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Theorem signswrid 33170
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 11149 . . . 4 0 ∈ V
21tpid2 4731 . . 3 0 ∈ {-1, 0, 1}
3 signsw.p . . . 4 = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
43signspval 33164 . . 3 ((𝑋 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑋 0) = if(0 = 0, 𝑋, 0))
52, 4mpan2 689 . 2 (𝑋 ∈ {-1, 0, 1} → (𝑋 0) = if(0 = 0, 𝑋, 0))
6 eqid 2736 . . 3 0 = 0
7 iftrue 4492 . . 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 2776 1 (𝑋 ∈ {-1, 0, 1} → (𝑋 0) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  ifcif 4486  {cpr 4588  {ctp 4590  cop 4592  cfv 6496  (class class class)co 7357  cmpo 7359  0cc0 11051  1c1 11052  -cneg 11386  ndxcnx 17065  Basecbs 17083  +gcplusg 17133
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-mulcl 11113  ax-i2m1 11119
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362
This theorem is referenced by:  signstfveq0  33189
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