Theorem signsw0glem 35293

Description: Neutral element property of ⨣. (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
signsw0glem	⊢ ∀𝑢 ∈ { -1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)

Distinct variable group:   𝑎,𝑏,𝑢

Allowed substitution hints:   ⨣ (𝑢, 𝑎, 𝑏)

Proof of Theorem signsw0glem

Step	Hyp	Ref	Expression
1		c0ex 11339	. . . . . 6 ⊢ 0 ∈ V
2	1	tpid2 4733	. . . . 5 ⊢ 0 ∈ { -1, 0, 1}
3		signsw.p	. . . . . 6 ⊢ ⨣ = (𝑎 ∈ { -1, 0, 1}, 𝑏 ∈ { -1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
4	3	signspval 35292	. . . . 5 ⊢ ((0 ∈ { -1, 0, 1} ∧ 𝑢 ∈ { -1, 0, 1}) → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢))
5	2, 4	mpan 703	. . . 4 ⊢ (𝑢 ∈ { -1, 0, 1} → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢))
6		iftrue 4491	. . . . . 6 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0)
7		id 23	. . . . . 6 ⊢ (𝑢 = 0 → 𝑢 = 0)
8	6, 7	eqtr4d 2801	. . . . 5 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
9		iffalse 4494	. . . . 5 ⊢ (¬ 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
10	8, 9	pm2.61i 184	. . . 4 ⊢ if(𝑢 = 0, 0, 𝑢) = 𝑢
11	5, 10	eqtrdi 2814	. . 3 ⊢ (𝑢 ∈ { -1, 0, 1} → (0 ⨣ 𝑢) = 𝑢)
12	3	signspval 35292	. . . . 5 ⊢ ((𝑢 ∈ { -1, 0, 1} ∧ 0 ∈ { -1, 0, 1}) → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0))
13	2, 12	mpan2 704	. . . 4 ⊢ (𝑢 ∈ { -1, 0, 1} → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0))
14		eqid 2763	. . . . 5 ⊢ 0 = 0
15	14	iftruei 4492	. . . 4 ⊢ if(0 = 0, 𝑢, 0) = 𝑢
16	13, 15	eqtrdi 2814	. . 3 ⊢ (𝑢 ∈ { -1, 0, 1} → (𝑢 ⨣ 0) = 𝑢)
17	11, 16	jca 521	. 2 ⊢ (𝑢 ∈ { -1, 0, 1} → ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢))
18	17	rgen 3082	1 ⊢ ∀𝑢 ∈ { -1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ifcif 4485  {ctp 4592  (class class class)co 7451   ∈ cmpo 7453  0cc0 11239  1c1 11240   -cneg 11581

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-mulcl 11301  ax-i2m1 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456

This theorem is used by:  signsw0g  35296  signswmnd  35297