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Theorem signswrid 32837
Description: The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-2018.)
Hypotheses
Ref Expression
signsw.p ⨣ = (π‘Ž ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, π‘Ž, 𝑏))
signsw.w π‘Š = {⟨(Baseβ€˜ndx), {-1, 0, 1}⟩, ⟨(+gβ€˜ndx), ⨣ ⟩}
Assertion
Ref Expression
signswrid (𝑋 ∈ {-1, 0, 1} β†’ (𝑋 ⨣ 0) = 𝑋)
Distinct variable group:   π‘Ž,𝑏,𝑋
Allowed substitution hints:   ⨣ (π‘Ž,𝑏)   π‘Š(π‘Ž,𝑏)

Proof of Theorem signswrid
StepHypRef Expression
1 c0ex 11070 . . . 4 0 ∈ V
21tpid2 4718 . . 3 0 ∈ {-1, 0, 1}
3 signsw.p . . . 4 ⨣ = (π‘Ž ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, π‘Ž, 𝑏))
43signspval 32831 . . 3 ((𝑋 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) β†’ (𝑋 ⨣ 0) = if(0 = 0, 𝑋, 0))
52, 4mpan2 688 . 2 (𝑋 ∈ {-1, 0, 1} β†’ (𝑋 ⨣ 0) = if(0 = 0, 𝑋, 0))
6 eqid 2736 . . 3 0 = 0
7 iftrue 4479 . . 3 (0 = 0 β†’ if(0 = 0, 𝑋, 0) = 𝑋)
86, 7mp1i 13 . 2 (𝑋 ∈ {-1, 0, 1} β†’ if(0 = 0, 𝑋, 0) = 𝑋)
95, 8eqtrd 2776 1 (𝑋 ∈ {-1, 0, 1} β†’ (𝑋 ⨣ 0) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1540   ∈ wcel 2105  ifcif 4473  {cpr 4575  {ctp 4577  βŸ¨cop 4579  β€˜cfv 6479  (class class class)co 7337   ∈ cmpo 7339  0cc0 10972  1c1 10973  -cneg 11307  ndxcnx 16991  Basecbs 17009  +gcplusg 17059
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 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-mulcl 11034  ax-i2m1 11040
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342
This theorem is referenced by:  signstfveq0  32856
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