Theorem signswmnd 33258

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 33253	. . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
3		ifcl 4536	. . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → if(𝑣 = 0, 𝑢, 𝑣) ∈ {-1, 0, 1})
4	2, 3	eqeltrd 2832	. . . 4 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 ⨣ 𝑣) ∈ {-1, 0, 1})
5	1	signspval 33253	. . . . . . . . . . . . 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 4497	. . . . . . . . . . . 12 ⊢ (𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = (𝑢 ⨣ 𝑣))
8	6, 7	sylan9eq 2791	. . . . . . . . . . 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 4497	. . . . . . . . . . 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 2775	. . . . . . . . 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 33253	. . . . . . . . . . . . . 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 4526	. . . . . . . . . . . . 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 2832	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 ⨣ 𝑤) ∈ {-1, 0, 1})
22	1	signspval 33253	. . . . . . . . . . . 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 4497	. . . . . . . . . . . . 13 ⊢ (𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣)
26	17, 25	sylan9eq 2791	. . . . . . . . . . . 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 2791	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 0)
29	28	iftrued 4499	. . . . . . . . . 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 2771	. . . . . . . . 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 2774	. . . . . . . 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 4500	. . . . . . . . . . . 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 2771	. . . . . . . . . 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 2775	. . . . . . . . 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 2771	. . . . . . . . . . . . 13 ⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 𝑣)
44	43	eqeq1d 2733	. . . . . . . . . . . 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 4502	. . . . . . . . . 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 2775	. . . . . . . . 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 2774	. . . . . . . 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 4500	. . . . . . . . 9 ⊢ (¬ 𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = 𝑤)
51	6, 50	sylan9eq 2791	. . . . . . . 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 4500	. . . . . . . . . . . . 13 ⊢ (¬ 𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑤)
55	17, 54	sylan9eq 2791	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑣 ⨣ 𝑤) = 𝑤)
56	55	eqeq1d 2733	. . . . . . . . . . 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 4502	. . . . . . . . 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 2775	. . . . . . . 8 ⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = 𝑤)
60	51, 59	eqtr4d 2774	. . . . . . 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 3139	. . . 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 3190	. 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 11158	. . . 4 ⊢ 0 ∈ V
67	66	tpid2 4736	. . 3 ⊢ 0 ∈ {-1, 0, 1}
68	1	signsw0glem 33254	. . 3 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)
69		oveq1 7369	. . . . . 6 ⊢ (𝑒 = 0 → (𝑒 ⨣ 𝑢) = (0 ⨣ 𝑢))
70	69	eqeq1d 2733	. . . . 5 ⊢ (𝑒 = 0 → ((𝑒 ⨣ 𝑢) = 𝑢 ↔ (0 ⨣ 𝑢) = 𝑢))
71	70	ovanraleqv 7386	. . . 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 3582	. . 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 33255	. . 3 ⊢ {-1, 0, 1} = (Base‘𝑊)
76	1, 74	signswplusg 33256	. . 3 ⊢ ⨣ = (+g‘𝑊)
77	75, 76	ismnd 18573	. 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 3060  ∃wrex 3069  ifcif 4491  {cpr 4593  {ctp 4595  ⟨cop 4597  ‘cfv 6501  (class class class)co 7362   ∈ cmpo 7364  0cc0 11060  1c1 11061  -cneg 11395  ndxcnx 17076  Basecbs 17094  +_gcplusg 17147  Mndcmnd 18570

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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-n0 12423  df-z 12509  df-uz 12773  df-fz 13435  df-struct 17030  df-slot 17065  df-ndx 17077  df-base 17095  df-plusg 17160  df-mgm 18511  df-sgrp 18560  df-mnd 18571

This theorem is referenced by:  signstcl  33266  signstf  33267  signstf0  33269  signstfvn  33270