Theorem signsw0glem 34532

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 11286	. . . . . 6 ⊢ 0 ∈ V
2	1	tpid2 4795	. . . . 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 34531	. . . . 5 ⊢ ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢))
5	2, 4	mpan 689	. . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢))
6		iftrue 4554	. . . . . 6 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0)
7		id 22	. . . . . 6 ⊢ (𝑢 = 0 → 𝑢 = 0)
8	6, 7	eqtr4d 2783	. . . . 5 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
9		iffalse 4557	. . . . 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 2796	. . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = 𝑢)
12	3	signspval 34531	. . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0))
13	2, 12	mpan2 690	. . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0))
14		eqid 2740	. . . . 5 ⊢ 0 = 0
15	14	iftruei 4555	. . . 4 ⊢ if(0 = 0, 𝑢, 0) = 𝑢
16	13, 15	eqtrdi 2796	. . 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 3069	1 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ifcif 4548  {ctp 4652  (class class class)co 7450   ∈ cmpo 7452  0cc0 11186  1c1 11187  -cneg 11523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-1cn 11244  ax-icn 11245  ax-addcl 11246  ax-mulcl 11248  ax-i2m1 11254

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455

This theorem is referenced by:  signsw0g  34535  signswmnd  34536