Theorem signswmnd 33169

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 33164	. . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
3		ifcl 4531	. . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → if(𝑣 = 0, 𝑢, 𝑣) ∈ {-1, 0, 1})
4	2, 3	eqeltrd 2838	. . . 4 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 ⨣ 𝑣) ∈ {-1, 0, 1})
5	1	signspval 33164	. . . . . . . . . . . . 13 ⊢ (((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤))
6	4, 5	stoic3 1778	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤))
7		iftrue 4492	. . . . . . . . . . . 12 ⊢ (𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = (𝑢 ⨣ 𝑣))
8	6, 7	sylan9eq 2796	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ 𝑣))
9	8	adantr 481	. . . . . . . . . 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 724	. . . . . . . . . 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 4492	. . . . . . . . . . 11 ⊢ (𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢)
13	12	adantl 482	. . . . . . . . . 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 2780	. . . . . . . . 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 33164	. . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . 14 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → 𝑣 ∈ {-1, 0, 1})
19		simpl3 1193	. . . . . . . . . . . . . 14 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ {-1, 0, 1})
20	18, 19	ifclda 4521	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 ⨣ 𝑤) ∈ {-1, 0, 1})
22	1	signspval 33164	. . . . . . . . . . . 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 724	. . . . . . . . . 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 4492	. . . . . . . . . . . . 13 ⊢ (𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣)
26	17, 25	sylan9eq 2796	. . . . . . . . . . . 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 2796	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 0)
29	28	iftrued 4494	. . . . . . . . . 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 2776	. . . . . . . . 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 2779	. . . . . . . 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 724	. . . . . . . . . 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 725	. . . . . . . . . 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 724	. . . . . . . . . . 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 4495	. . . . . . . . . . . 12 ⊢ (¬ 𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣)
36	35	adantl 482	. . . . . . . . . . 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 2776	. . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑣)
39	23	ad2antrr 724	. . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ 𝑣 = 0)
41	17	ad2antrr 724	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 13 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 𝑣)
44	43	eqeq1d 2738	. . . . . . . . . . . 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 324	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ (𝑣 ⨣ 𝑤) = 0)
46	45	iffalsed 4497	. . . . . . . . . 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 2780	. . . . . . . . 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 2779	. . . . . . . 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 811	. . . . . . 7 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
50		iffalse 4495	. . . . . . . . 9 ⊢ (¬ 𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = 𝑤)
51	6, 50	sylan9eq 2796	. . . . . . . 8 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑤)
52	23	adantr 481	. . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ 𝑤 = 0)
54		iffalse 4495	. . . . . . . . . . . . 13 ⊢ (¬ 𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑤)
55	17, 54	sylan9eq 2796	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑣 ⨣ 𝑤) = 𝑤)
56	55	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑣 ⨣ 𝑤) = 0 ↔ 𝑤 = 0))
57	53, 56	mtbird 324	. . . . . . . . . 10 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ (𝑣 ⨣ 𝑤) = 0)
58	57	iffalsed 4497	. . . . . . . . 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 2780	. . . . . . . 8 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = 𝑤)
60	51, 59	eqtr4d 2779	. . . . . . 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 811	. . . . . 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 3143	. . . 4 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → ∀𝑤 ∈ {-1, 0, 1} ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))
64	4, 63	jca 512	. . 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 3194	. 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 11149	. . . 4 ⊢ 0 ∈ V
67	66	tpid2 4731	. . 3 ⊢ 0 ∈ {-1, 0, 1}
68	1	signsw0glem 33165	. . 3 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)
69		oveq1 7364	. . . . . 6 ⊢ (𝑒 = 0 → (𝑒 ⨣ 𝑢) = (0 ⨣ 𝑢))
70	69	eqeq1d 2738	. . . . 5 ⊢ (𝑒 = 0 → ((𝑒 ⨣ 𝑢) = 𝑢 ↔ (0 ⨣ 𝑢) = 𝑢))
71	70	ovanraleqv 7381	. . . 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 3581	. . 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 690	. 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 33166	. . 3 ⊢ {-1, 0, 1} = (Base‘𝑊)
76	1, 74	signswplusg 33167	. . 3 ⊢ ⨣ = (+g‘𝑊)
77	75, 76	ismnd 18559	. 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 709	1 ⊢ 𝑊 ∈ Mnd

Syntax hints:  ¬ wn 3   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  ifcif 4486  {cpr 4588  {ctp 4590  ⟨cop 4592  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  0cc0 11051  1c1 11052  -cneg 11386  ndxcnx 17065  Basecbs 17083  +_gcplusg 17133  Mndcmnd 18556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mgm 18497  df-sgrp 18546  df-mnd 18557

This theorem is referenced by:  signstcl  33177  signstf  33178  signstf0  33180  signstfvn  33181