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Theorem signspval 34093
Description: The value of the skipping 0 sign operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)
Hypothesis
Ref Expression
signsw.p âĻĢ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} â†Ķ if(𝑏 = 0, 𝑎, 𝑏))
Assertion
Ref Expression
signspval ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 âĻĢ ð‘Œ) = if(𝑌 = 0, 𝑋, 𝑌))
Distinct variable groups:   𝑎,𝑏,𝑋   𝑌,𝑎,𝑏
Allowed substitution hints:   âĻĢ (𝑎,𝑏)

Proof of Theorem signspval
StepHypRef Expression
1 ifcl 4568 . 2 ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → if(𝑌 = 0, 𝑋, 𝑌) ∈ {-1, 0, 1})
2 ifeq1 4527 . . 3 (𝑎 = 𝑋 → if(𝑏 = 0, 𝑎, 𝑏) = if(𝑏 = 0, 𝑋, 𝑏))
3 eqeq1 2730 . . . 4 (𝑏 = 𝑌 → (𝑏 = 0 ↔ 𝑌 = 0))
4 id 22 . . . 4 (𝑏 = 𝑌 → 𝑏 = 𝑌)
53, 4ifbieq2d 4549 . . 3 (𝑏 = 𝑌 → if(𝑏 = 0, 𝑋, 𝑏) = if(𝑌 = 0, 𝑋, 𝑌))
6 signsw.p . . 3 âĻĢ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} â†Ķ if(𝑏 = 0, 𝑎, 𝑏))
72, 5, 6ovmpog 7563 . 2 ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1} ∧ if(𝑌 = 0, 𝑋, 𝑌) ∈ {-1, 0, 1}) → (𝑋 âĻĢ ð‘Œ) = if(𝑌 = 0, 𝑋, 𝑌))
81, 7mpd3an3 1458 1 ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 âĻĢ ð‘Œ) = if(𝑌 = 0, 𝑋, 𝑌))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ifcif 4523  {ctp 4627  (class class class)co 7405   ∈ cmpo 7407  0cc0 11112  1c1 11113  -cneg 11449
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-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:  signsw0glem  34094  signswmnd  34098  signswrid  34099  signswlid  34100  signswn0  34101  signswch  34102
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