5.32i 668 . . . . . 6 ((𝜑 ∧ 0 = (sgn‘𝐴)) ↔ (𝜑 ∧ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
6 sgn3da.1 . . . . . . . . 9 ((sgn‘𝐴) = 0 → (𝜓𝜒))
76eqcoms 2634 . . . . . . . 8 (0 = (sgn‘𝐴) → (𝜓𝜒))
87bicomd 213 . . . . . . 7 (0 = (sgn‘𝐴) → (𝜒𝜓))
98adantl 482 . . . . . 6 ((𝜑 ∧ 0 = (sgn‘𝐴)) → (𝜒𝜓))
105, 9sylbir 225 . . . . 5 ((𝜑 ∧ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (𝜒𝜓))
1110expcom 451 . . . 4 (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → (𝜑 → (𝜒𝜓)))
1211pm5.74d 262 . . 3 (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → ((𝜑𝜒) ↔ (𝜑𝜓)))
133eqeq2d 2636 . . . . . . 7 (𝜑 → (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
1413pm5.32i 668 . . . . . 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 2630 . . . . . . . . 9 (-1 = if(𝐴 < 0, -1, 1) → (-1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
1615imbi1d 331 . . . . . . . 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 2630 . . . . . . . . 9 (1 = if(𝐴 < 0, -1, 1) → (1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
1817imbi1d 331 . . . . . . . 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 ((𝜑𝐴 < 0) → 𝜏)
2019adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝐴 < 0) ∧ (𝐴 < 0 → 𝜏)) → 𝜏)
21 simp2 1060 . . . . . . . . . . . . . . 15 (((𝜑𝐴 < 0) ∧ 𝜏𝐴 < 0) → 𝜏)
22213expia 1264 . . . . . . . . . . . . . 14 (((𝜑𝐴 < 0) ∧ 𝜏) → (𝐴 < 0 → 𝜏))
2320, 22impbida 876 . . . . . . . . . . . . 13 ((𝜑𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ 𝜏))
24 pm3.24 925 . . . . . . . . . . . . . . . . 17 ¬ (𝐴 < 0 ∧ ¬ 𝐴 < 0)
2524pm2.21i 116 . . . . . . . . . . . . . . . 16 ((𝐴 < 0 ∧ ¬ 𝐴 < 0) → 𝜃)
2625adantl 482 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 < 0 ∧ ¬ 𝐴 < 0)) → 𝜃)
2726expr 642 . . . . . . . . . . . . . 14 ((𝜑𝐴 < 0) → (¬ 𝐴 < 0 → 𝜃))
28 tbtru 1491 . . . . . . . . . . . . . 14 ((¬ 𝐴 < 0 → 𝜃) ↔ ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
2927, 28sylib 208 . . . . . . . . . . . . 13 ((𝜑𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
3023, 29anbi12d 746 . . . . . . . . . . . 12 ((𝜑𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (𝜏 ∧ ⊤)))
31 ancom 466 . . . . . . . . . . . . 13 ((𝜏 ∧ ⊤) ↔ (⊤ ∧ 𝜏))
32 truan 1498 . . . . . . . . . . . . 13 ((⊤ ∧ 𝜏) ↔ 𝜏)
3331, 32bitri 264 . . . . . . . . . . . 12 ((𝜏 ∧ ⊤) ↔ 𝜏)
3430, 33syl6bb 276 . . . . . . . . . . 11 ((𝜑𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
35343adant3 1079 . . . . . . . . . 10 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
36 sgn3da.3 . . . . . . . . . . . 12 ((sgn‘𝐴) = -1 → (𝜓𝜏))
3736eqcoms 2634 . . . . . . . . . . 11 (-1 = (sgn‘𝐴) → (𝜓𝜏))
38373ad2ant3 1082 . . . . . . . . . 10 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (𝜓𝜏))
3935, 38bitr4d 271 . . . . . . . . 9 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
40393expia 1264 . . . . . . . 8 ((𝜑𝐴 < 0) → (-1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
41193adant2 1078 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 𝐴 < 0) → 𝜏)
42413expia 1264 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝐴 < 0) → (𝐴 < 0 → 𝜏))
43 tbtru 1491 . . . . . . . . . . . . . 14 ((𝐴 < 0 → 𝜏) ↔ ((𝐴 < 0 → 𝜏) ↔ ⊤))
4442, 43sylib 208 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ ⊤))
45 pm3.35 610 . . . . . . . . . . . . . . 15 ((¬ 𝐴 < 0 ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
4645adantll 749 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
47 simp2 1060 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃 ∧ ¬ 𝐴 < 0) → 𝜃)
48473expia 1264 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃) → (¬ 𝐴 < 0 → 𝜃))
4946, 48impbida 876 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ 𝜃))
5044, 49anbi12d 746 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (⊤ ∧ 𝜃)))
51 truan 1498 . . . . . . . . . . . 12 ((⊤ ∧ 𝜃) ↔ 𝜃)
5250, 51syl6bb 276 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
53523adant3 1079 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
54 sgn3da.2 . . . . . . . . . . . 12 ((sgn‘𝐴) = 1 → (𝜓𝜃))
5554eqcoms 2634 . . . . . . . . . . 11 (1 = (sgn‘𝐴) → (𝜓𝜃))
56553ad2ant3 1082 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (𝜓𝜃))
5753, 56bitr4d 271 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
58573expia 1264 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 < 0) → (1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
5916, 18, 40, 58ifbothda 4100 . . . . . . 7 (𝜑 → (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
6059imp 445 . . . . . 6 ((𝜑 ∧ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
6114, 60sylbir 225 . . . . 5 ((𝜑 ∧ if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
6261expcom 451 . . . 4 (if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → (𝜑 → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
6362pm5.74d 262 . . 3 (if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → ((𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))) ↔ (𝜑𝜓)))
64 sgn3da.4 . . . . 5 ((𝜑𝐴 = 0) → 𝜒)
6564expcom 451 . . . 4 (𝐴 = 0 → (𝜑𝜒))
6665adantl 482 . . 3 ((⊤ ∧ 𝐴 = 0) → (𝜑𝜒))
6719ex 450 . . . . . . 7 (𝜑 → (𝐴 < 0 → 𝜏))
6867adantr 481 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 → 𝜏))
69 simp1 1059 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜑)
70 df-ne 2797 . . . . . . . . . . . 12 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
71 0xr 10031 . . . . . . . . . . . . 13 0 ∈ ℝ*
72 xrlttri2 11919 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
731, 71, 72sylancl 693 . . . . . . . . . . . 12 (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
7470, 73syl5bbr 274 . . . . . . . . . . 11 (𝜑 → (¬ 𝐴 = 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
7574biimpa 501 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 ∨ 0 < 𝐴))
7675ord 392 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 0 < 𝐴))
77763impia 1258 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 0 < 𝐴)
78 sgn3da.5 . . . . . . . 8 ((𝜑 ∧ 0 < 𝐴) → 𝜃)
7969, 77, 78syl2anc 692 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜃)
80793expia 1264 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 𝜃))
8168, 80jca 554 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 0) → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)))
8281expcom 451 . . . 4 𝐴 = 0 → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
8382adantl 482 . . 3 ((⊤ ∧ ¬ 𝐴 = 0) → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
8412, 63, 66, 83ifbothda 4100 . 2 (⊤ → (𝜑𝜓))
8584trud 1490 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wtru 1481  wcel 1992  wne 2796  ifcif 4063   class class class wbr 4618  cfv 5850  0cc0 9881  1c1 9882  *cxr 10018   < clt 10019  -cneg 10212  sgncsgn 13755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-i2m1 9949  ax-1ne0 9950  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-neg 10214  df-sgn 13756
This theorem is referenced by:  sgnmul  30377  sgnsub  30379  sgnnbi  30380  sgnpbi  30381  sgn0bi  30382  sgnsgn  30383
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