Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sgn3da Structured version   Visualization version   GIF version

Theorem sgn3da 32244
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 14675 . . . . . . . . 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 2749 . . . . . . 7 (𝜑 → (0 = (sgn‘𝐴) ↔ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
54pm5.32i 578 . . . . . 6 ((𝜑 ∧ 0 = (sgn‘𝐴)) ↔ (𝜑 ∧ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
6 sgn3da.1 . . . . . . . . 9 ((sgn‘𝐴) = 0 → (𝜓𝜒))
76eqcoms 2746 . . . . . . . 8 (0 = (sgn‘𝐴) → (𝜓𝜒))
87bicomd 226 . . . . . . 7 (0 = (sgn‘𝐴) → (𝜒𝜓))
98adantl 485 . . . . . 6 ((𝜑 ∧ 0 = (sgn‘𝐴)) → (𝜒𝜓))
105, 9sylbir 238 . . . . 5 ((𝜑 ∧ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (𝜒𝜓))
1110expcom 417 . . . 4 (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → (𝜑 → (𝜒𝜓)))
1211pm5.74d 276 . . 3 (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → ((𝜑𝜒) ↔ (𝜑𝜓)))
133eqeq2d 2749 . . . . . . 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 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 2742 . . . . . . . . 9 (-1 = if(𝐴 < 0, -1, 1) → (-1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
1615imbi1d 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 2742 . . . . . . . . 9 (1 = if(𝐴 < 0, -1, 1) → (1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
1817imbi1d 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 ((𝜑𝐴 < 0) → 𝜏)
2019adantr 484 . . . . . . . . . . . . . 14 (((𝜑𝐴 < 0) ∧ (𝐴 < 0 → 𝜏)) → 𝜏)
21 simp2 1139 . . . . . . . . . . . . . . 15 (((𝜑𝐴 < 0) ∧ 𝜏𝐴 < 0) → 𝜏)
22213expia 1123 . . . . . . . . . . . . . 14 (((𝜑𝐴 < 0) ∧ 𝜏) → (𝐴 < 0 → 𝜏))
2320, 22impbida 801 . . . . . . . . . . . . 13 ((𝜑𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ 𝜏))
24 pm3.24 406 . . . . . . . . . . . . . . . . 17 ¬ (𝐴 < 0 ∧ ¬ 𝐴 < 0)
2524pm2.21i 119 . . . . . . . . . . . . . . . 16 ((𝐴 < 0 ∧ ¬ 𝐴 < 0) → 𝜃)
2625adantl 485 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 < 0 ∧ ¬ 𝐴 < 0)) → 𝜃)
2726expr 460 . . . . . . . . . . . . . 14 ((𝜑𝐴 < 0) → (¬ 𝐴 < 0 → 𝜃))
28 tbtru 1551 . . . . . . . . . . . . . 14 ((¬ 𝐴 < 0 → 𝜃) ↔ ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
2927, 28sylib 221 . . . . . . . . . . . . 13 ((𝜑𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
3023, 29anbi12d 634 . . . . . . . . . . . 12 ((𝜑𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (𝜏 ∧ ⊤)))
31 ancom 464 . . . . . . . . . . . . 13 ((𝜏 ∧ ⊤) ↔ (⊤ ∧ 𝜏))
32 truan 1554 . . . . . . . . . . . . 13 ((⊤ ∧ 𝜏) ↔ 𝜏)
3331, 32bitri 278 . . . . . . . . . . . 12 ((𝜏 ∧ ⊤) ↔ 𝜏)
3430, 33bitrdi 290 . . . . . . . . . . 11 ((𝜑𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
35343adant3 1134 . . . . . . . . . 10 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
36 sgn3da.3 . . . . . . . . . . . 12 ((sgn‘𝐴) = -1 → (𝜓𝜏))
3736eqcoms 2746 . . . . . . . . . . 11 (-1 = (sgn‘𝐴) → (𝜓𝜏))
38373ad2ant3 1137 . . . . . . . . . 10 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (𝜓𝜏))
3935, 38bitr4d 285 . . . . . . . . 9 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
40393expia 1123 . . . . . . . 8 ((𝜑𝐴 < 0) → (-1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
41193adant2 1133 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 𝐴 < 0) → 𝜏)
42413expia 1123 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝐴 < 0) → (𝐴 < 0 → 𝜏))
43 tbtru 1551 . . . . . . . . . . . . . 14 ((𝐴 < 0 → 𝜏) ↔ ((𝐴 < 0 → 𝜏) ↔ ⊤))
4442, 43sylib 221 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ ⊤))
45 pm3.35 803 . . . . . . . . . . . . . . 15 ((¬ 𝐴 < 0 ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
4645adantll 714 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ (¬ 𝐴 < 0 → 𝜃)) → 𝜃)
47 simp2 1139 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃 ∧ ¬ 𝐴 < 0) → 𝜃)
48473expia 1123 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃) → (¬ 𝐴 < 0 → 𝜃))
4946, 48impbida 801 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ 𝜃))
5044, 49anbi12d 634 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (⊤ ∧ 𝜃)))
51 truan 1554 . . . . . . . . . . . 12 ((⊤ ∧ 𝜃) ↔ 𝜃)
5250, 51bitrdi 290 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
53523adant3 1134 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
54 sgn3da.2 . . . . . . . . . . . 12 ((sgn‘𝐴) = 1 → (𝜓𝜃))
5554eqcoms 2746 . . . . . . . . . . 11 (1 = (sgn‘𝐴) → (𝜓𝜃))
56553ad2ant3 1137 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (𝜓𝜃))
5753, 56bitr4d 285 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
58573expia 1123 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 < 0) → (1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
5916, 18, 40, 58ifbothda 4491 . . . . . . 7 (𝜑 → (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
6059imp 410 . . . . . 6 ((𝜑 ∧ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
6114, 60sylbir 238 . . . . 5 ((𝜑 ∧ if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
6261expcom 417 . . . 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 276 . . 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 417 . . . 4 (𝐴 = 0 → (𝜑𝜒))
6665adantl 485 . . 3 ((⊤ ∧ 𝐴 = 0) → (𝜑𝜒))
6719ex 416 . . . . . . 7 (𝜑 → (𝐴 < 0 → 𝜏))
6867adantr 484 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 → 𝜏))
69 simp1 1138 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜑)
70 df-ne 2942 . . . . . . . . . . . 12 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
71 0xr 10904 . . . . . . . . . . . . 13 0 ∈ ℝ*
72 xrlttri2 12756 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
731, 71, 72sylancl 589 . . . . . . . . . . . 12 (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
7470, 73bitr3id 288 . . . . . . . . . . 11 (𝜑 → (¬ 𝐴 = 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
7574biimpa 480 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 ∨ 0 < 𝐴))
7675ord 864 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 0 < 𝐴))
77763impia 1119 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 0 < 𝐴)
78 sgn3da.5 . . . . . . . 8 ((𝜑 ∧ 0 < 𝐴) → 𝜃)
7969, 77, 78syl2anc 587 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜃)
80793expia 1123 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 𝜃))
8168, 80jca 515 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 0) → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)))
8281expcom 417 . . . 4 𝐴 = 0 → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
8382adantl 485 . . 3 ((⊤ ∧ ¬ 𝐴 = 0) → (𝜑 → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃))))
8412, 63, 66, 83ifbothda 4491 . 2 (⊤ → (𝜑𝜓))
8584mptru 1550 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847  w3a 1089   = wceq 1543  wtru 1544  wcel 2111  wne 2941  ifcif 4453   class class class wbr 5067  cfv 6397  0cc0 10753  1c1 10754  *cxr 10890   < clt 10891  -cneg 11087  sgncsgn 14673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pow 5272  ax-pr 5336  ax-un 7541  ax-cnex 10809  ax-resscn 10810  ax-1cn 10811  ax-icn 10812  ax-addcl 10813  ax-addrcl 10814  ax-mulcl 10815  ax-i2m1 10821  ax-rnegex 10824  ax-cnre 10826  ax-pre-lttri 10827  ax-pre-lttrn 10828
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-sbc 3709  df-csb 3826  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-pw 4529  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-mpt 5150  df-id 5469  df-po 5482  df-so 5483  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fo 6403  df-f1o 6404  df-fv 6405  df-ov 7234  df-er 8411  df-en 8647  df-dom 8648  df-sdom 8649  df-pnf 10893  df-mnf 10894  df-xr 10895  df-ltxr 10896  df-neg 11089  df-sgn 14674
This theorem is referenced by:  sgnmul  32245  sgnsub  32247  sgnnbi  32248  sgnpbi  32249  sgn0bi  32250  sgnsgn  32251
  Copyright terms: Public domain W3C validator