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Theorem sgn3da 29930
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 13676 . . . . . . . . 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 2620 . . . . . . 7 (𝜑 → (0 = (sgn‘𝐴) ↔ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
54pm5.32i 667 . . . . . 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 2618 . . . . . . . 8 (0 = (sgn‘𝐴) → (𝜓𝜒))
87bicomd 212 . . . . . . 7 (0 = (sgn‘𝐴) → (𝜒𝜓))
98adantl 481 . . . . . 6 ((𝜑 ∧ 0 = (sgn‘𝐴)) → (𝜒𝜓))
105, 9sylbir 224 . . . . 5 ((𝜑 ∧ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (𝜒𝜓))
1110expcom 450 . . . 4 (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → (𝜑 → (𝜒𝜓)))
1211pm5.74d 261 . . 3 (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → ((𝜑𝜒) ↔ (𝜑𝜓)))
133eqeq2d 2620 . . . . . . 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 667 . . . . . 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 2614 . . . . . . . . 9 (-1 = if(𝐴 < 0, -1, 1) → (-1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
1615imbi1d 330 . . . . . . . 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 2614 . . . . . . . . 9 (1 = if(𝐴 < 0, -1, 1) → (1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
1817imbi1d 330 . . . . . . . 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 480 . . . . . . . . . . . . . 14 (((𝜑𝐴 < 0) ∧ (𝐴 < 0 → 𝜏)) → 𝜏)
21 simp2 1055 . . . . . . . . . . . . . . 15 (((𝜑𝐴 < 0) ∧ 𝜏𝐴 < 0) → 𝜏)
22213expia 1259 . . . . . . . . . . . . . 14 (((𝜑𝐴 < 0) ∧ 𝜏) → (𝐴 < 0 → 𝜏))
2320, 22impbida 873 . . . . . . . . . . . . 13 ((𝜑𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ 𝜏))
24 pm3.24 922 . . . . . . . . . . . . . . . . 17 ¬ (𝐴 < 0 ∧ ¬ 𝐴 < 0)
2524pm2.21i 115 . . . . . . . . . . . . . . . 16 ((𝐴 < 0 ∧ ¬ 𝐴 < 0) → 𝜃)
2625adantl 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 < 0 ∧ ¬ 𝐴 < 0)) → 𝜃)
2726expr 641 . . . . . . . . . . . . . 14 ((𝜑𝐴 < 0) → (¬ 𝐴 < 0 → 𝜃))
28 tbtru 1485 . . . . . . . . . . . . . 14 ((¬ 𝐴 < 0 → 𝜃) ↔ ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
2927, 28sylib 207 . . . . . . . . . . . . 13 ((𝜑𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
3023, 29anbi12d 743 . . . . . . . . . . . 12 ((𝜑𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (𝜏 ∧ ⊤)))
31 ancom 465 . . . . . . . . . . . . 13 ((𝜏 ∧ ⊤) ↔ (⊤ ∧ 𝜏))
32 truan 1492 . . . . . . . . . . . . 13 ((⊤ ∧ 𝜏) ↔ 𝜏)
3331, 32bitri 263 . . . . . . . . . . . 12 ((𝜏 ∧ ⊤) ↔ 𝜏)
3430, 33syl6bb 275 . . . . . . . . . . 11 ((𝜑𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
35343adant3 1074 . . . . . . . . . 10 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
36 sgn3da.3 . . . . . . . . . . . 12 ((sgn‘𝐴) = -1 → (𝜓𝜏))
3736eqcoms 2618 . . . . . . . . . . 11 (-1 = (sgn‘𝐴) → (𝜓𝜏))
38373ad2ant3 1077 . . . . . . . . . 10 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (𝜓𝜏))
3935, 38bitr4d 270 . . . . . . . . 9 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
40393expia 1259 . . . . . . . 8 ((𝜑𝐴 < 0) → (-1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
41193adant2 1073 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 𝐴 < 0) → 𝜏)
42413expia 1259 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝐴 < 0) → (𝐴 < 0 → 𝜏))
43 tbtru 1485 . . . . . . . . . . . . . 14 ((𝐴 < 0 → 𝜏) ↔ ((𝐴 < 0 → 𝜏) ↔ ⊤))
4442, 43sylib 207 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ ⊤))
45 pm3.35 609 . . . . . . . . . . . . . . 15 ((¬ 𝐴 < 0 ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
4645adantll 746 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
47 simp2 1055 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃 ∧ ¬ 𝐴 < 0) → 𝜃)
48473expia 1259 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃) → (¬ 𝐴 < 0 → 𝜃))
4946, 48impbida 873 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ 𝜃))
5044, 49anbi12d 743 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (⊤ ∧ 𝜃)))
51 truan 1492 . . . . . . . . . . . 12 ((⊤ ∧ 𝜃) ↔ 𝜃)
5250, 51syl6bb 275 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
53523adant3 1074 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
54 sgn3da.2 . . . . . . . . . . . 12 ((sgn‘𝐴) = 1 → (𝜓𝜃))
5554eqcoms 2618 . . . . . . . . . . 11 (1 = (sgn‘𝐴) → (𝜓𝜃))
56553ad2ant3 1077 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (𝜓𝜃))
5753, 56bitr4d 270 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
58573expia 1259 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 < 0) → (1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
5916, 18, 40, 58ifbothda 4073 . . . . . . 7 (𝜑 → (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
6059imp 444 . . . . . 6 ((𝜑 ∧ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
6114, 60sylbir 224 . . . . 5 ((𝜑 ∧ if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
6261expcom 450 . . . 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 261 . . 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 450 . . . 4 (𝐴 = 0 → (𝜑𝜒))
6665adantl 481 . . 3 ((⊤ ∧ 𝐴 = 0) → (𝜑𝜒))
6719ex 449 . . . . . . 7 (𝜑 → (𝐴 < 0 → 𝜏))
6867adantr 480 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 → 𝜏))
69 simp1 1054 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜑)
70 df-ne 2782 . . . . . . . . . . . 12 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
71 0xr 9965 . . . . . . . . . . . . 13 0 ∈ ℝ*
72 xrlttri2 11851 . . . . . . . . . . . . 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 273 . . . . . . . . . . 11 (𝜑 → (¬ 𝐴 = 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
7574biimpa 500 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 ∨ 0 < 𝐴))
7675ord 391 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 0 < 𝐴))
77763impia 1253 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 0 < 𝐴)
78 sgn3da.5 . . . . . . . 8 ((𝜑 ∧ 0 < 𝐴) → 𝜃)
7969, 77, 78syl2anc 691 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜃)
80793expia 1259 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 𝜃))
8168, 80jca 553 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 0) → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)))
8281expcom 450 . . . 4 𝐴 = 0 → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
8382adantl 481 . . 3 ((⊤ ∧ ¬ 𝐴 = 0) → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
8412, 63, 66, 83ifbothda 4073 . 2 (⊤ → (𝜑𝜓))
8584trud 1484 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wtru 1476  wcel 1977  wne 2780  ifcif 4036   class class class wbr 4583  cfv 5804  0cc0 9815  1c1 9816  *cxr 9952   < clt 9953  -cneg 10146  sgncsgn 13674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-neg 10148  df-sgn 13675
This theorem is referenced by:  sgnmul  29931  sgnsub  29933  sgnnbi  29934  sgnpbi  29935  sgn0bi  29936  sgnsgn  29937
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