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Theorem sgn3da 34544
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 15127 . . . . . . . . 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 2748 . . . . . . 7 (𝜑 → (0 = (sgn‘𝐴) ↔ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))))
54pm5.32i 574 . . . . . 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 2745 . . . . . . . 8 (0 = (sgn‘𝐴) → (𝜓𝜒))
87bicomd 223 . . . . . . 7 (0 = (sgn‘𝐴) → (𝜒𝜓))
98adantl 481 . . . . . 6 ((𝜑 ∧ 0 = (sgn‘𝐴)) → (𝜒𝜓))
105, 9sylbir 235 . . . . 5 ((𝜑 ∧ 0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (𝜒𝜓))
1110expcom 413 . . . 4 (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → (𝜑 → (𝜒𝜓)))
1211pm5.74d 273 . . 3 (0 = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) → ((𝜑𝜒) ↔ (𝜑𝜓)))
133eqeq2d 2748 . . . . . . 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 574 . . . . . 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 2741 . . . . . . . . 9 (-1 = if(𝐴 < 0, -1, 1) → (-1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
1615imbi1d 341 . . . . . . . 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 2741 . . . . . . . . 9 (1 = if(𝐴 < 0, -1, 1) → (1 = (sgn‘𝐴) ↔ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)))
1817imbi1d 341 . . . . . . . 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 1138 . . . . . . . . . . . . . . 15 (((𝜑𝐴 < 0) ∧ 𝜏𝐴 < 0) → 𝜏)
22213expia 1122 . . . . . . . . . . . . . 14 (((𝜑𝐴 < 0) ∧ 𝜏) → (𝐴 < 0 → 𝜏))
2320, 22impbida 801 . . . . . . . . . . . . 13 ((𝜑𝐴 < 0) → ((𝐴 < 0 → 𝜏) ↔ 𝜏))
24 pm3.24 402 . . . . . . . . . . . . . . . . 17 ¬ (𝐴 < 0 ∧ ¬ 𝐴 < 0)
2524pm2.21i 119 . . . . . . . . . . . . . . . 16 ((𝐴 < 0 ∧ ¬ 𝐴 < 0) → 𝜃)
2625adantl 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 < 0 ∧ ¬ 𝐴 < 0)) → 𝜃)
2726expr 456 . . . . . . . . . . . . . 14 ((𝜑𝐴 < 0) → (¬ 𝐴 < 0 → 𝜃))
28 tbtru 1548 . . . . . . . . . . . . . 14 ((¬ 𝐴 < 0 → 𝜃) ↔ ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
2927, 28sylib 218 . . . . . . . . . . . . 13 ((𝜑𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ ⊤))
3023, 29anbi12d 632 . . . . . . . . . . . 12 ((𝜑𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (𝜏 ∧ ⊤)))
31 ancom 460 . . . . . . . . . . . . 13 ((𝜏 ∧ ⊤) ↔ (⊤ ∧ 𝜏))
32 truan 1551 . . . . . . . . . . . . 13 ((⊤ ∧ 𝜏) ↔ 𝜏)
3331, 32bitri 275 . . . . . . . . . . . 12 ((𝜏 ∧ ⊤) ↔ 𝜏)
3430, 33bitrdi 287 . . . . . . . . . . 11 ((𝜑𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
35343adant3 1133 . . . . . . . . . 10 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜏))
36 sgn3da.3 . . . . . . . . . . . 12 ((sgn‘𝐴) = -1 → (𝜓𝜏))
3736eqcoms 2745 . . . . . . . . . . 11 (-1 = (sgn‘𝐴) → (𝜓𝜏))
38373ad2ant3 1136 . . . . . . . . . 10 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (𝜓𝜏))
3935, 38bitr4d 282 . . . . . . . . 9 ((𝜑𝐴 < 0 ∧ -1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
40393expia 1122 . . . . . . . 8 ((𝜑𝐴 < 0) → (-1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
41193adant2 1132 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 𝐴 < 0) → 𝜏)
42413expia 1122 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝐴 < 0) → (𝐴 < 0 → 𝜏))
43 tbtru 1548 . . . . . . . . . . . . . 14 ((𝐴 < 0 → 𝜏) ↔ ((𝐴 < 0 → 𝜏) ↔ ⊤))
4442, 43sylib 218 . . . . . . . . . . . . 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 1138 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃 ∧ ¬ 𝐴 < 0) → 𝜃)
48473expia 1122 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝐴 < 0) ∧ 𝜃) → (¬ 𝐴 < 0 → 𝜃))
4946, 48impbida 801 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝐴 < 0) → ((¬ 𝐴 < 0 → 𝜃) ↔ 𝜃))
5044, 49anbi12d 632 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ (⊤ ∧ 𝜃)))
51 truan 1551 . . . . . . . . . . . 12 ((⊤ ∧ 𝜃) ↔ 𝜃)
5250, 51bitrdi 287 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐴 < 0) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
53523adant3 1133 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜃))
54 sgn3da.2 . . . . . . . . . . . 12 ((sgn‘𝐴) = 1 → (𝜓𝜃))
5554eqcoms 2745 . . . . . . . . . . 11 (1 = (sgn‘𝐴) → (𝜓𝜃))
56553ad2ant3 1136 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (𝜓𝜃))
5753, 56bitr4d 282 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐴 < 0 ∧ 1 = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
58573expia 1122 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 < 0) → (1 = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
5916, 18, 40, 58ifbothda 4564 . . . . . . 7 (𝜑 → (if(𝐴 < 0, -1, 1) = (sgn‘𝐴) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓)))
6059imp 406 . . . . . 6 ((𝜑 ∧ if(𝐴 < 0, -1, 1) = (sgn‘𝐴)) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
6114, 60sylbir 235 . . . . 5 ((𝜑 ∧ if(𝐴 < 0, -1, 1) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) → (((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)) ↔ 𝜓))
6261expcom 413 . . . 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 273 . . 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 413 . . . 4 (𝐴 = 0 → (𝜑𝜒))
6665adantl 481 . . 3 ((⊤ ∧ 𝐴 = 0) → (𝜑𝜒))
6719ex 412 . . . . . . 7 (𝜑 → (𝐴 < 0 → 𝜏))
6867adantr 480 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 → 𝜏))
69 simp1 1137 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜑)
70 df-ne 2941 . . . . . . . . . . . 12 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
71 0xr 11308 . . . . . . . . . . . . 13 0 ∈ ℝ*
72 xrlttri2 13184 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
731, 71, 72sylancl 586 . . . . . . . . . . . 12 (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
7470, 73bitr3id 285 . . . . . . . . . . 11 (𝜑 → (¬ 𝐴 = 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴)))
7574biimpa 476 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐴 = 0) → (𝐴 < 0 ∨ 0 < 𝐴))
7675ord 865 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 0 < 𝐴))
77763impia 1118 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 0 < 𝐴)
78 sgn3da.5 . . . . . . . 8 ((𝜑 ∧ 0 < 𝐴) → 𝜃)
7969, 77, 78syl2anc 584 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴 = 0 ∧ ¬ 𝐴 < 0) → 𝜃)
80793expia 1122 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 0) → (¬ 𝐴 < 0 → 𝜃))
8168, 80jca 511 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 0) → ((𝐴 < 0 → 𝜏) ∧ (¬ 𝐴 < 0 → 𝜃)))
8281expcom 413 . . . 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 4564 . 2 (⊤ → (𝜑𝜓))
8584mptru 1547 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wtru 1541  wcel 2108  wne 2940  ifcif 4525   class class class wbr 5143  cfv 6561  0cc0 11155  1c1 11156  *cxr 11294   < clt 11295  -cneg 11493  sgncsgn 15125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-i2m1 11223  ax-rnegex 11226  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-neg 11495  df-sgn 15126
This theorem is referenced by:  sgnmul  34545  sgnsub  34547  sgnnbi  34548  sgnpbi  34549  sgn0bi  34550  sgnsgn  34551
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