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Theorem signsw0glem 33633
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 11210 . . . . . 6 0 ∈ V
21tpid2 4774 . . . . 5 0 ∈ {-1, 0, 1}
3 signsw.p . . . . . 6 = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
43signspval 33632 . . . . 5 ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 𝑢) = if(𝑢 = 0, 0, 𝑢))
52, 4mpan 688 . . . 4 (𝑢 ∈ {-1, 0, 1} → (0 𝑢) = if(𝑢 = 0, 0, 𝑢))
6 iftrue 4534 . . . . . 6 (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0)
7 id 22 . . . . . 6 (𝑢 = 0 → 𝑢 = 0)
86, 7eqtr4d 2775 . . . . 5 (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
9 iffalse 4537 . . . . 5 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
108, 9pm2.61i 182 . . . 4 if(𝑢 = 0, 0, 𝑢) = 𝑢
115, 10eqtrdi 2788 . . 3 (𝑢 ∈ {-1, 0, 1} → (0 𝑢) = 𝑢)
123signspval 33632 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 0) = if(0 = 0, 𝑢, 0))
132, 12mpan2 689 . . . 4 (𝑢 ∈ {-1, 0, 1} → (𝑢 0) = if(0 = 0, 𝑢, 0))
14 eqid 2732 . . . . 5 0 = 0
1514iftruei 4535 . . . 4 if(0 = 0, 𝑢, 0) = 𝑢
1613, 15eqtrdi 2788 . . 3 (𝑢 ∈ {-1, 0, 1} → (𝑢 0) = 𝑢)
1711, 16jca 512 . 2 (𝑢 ∈ {-1, 0, 1} → ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢))
1817rgen 3063 1 𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  wral 3061  ifcif 4528  {ctp 4632  (class class class)co 7411  cmpo 7413  0cc0 11112  1c1 11113  -cneg 11447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-mulcl 11174  ax-i2m1 11180
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  signsw0g  33636  signswmnd  33637
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