Theorem signspval 35228

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

Step	Hyp	Ref	Expression
1		ifcl 4527	. 2 ⊢ ((𝑋 ∈ { -1, 0, 1} ∧ 𝑌 ∈ { -1, 0, 1}) → if(𝑌 = 0, 𝑋, 𝑌) ∈ { -1, 0, 1})
2		ifeq1 4485	. . 3 ⊢ (𝑎 = 𝑋 → if(𝑏 = 0, 𝑎, 𝑏) = if(𝑏 = 0, 𝑋, 𝑏))
3		eqeq1 2764	. . . 4 ⊢ (𝑏 = 𝑌 → (𝑏 = 0 ↔ 𝑌 = 0))
4		id 23	. . . 4 ⊢ (𝑏 = 𝑌 → 𝑏 = 𝑌)
5	3, 4	ifbieq2d 4508	. . 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))
7	2, 5, 6	ovmpog 7580	. 2 ⊢ ((𝑋 ∈ { -1, 0, 1} ∧ 𝑌 ∈ { -1, 0, 1} ∧ if(𝑌 = 0, 𝑋, 𝑌) ∈ { -1, 0, 1}) → (𝑋 ⨣ 𝑌) = if(𝑌 = 0, 𝑋, 𝑌))
8	1, 7	mpd3an3 1491	1 ⊢ ((𝑋 ∈ { -1, 0, 1} ∧ 𝑌 ∈ { -1, 0, 1}) → (𝑋 ⨣ 𝑌) = if(𝑌 = 0, 𝑋, 𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ifcif 4481  {ctp 4588  (class class class)co 7421   ∈ cmpo 7423  0cc0 11204  1c1 11205   -cneg 11546

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426

This theorem is used by:  signsw0glem  35229  signswmnd  35233  signswrid  35234  signswlid  35235  signswn0  35236  signswch  35237