Theorem signsw0glem 35232

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 11307	. . . . . 6 ⊢ 0 ∈ V
2	1	tpid2 4727	. . . . 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 35231	. . . . 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 4487	. . . . . 6 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0)
7		id 23	. . . . . 6 ⊢ (𝑢 = 0 → 𝑢 = 0)
8	6, 7	eqtr4d 2798	. . . . 5 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢)
9		iffalse 4490	. . . . 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 2811	. . 3 ⊢ (𝑢 ∈ { -1, 0, 1} → (0 ⨣ 𝑢) = 𝑢)
12	3	signspval 35231	. . . . 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 2760	. . . . 5 ⊢ 0 = 0
15	14	iftruei 4488	. . . 4 ⊢ if(0 = 0, 𝑢, 0) = 𝑢
16	13, 15	eqtrdi 2811	. . 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 3078	1 ⊢ ∀𝑢 ∈ { -1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ifcif 4481  {ctp 4588  (class class class)co 7424   ∈ cmpo 7426  0cc0 11207  1c1 11208   -cneg 11549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5391  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-mulcl 11269  ax-i2m1 11275

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429

This theorem is used by:  signsw0g  35235  signswmnd  35236