Theorem signspval 30629

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 4130	. 2 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → if(𝑌 = 0, 𝑋, 𝑌) ∈ {-1, 0, 1})
2		ifeq1 4090	. . 3 ⊢ (𝑎 = 𝑋 → if(𝑏 = 0, 𝑎, 𝑏) = if(𝑏 = 0, 𝑋, 𝑏))
3		eqeq1 2626	. . . 4 ⊢ (𝑏 = 𝑌 → (𝑏 = 0 ↔ 𝑌 = 0))
4		id 22	. . . 4 ⊢ (𝑏 = 𝑌 → 𝑏 = 𝑌)
5	3, 4	ifbieq2d 4111	. . 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	ovmpt2g 6795	. 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 1425	1 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 ⨣ 𝑌) = if(𝑌 = 0, 𝑋, 𝑌))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ifcif 4086  {ctp 4181  (class class class)co 6650   ↦ cmpt2 6652  0cc0 9936  1c1 9937  -cneg 10267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  signsw0glem  30630  signswmnd  30634  signswrid  30635  signswlid  30636  signswn0  30637  signswch  30638