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Theorem signswrid 31836
Description: The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-2018.)
Hypotheses
Ref Expression
signsw.p = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
signsw.w 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}
Assertion
Ref Expression
signswrid (𝑋 ∈ {-1, 0, 1} → (𝑋 0) = 𝑋)
Distinct variable group:   𝑎,𝑏,𝑋
Allowed substitution hints:   (𝑎,𝑏)   𝑊(𝑎,𝑏)

Proof of Theorem signswrid
StepHypRef Expression
1 c0ex 10612 . . . 4 0 ∈ V
21tpid2 4679 . . 3 0 ∈ {-1, 0, 1}
3 signsw.p . . . 4 = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
43signspval 31830 . . 3 ((𝑋 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑋 0) = if(0 = 0, 𝑋, 0))
52, 4mpan2 690 . 2 (𝑋 ∈ {-1, 0, 1} → (𝑋 0) = if(0 = 0, 𝑋, 0))
6 eqid 2821 . . 3 0 = 0
7 iftrue 4446 . . 3 (0 = 0 → if(0 = 0, 𝑋, 0) = 𝑋)
86, 7mp1i 13 . 2 (𝑋 ∈ {-1, 0, 1} → if(0 = 0, 𝑋, 0) = 𝑋)
95, 8eqtrd 2856 1 (𝑋 ∈ {-1, 0, 1} → (𝑋 0) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  ifcif 4440  {cpr 4542  {ctp 4544  cop 4546  cfv 6328  (class class class)co 7130  cmpo 7132  0cc0 10514  1c1 10515  -cneg 10848  ndxcnx 16459  Basecbs 16462  +gcplusg 16544
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-mulcl 10576  ax-i2m1 10582
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135
This theorem is referenced by:  signstfveq0  31855
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