Theorem signsw0glem 34019

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 11204	. . . . . 6 ⊢ 0 ∈ V
2	1	tpid2 4766	. . . . 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 34018	. . . . 5 ⊢ ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢))
5	2, 4	mpan 687	. . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢))
6		iftrue 4526	. . . . . 6 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0)
7		id 22	. . . . . 6 ⊢ (𝑢 = 0 → 𝑢 = 0)
8	6, 7	eqtr4d 2767	. . . . 5 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
9		iffalse 4529	. . . . 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 2780	. . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = 𝑢)
12	3	signspval 34018	. . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0))
13	2, 12	mpan2 688	. . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0))
14		eqid 2724	. . . . 5 ⊢ 0 = 0
15	14	iftruei 4527	. . . 4 ⊢ if(0 = 0, 𝑢, 0) = 𝑢
16	13, 15	eqtrdi 2780	. . 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 3055	1 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)

Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053  ifcif 4520  {ctp 4624  (class class class)co 7401   ∈ cmpo 7403  0cc0 11105  1c1 11106  -cneg 11441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-mulcl 11167  ax-i2m1 11173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406

This theorem is referenced by:  signsw0g  34022  signswmnd  34023