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Theorem signswmnd 30964
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.
StepHypRef Expression
1 signsw.p . . . . . 6 = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
21signspval 30959 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
3 ifcl 4274 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → if(𝑣 = 0, 𝑢, 𝑣) ∈ {-1, 0, 1})
42, 3eqeltrd 2839 . . . 4 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 𝑣) ∈ {-1, 0, 1})
51signspval 30959 . . . . . . . . . . . . 13 (((𝑢 𝑣) ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = if(𝑤 = 0, (𝑢 𝑣), 𝑤))
64, 5stoic3 1850 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = if(𝑤 = 0, (𝑢 𝑣), 𝑤))
7 iftrue 4236 . . . . . . . . . . . 12 (𝑤 = 0 → if(𝑤 = 0, (𝑢 𝑣), 𝑤) = (𝑢 𝑣))
86, 7sylan9eq 2814 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 𝑣))
98adantr 472 . . . . . . . . . 10 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 𝑣))
1023adant3 1127 . . . . . . . . . . 11 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑢 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
1110ad2antrr 764 . . . . . . . . . 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 4236 . . . . . . . . . . 11 (𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢)
1312adantl 473 . . . . . . . . . 10 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢)
149, 11, 133eqtrd 2798 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = 𝑢)
15 simp1 1131 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → 𝑢 ∈ {-1, 0, 1})
161signspval 30959 . . . . . . . . . . . . . 14 ((𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 𝑤) = if(𝑤 = 0, 𝑣, 𝑤))
17163adant1 1125 . . . . . . . . . . . . 13 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 𝑤) = if(𝑤 = 0, 𝑣, 𝑤))
18 simpl2 1230 . . . . . . . . . . . . . 14 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → 𝑣 ∈ {-1, 0, 1})
19 simpl3 1232 . . . . . . . . . . . . . 14 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ {-1, 0, 1})
2018, 19ifclda 4264 . . . . . . . . . . . . 13 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → if(𝑤 = 0, 𝑣, 𝑤) ∈ {-1, 0, 1})
2117, 20eqeltrd 2839 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 𝑤) ∈ {-1, 0, 1})
221signspval 30959 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ (𝑣 𝑤) ∈ {-1, 0, 1}) → (𝑢 (𝑣 𝑤)) = if((𝑣 𝑤) = 0, 𝑢, (𝑣 𝑤)))
2315, 21, 22syl2anc 696 . . . . . . . . . . 11 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑢 (𝑣 𝑤)) = if((𝑣 𝑤) = 0, 𝑢, (𝑣 𝑤)))
2423ad2antrr 764 . . . . . . . . . 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 4236 . . . . . . . . . . . . 13 (𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣)
2617, 25sylan9eq 2814 . . . . . . . . . . . 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)
2826, 27sylan9eq 2814 . . . . . . . . . . 11 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑣 𝑤) = 0)
2928iftrued 4238 . . . . . . . . . 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)
3024, 29eqtrd 2794 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 (𝑣 𝑤)) = 𝑢)
3114, 30eqtr4d 2797 . . . . . . . 8 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
326ad2antrr 764 . . . . . . . . . 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))
337ad2antlr 765 . . . . . . . . . 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))
3410ad2antrr 764 . . . . . . . . . . 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 4239 . . . . . . . . . . . 12 𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣)
3635adantl 473 . . . . . . . . . . 11 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣)
3734, 36eqtrd 2794 . . . . . . . . . 10 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 𝑣) = 𝑣)
3832, 33, 373eqtrd 2798 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = 𝑣)
3923ad2antrr 764 . . . . . . . . . 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 479 . . . . . . . . . . . 12 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ 𝑣 = 0)
4117ad2antrr 764 . . . . . . . . . . . . . 14 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 𝑤) = if(𝑤 = 0, 𝑣, 𝑤))
4225ad2antlr 765 . . . . . . . . . . . . . 14 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣)
4341, 42eqtrd 2794 . . . . . . . . . . . . 13 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 𝑤) = 𝑣)
4443eqeq1d 2762 . . . . . . . . . . . 12 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑣 𝑤) = 0 ↔ 𝑣 = 0))
4540, 44mtbird 314 . . . . . . . . . . 11 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ (𝑣 𝑤) = 0)
4645iffalsed 4241 . . . . . . . . . 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))
4739, 46, 433eqtrd 2798 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 (𝑣 𝑤)) = 𝑣)
4838, 47eqtr4d 2797 . . . . . . . 8 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
4931, 48pm2.61dan 867 . . . . . . 7 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
50 iffalse 4239 . . . . . . . . 9 𝑤 = 0 → if(𝑤 = 0, (𝑢 𝑣), 𝑤) = 𝑤)
516, 50sylan9eq 2814 . . . . . . . 8 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = 𝑤)
5223adantr 472 . . . . . . . . 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 479 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ 𝑤 = 0)
54 iffalse 4239 . . . . . . . . . . . . 13 𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑤)
5517, 54sylan9eq 2814 . . . . . . . . . . . 12 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑣 𝑤) = 𝑤)
5655eqeq1d 2762 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑣 𝑤) = 0 ↔ 𝑤 = 0))
5753, 56mtbird 314 . . . . . . . . . 10 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ (𝑣 𝑤) = 0)
5857iffalsed 4241 . . . . . . . . 9 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → if((𝑣 𝑤) = 0, 𝑢, (𝑣 𝑤)) = (𝑣 𝑤))
5952, 58, 553eqtrd 2798 . . . . . . . 8 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑢 (𝑣 𝑤)) = 𝑤)
6051, 59eqtr4d 2797 . . . . . . 7 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
6149, 60pm2.61dan 867 . . . . . 6 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
62613expa 1112 . . . . 5 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
6362ralrimiva 3104 . . . 4 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → ∀𝑤 ∈ {-1, 0, 1} ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
644, 63jca 555 . . 3 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → ((𝑢 𝑣) ∈ {-1, 0, 1} ∧ ∀𝑤 ∈ {-1, 0, 1} ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤))))
6564rgen2a 3115 . 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 10246 . . . 4 0 ∈ V
6766tpid2 4448 . . 3 0 ∈ {-1, 0, 1}
681signsw0glem 30960 . . 3 𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)
69 oveq1 6821 . . . . . . 7 (𝑒 = 0 → (𝑒 𝑢) = (0 𝑢))
7069eqeq1d 2762 . . . . . 6 (𝑒 = 0 → ((𝑒 𝑢) = 𝑢 ↔ (0 𝑢) = 𝑢))
71 oveq2 6822 . . . . . . 7 (𝑒 = 0 → (𝑢 𝑒) = (𝑢 0))
7271eqeq1d 2762 . . . . . 6 (𝑒 = 0 → ((𝑢 𝑒) = 𝑢 ↔ (𝑢 0) = 𝑢))
7370, 72anbi12d 749 . . . . 5 (𝑒 = 0 → (((𝑒 𝑢) = 𝑢 ∧ (𝑢 𝑒) = 𝑢) ↔ ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)))
7473ralbidv 3124 . . . 4 (𝑒 = 0 → (∀𝑢 ∈ {-1, 0, 1} ((𝑒 𝑢) = 𝑢 ∧ (𝑢 𝑒) = 𝑢) ↔ ∀𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)))
7574rspcev 3449 . . 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))
7667, 68, 75mp2an 710 . 2 𝑒 ∈ {-1, 0, 1}∀𝑢 ∈ {-1, 0, 1} ((𝑒 𝑢) = 𝑢 ∧ (𝑢 𝑒) = 𝑢)
77 signsw.w . . . 4 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}
781, 77signswbase 30961 . . 3 {-1, 0, 1} = (Base‘𝑊)
791, 77signswplusg 30962 . . 3 = (+g𝑊)
8078, 79ismnd 17518 . 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)))
8165, 76, 80mpbir2an 993 1 𝑊 ∈ Mnd
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  ifcif 4230  {cpr 4323  {ctp 4325  cop 4327  cfv 6049  (class class class)co 6814  cmpt2 6816  0cc0 10148  1c1 10149  -cneg 10479  ndxcnx 16076  Basecbs 16079  +gcplusg 16163  Mndcmnd 17515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-plusg 16176  df-mgm 17463  df-sgrp 17505  df-mnd 17516
This theorem is referenced by:  signstcl  30972  signstf  30973  signstf0  30975  signstfvn  30976
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