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Theorem signsw0glem 33859
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 11213 . . . . . 6 0 ∈ V
21tpid2 4775 . . . . 5 0 ∈ {-1, 0, 1}
3 signsw.p . . . . . 6 = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
43signspval 33858 . . . . 5 ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 𝑢) = if(𝑢 = 0, 0, 𝑢))
52, 4mpan 687 . . . 4 (𝑢 ∈ {-1, 0, 1} → (0 𝑢) = if(𝑢 = 0, 0, 𝑢))
6 iftrue 4535 . . . . . 6 (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0)
7 id 22 . . . . . 6 (𝑢 = 0 → 𝑢 = 0)
86, 7eqtr4d 2774 . . . . 5 (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
9 iffalse 4538 . . . . 5 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
108, 9pm2.61i 182 . . . 4 if(𝑢 = 0, 0, 𝑢) = 𝑢
115, 10eqtrdi 2787 . . 3 (𝑢 ∈ {-1, 0, 1} → (0 𝑢) = 𝑢)
123signspval 33858 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 0) = if(0 = 0, 𝑢, 0))
132, 12mpan2 688 . . . 4 (𝑢 ∈ {-1, 0, 1} → (𝑢 0) = if(0 = 0, 𝑢, 0))
14 eqid 2731 . . . . 5 0 = 0
1514iftruei 4536 . . . 4 if(0 = 0, 𝑢, 0) = 𝑢
1613, 15eqtrdi 2787 . . 3 (𝑢 ∈ {-1, 0, 1} → (𝑢 0) = 𝑢)
1711, 16jca 511 . 2 (𝑢 ∈ {-1, 0, 1} → ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢))
1817rgen 3062 1 𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2105  wral 3060  ifcif 4529  {ctp 4633  (class class class)co 7412  cmpo 7414  0cc0 11113  1c1 11114  -cneg 11450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-mulcl 11175  ax-i2m1 11181
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417
This theorem is referenced by:  signsw0g  33862  signswmnd  33863
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