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Theorem signspval 31176
 Description: The value of the skipping 0 sign operation. (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
signspval ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 𝑌) = if(𝑌 = 0, 𝑋, 𝑌))
Distinct variable groups:   𝑎,𝑏,𝑋   𝑌,𝑎,𝑏
Allowed substitution hints:   (𝑎,𝑏)

Proof of Theorem signspval
StepHypRef Expression
1 ifcl 4350 . 2 ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → if(𝑌 = 0, 𝑋, 𝑌) ∈ {-1, 0, 1})
2 ifeq1 4310 . . 3 (𝑎 = 𝑋 → if(𝑏 = 0, 𝑎, 𝑏) = if(𝑏 = 0, 𝑋, 𝑏))
3 eqeq1 2829 . . . 4 (𝑏 = 𝑌 → (𝑏 = 0 ↔ 𝑌 = 0))
4 id 22 . . . 4 (𝑏 = 𝑌𝑏 = 𝑌)
53, 4ifbieq2d 4331 . . 3 (𝑏 = 𝑌 → if(𝑏 = 0, 𝑋, 𝑏) = if(𝑌 = 0, 𝑋, 𝑌))
6 signsw.p . . 3 = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
72, 5, 6ovmpt2g 7055 . 2 ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1} ∧ if(𝑌 = 0, 𝑋, 𝑌) ∈ {-1, 0, 1}) → (𝑋 𝑌) = if(𝑌 = 0, 𝑋, 𝑌))
81, 7mpd3an3 1592 1 ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 𝑌) = if(𝑌 = 0, 𝑋, 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ifcif 4306  {ctp 4401  (class class class)co 6905   ↦ cmpt2 6907  0cc0 10252  1c1 10253  -cneg 10586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910 This theorem is referenced by:  signsw0glem  31177  signswmnd  31181  signswrid  31182  signswlid  31183  signswn0  31184  signswch  31185
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