Theorem signspval 31932

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 4469	. 2 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → if(𝑌 = 0, 𝑋, 𝑌) ∈ {-1, 0, 1})
2		ifeq1 4429	. . 3 ⊢ (𝑎 = 𝑋 → if(𝑏 = 0, 𝑎, 𝑏) = if(𝑏 = 0, 𝑋, 𝑏))
3		eqeq1 2802	. . . 4 ⊢ (𝑏 = 𝑌 → (𝑏 = 0 ↔ 𝑌 = 0))
4		id 22	. . . 4 ⊢ (𝑏 = 𝑌 → 𝑏 = 𝑌)
5	3, 4	ifbieq2d 4450	. . 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 7288	. 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 1459	1 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 ⨣ 𝑌) = if(𝑌 = 0, 𝑋, 𝑌))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ifcif 4425  {ctp 4529  (class class class)co 7135   ∈ cmpo 7137  0cc0 10526  1c1 10527  -cneg 10860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140

This theorem is referenced by:  signsw0glem  31933  signswmnd  31937  signswrid  31938  signswlid  31939  signswn0  31940  signswch  31941