Theorem signswmnd 34525

Description: 𝑊 is a monoid structure on {-1, 0, 1} which operation retains the right side, but skips zeroes. This will be used for skipping zeroes when counting sign changes. (Contributed by Thierry Arnoux, 9-Sep-2018.)

Hypotheses

Ref	Expression
signsw.p	⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
signsw.w	⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⨣ ⟩}

Assertion

Ref	Expression
signswmnd	⊢ 𝑊 ∈ Mnd

Distinct variable group:   𝑎,𝑏, ⨣

Allowed substitution hints:   𝑊(𝑎,𝑏)

Proof of Theorem signswmnd

Dummy variables 𝑢 𝑒 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		signsw.p	. . . . . 6 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
2	1	signspval 34520	. . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
3		ifcl 4522	. . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → if(𝑣 = 0, 𝑢, 𝑣) ∈ {-1, 0, 1})
4	2, 3	eqeltrd 2828	. . . 4 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 ⨣ 𝑣) ∈ {-1, 0, 1})
5	1	signspval 34520	. . . . . . . . . . . . 13 ⊢ (((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤))
6	4, 5	stoic3 1776	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤))
7		iftrue 4482	. . . . . . . . . . . 12 ⊢ (𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = (𝑢 ⨣ 𝑣))
8	6, 7	sylan9eq 2784	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ 𝑣))
9	8	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ 𝑣))
10	2	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
11	10	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
12		iftrue 4482	. . . . . . . . . . 11 ⊢ (𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢)
13	12	adantl 481	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢)
14	9, 11, 13	3eqtrd 2768	. . . . . . . . 9 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑢)
15		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → 𝑢 ∈ {-1, 0, 1})
16	1	signspval 34520	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 ⨣ 𝑤) = if(𝑤 = 0, 𝑣, 𝑤))
17	16	3adant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 ⨣ 𝑤) = if(𝑤 = 0, 𝑣, 𝑤))
18		simpl2 1193	. . . . . . . . . . . . . 14 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → 𝑣 ∈ {-1, 0, 1})
19		simpl3 1194	. . . . . . . . . . . . . 14 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ {-1, 0, 1})
20	18, 19	ifclda 4512	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → if(𝑤 = 0, 𝑣, 𝑤) ∈ {-1, 0, 1})
21	17, 20	eqeltrd 2828	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 ⨣ 𝑤) ∈ {-1, 0, 1})
22	1	signspval 34520	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ (𝑣 ⨣ 𝑤) ∈ {-1, 0, 1}) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)))
23	15, 21, 22	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)))
24	23	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)))
25		iftrue 4482	. . . . . . . . . . . . 13 ⊢ (𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣)
26	17, 25	sylan9eq 2784	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → (𝑣 ⨣ 𝑤) = 𝑣)
27		id 22	. . . . . . . . . . . 12 ⊢ (𝑣 = 0 → 𝑣 = 0)
28	26, 27	sylan9eq 2784	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 0)
29	28	iftrued 4484	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)) = 𝑢)
30	24, 29	eqtrd 2764	. . . . . . . . 9 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = 𝑢)
31	14, 30	eqtr4d 2767	. . . . . . . 8 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
32	6	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤))
33	7	ad2antlr 727	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = (𝑢 ⨣ 𝑣))
34	10	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
35		iffalse 4485	. . . . . . . . . . . 12 ⊢ (¬ 𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣)
36	35	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣)
37	34, 36	eqtrd 2764	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 ⨣ 𝑣) = 𝑣)
38	32, 33, 37	3eqtrd 2768	. . . . . . . . 9 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑣)
39	23	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)))
40		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ 𝑣 = 0)
41	17	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = if(𝑤 = 0, 𝑣, 𝑤))
42	25	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣)
43	41, 42	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 𝑣)
44	43	eqeq1d 2731	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑣 ⨣ 𝑤) = 0 ↔ 𝑣 = 0))
45	40, 44	mtbird 325	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ (𝑣 ⨣ 𝑤) = 0)
46	45	iffalsed 4487	. . . . . . . . . 10 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)) = (𝑣 ⨣ 𝑤))
47	39, 46, 43	3eqtrd 2768	. . . . . . . . 9 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = 𝑣)
48	38, 47	eqtr4d 2767	. . . . . . . 8 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
49	31, 48	pm2.61dan 812	. . . . . . 7 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
50		iffalse 4485	. . . . . . . . 9 ⊢ (¬ 𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = 𝑤)
51	6, 50	sylan9eq 2784	. . . . . . . 8 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑤)
52	23	adantr 480	. . . . . . . . 9 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)))
53		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ 𝑤 = 0)
54		iffalse 4485	. . . . . . . . . . . . 13 ⊢ (¬ 𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑤)
55	17, 54	sylan9eq 2784	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑣 ⨣ 𝑤) = 𝑤)
56	55	eqeq1d 2731	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑣 ⨣ 𝑤) = 0 ↔ 𝑤 = 0))
57	53, 56	mtbird 325	. . . . . . . . . 10 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ (𝑣 ⨣ 𝑤) = 0)
58	57	iffalsed 4487	. . . . . . . . 9 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)) = (𝑣 ⨣ 𝑤))
59	52, 58, 55	3eqtrd 2768	. . . . . . . 8 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = 𝑤)
60	51, 59	eqtr4d 2767	. . . . . . 7 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
61	49, 60	pm2.61dan 812	. . . . . 6 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
62	61	3expa 1118	. . . . 5 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
63	62	ralrimiva 3121	. . . 4 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → ∀𝑤 ∈ {-1, 0, 1} ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
64	4, 63	jca 511	. . 3 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → ((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧ ∀𝑤 ∈ {-1, 0, 1} ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))))
65	64	rgen2 3169	. 2 ⊢ ∀𝑢 ∈ {-1, 0, 1}∀𝑣 ∈ {-1, 0, 1} ((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧ ∀𝑤 ∈ {-1, 0, 1} ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
66		c0ex 11109	. . . 4 ⊢ 0 ∈ V
67	66	tpid2 4722	. . 3 ⊢ 0 ∈ {-1, 0, 1}
68	1	signsw0glem 34521	. . 3 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)
69		oveq1 7356	. . . . . 6 ⊢ (𝑒 = 0 → (𝑒 ⨣ 𝑢) = (0 ⨣ 𝑢))
70	69	eqeq1d 2731	. . . . 5 ⊢ (𝑒 = 0 → ((𝑒 ⨣ 𝑢) = 𝑢 ↔ (0 ⨣ 𝑢) = 𝑢))
71	70	ovanraleqv 7373	. . . 4 ⊢ (𝑒 = 0 → (∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢) ↔ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)))
72	71	rspcev 3577	. . 3 ⊢ ((0 ∈ {-1, 0, 1} ∧ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) → ∃𝑒 ∈ {-1, 0, 1}∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢))
73	67, 68, 72	mp2an 692	. 2 ⊢ ∃𝑒 ∈ {-1, 0, 1}∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢)
74		signsw.w	. . . 4 ⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⨣ ⟩}
75	1, 74	signswbase 34522	. . 3 ⊢ {-1, 0, 1} = (Base‘𝑊)
76	1, 74	signswplusg 34523	. . 3 ⊢ ⨣ = (+g‘𝑊)
77	75, 76	ismnd 18611	. 2 ⊢ (𝑊 ∈ Mnd ↔ (∀𝑢 ∈ {-1, 0, 1}∀𝑣 ∈ {-1, 0, 1} ((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧ ∀𝑤 ∈ {-1, 0, 1} ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) ∧ ∃𝑒 ∈ {-1, 0, 1}∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢)))
78	65, 73, 77	mpbir2an 711	1 ⊢ 𝑊 ∈ Mnd

Syntax hints:  ¬ wn 3   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ifcif 4476  {cpr 4579  {ctp 4581  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  0cc0 11009  1c1 11010  -cneg 11348  ndxcnx 17104  Basecbs 17120  +_gcplusg 17161  Mndcmnd 18608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-mgm 18514  df-sgrp 18593  df-mnd 18609

This theorem is referenced by:  signstcl  34533  signstf  34534  signstf0  34536  signstfvn  34537