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Theorem signswlid 34100
Description: The zero-skipping operation keeps nonzeros. (Contributed by Thierry Arnoux, 12-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
signswlid (((𝑋 ∈ {-1, 0, 1} ∧ π‘Œ ∈ {-1, 0, 1}) ∧ π‘Œ β‰  0) β†’ (𝑋 ⨣ π‘Œ) = π‘Œ)
Distinct variable groups:   π‘Ž,𝑏,𝑋   π‘Œ,π‘Ž,𝑏
Allowed substitution hints:   ⨣ (π‘Ž,𝑏)   π‘Š(π‘Ž,𝑏)

Proof of Theorem signswlid
StepHypRef Expression
1 signsw.p . . . 4 ⨣ = (π‘Ž ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, π‘Ž, 𝑏))
21signspval 34093 . . 3 ((𝑋 ∈ {-1, 0, 1} ∧ π‘Œ ∈ {-1, 0, 1}) β†’ (𝑋 ⨣ π‘Œ) = if(π‘Œ = 0, 𝑋, π‘Œ))
32adantr 480 . 2 (((𝑋 ∈ {-1, 0, 1} ∧ π‘Œ ∈ {-1, 0, 1}) ∧ π‘Œ β‰  0) β†’ (𝑋 ⨣ π‘Œ) = if(π‘Œ = 0, 𝑋, π‘Œ))
4 simpr 484 . . . 4 (((𝑋 ∈ {-1, 0, 1} ∧ π‘Œ ∈ {-1, 0, 1}) ∧ π‘Œ β‰  0) β†’ π‘Œ β‰  0)
54neneqd 2939 . . 3 (((𝑋 ∈ {-1, 0, 1} ∧ π‘Œ ∈ {-1, 0, 1}) ∧ π‘Œ β‰  0) β†’ Β¬ π‘Œ = 0)
65iffalsed 4534 . 2 (((𝑋 ∈ {-1, 0, 1} ∧ π‘Œ ∈ {-1, 0, 1}) ∧ π‘Œ β‰  0) β†’ if(π‘Œ = 0, 𝑋, π‘Œ) = π‘Œ)
73, 6eqtrd 2766 1 (((𝑋 ∈ {-1, 0, 1} ∧ π‘Œ ∈ {-1, 0, 1}) ∧ π‘Œ β‰  0) β†’ (𝑋 ⨣ π‘Œ) = π‘Œ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  ifcif 4523  {cpr 4625  {ctp 4627  βŸ¨cop 4629  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  0cc0 11112  1c1 11113  -cneg 11449  ndxcnx 17135  Basecbs 17153  +gcplusg 17206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410
This theorem is referenced by:  signsvtn0  34111
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