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Theorem signsw0glem 34522
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 11280 . . . . . 6 0 ∈ V
21tpid2 4795 . . . . 5 0 ∈ {-1, 0, 1}
3 signsw.p . . . . . 6 = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
43signspval 34521 . . . . 5 ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 𝑢) = if(𝑢 = 0, 0, 𝑢))
52, 4mpan 689 . . . 4 (𝑢 ∈ {-1, 0, 1} → (0 𝑢) = if(𝑢 = 0, 0, 𝑢))
6 iftrue 4554 . . . . . 6 (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0)
7 id 22 . . . . . 6 (𝑢 = 0 → 𝑢 = 0)
86, 7eqtr4d 2777 . . . . 5 (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
9 iffalse 4557 . . . . 5 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
108, 9pm2.61i 182 . . . 4 if(𝑢 = 0, 0, 𝑢) = 𝑢
115, 10eqtrdi 2790 . . 3 (𝑢 ∈ {-1, 0, 1} → (0 𝑢) = 𝑢)
123signspval 34521 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 0) = if(0 = 0, 𝑢, 0))
132, 12mpan2 690 . . . 4 (𝑢 ∈ {-1, 0, 1} → (𝑢 0) = if(0 = 0, 𝑢, 0))
14 eqid 2734 . . . . 5 0 = 0
1514iftruei 4555 . . . 4 if(0 = 0, 𝑢, 0) = 𝑢
1613, 15eqtrdi 2790 . . 3 (𝑢 ∈ {-1, 0, 1} → (𝑢 0) = 𝑢)
1711, 16jca 511 . 2 (𝑢 ∈ {-1, 0, 1} → ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢))
1817rgen 3065 1 𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2103  wral 3063  ifcif 4548  {ctp 4652  (class class class)co 7445  cmpo 7447  0cc0 11180  1c1 11181  -cneg 11517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-mulcl 11242  ax-i2m1 11248
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450
This theorem is referenced by:  signsw0g  34525  signswmnd  34526
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