Theorem signsw0glem 34566

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 11106	. . . . . 6 ⊢ 0 ∈ V
2	1	tpid2 4720	. . . . 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 34565	. . . . 5 ⊢ ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢))
5	2, 4	mpan 690	. . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢))
6		iftrue 4478	. . . . . 6 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0)
7		id 22	. . . . . 6 ⊢ (𝑢 = 0 → 𝑢 = 0)
8	6, 7	eqtr4d 2769	. . . . 5 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
9		iffalse 4481	. . . . 5 ⊢ (¬ 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
10	8, 9	pm2.61i 182	. . . 4 ⊢ if(𝑢 = 0, 0, 𝑢) = 𝑢
11	5, 10	eqtrdi 2782	. . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = 𝑢)
12	3	signspval 34565	. . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0))
13	2, 12	mpan2 691	. . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0))
14		eqid 2731	. . . . 5 ⊢ 0 = 0
15	14	iftruei 4479	. . . 4 ⊢ if(0 = 0, 𝑢, 0) = 𝑢
16	13, 15	eqtrdi 2782	. . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = 𝑢)
17	11, 16	jca 511	. 2 ⊢ (𝑢 ∈ {-1, 0, 1} → ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢))
18	17	rgen 3049	1 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ifcif 4472  {ctp 4577  (class class class)co 7346   ∈ cmpo 7348  0cc0 11006  1c1 11007  -cneg 11345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-mulcl 11068  ax-i2m1 11074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351

This theorem is referenced by:  signsw0g  34569  signswmnd  34570