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Theorem signsw0glem 29956
 Description: Neutral element property of ⨣. (Contributed by Thierry Arnoux, 9-Sep-2018.)
Hypothesis
Ref Expression
signsw.p = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
Assertion
Ref Expression
signsw0glem 𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)
Distinct variable group:   𝑎,𝑏,𝑢
Allowed substitution hints:   (𝑢,𝑎,𝑏)

Proof of Theorem signsw0glem
StepHypRef Expression
1 c0ex 9913 . . . . . 6 0 ∈ V
21tpid2 4247 . . . . 5 0 ∈ {-1, 0, 1}
3 signsw.p . . . . . 6 = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
43signspval 29955 . . . . 5 ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 𝑢) = if(𝑢 = 0, 0, 𝑢))
52, 4mpan 702 . . . 4 (𝑢 ∈ {-1, 0, 1} → (0 𝑢) = if(𝑢 = 0, 0, 𝑢))
6 iftrue 4042 . . . . . 6 (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0)
7 id 22 . . . . . 6 (𝑢 = 0 → 𝑢 = 0)
86, 7eqtr4d 2647 . . . . 5 (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
9 iffalse 4045 . . . . 5 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
108, 9pm2.61i 175 . . . 4 if(𝑢 = 0, 0, 𝑢) = 𝑢
115, 10syl6eq 2660 . . 3 (𝑢 ∈ {-1, 0, 1} → (0 𝑢) = 𝑢)
123signspval 29955 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 0) = if(0 = 0, 𝑢, 0))
132, 12mpan2 703 . . . 4 (𝑢 ∈ {-1, 0, 1} → (𝑢 0) = if(0 = 0, 𝑢, 0))
14 eqid 2610 . . . . 5 0 = 0
1514iftruei 4043 . . . 4 if(0 = 0, 𝑢, 0) = 𝑢
1613, 15syl6eq 2660 . . 3 (𝑢 ∈ {-1, 0, 1} → (𝑢 0) = 𝑢)
1711, 16jca 553 . 2 (𝑢 ∈ {-1, 0, 1} → ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢))
1817rgen 2906 1 𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ifcif 4036  {ctp 4129  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816  -cneg 10146 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883 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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  signsw0g  29959  signswmnd  29960
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