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Theorem signswmnd 30414
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 30409 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
3 ifcl 4102 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → if(𝑣 = 0, 𝑢, 𝑣) ∈ {-1, 0, 1})
42, 3eqeltrd 2698 . . . 4 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 𝑣) ∈ {-1, 0, 1})
51signspval 30409 . . . . . . . . . . . . 13 (((𝑢 𝑣) ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = if(𝑤 = 0, (𝑢 𝑣), 𝑤))
64, 5stoic3 1698 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = if(𝑤 = 0, (𝑢 𝑣), 𝑤))
7 iftrue 4064 . . . . . . . . . . . 12 (𝑤 = 0 → if(𝑤 = 0, (𝑢 𝑣), 𝑤) = (𝑢 𝑣))
86, 7sylan9eq 2675 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 𝑣))
98adantr 481 . . . . . . . . . 10 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 𝑣))
1023adant3 1079 . . . . . . . . . . 11 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑢 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
1110ad2antrr 761 . . . . . . . . . 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 4064 . . . . . . . . . . 11 (𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢)
1312adantl 482 . . . . . . . . . 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 2659 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = 𝑢)
15 simp1 1059 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → 𝑢 ∈ {-1, 0, 1})
161signspval 30409 . . . . . . . . . . . . . 14 ((𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 𝑤) = if(𝑤 = 0, 𝑣, 𝑤))
17163adant1 1077 . . . . . . . . . . . . 13 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 𝑤) = if(𝑤 = 0, 𝑣, 𝑤))
18 simpl2 1063 . . . . . . . . . . . . . 14 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → 𝑣 ∈ {-1, 0, 1})
19 simpl3 1064 . . . . . . . . . . . . . 14 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ {-1, 0, 1})
2018, 19ifclda 4092 . . . . . . . . . . . . 13 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → if(𝑤 = 0, 𝑣, 𝑤) ∈ {-1, 0, 1})
2117, 20eqeltrd 2698 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 𝑤) ∈ {-1, 0, 1})
221signspval 30409 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ (𝑣 𝑤) ∈ {-1, 0, 1}) → (𝑢 (𝑣 𝑤)) = if((𝑣 𝑤) = 0, 𝑢, (𝑣 𝑤)))
2315, 21, 22syl2anc 692 . . . . . . . . . . 11 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑢 (𝑣 𝑤)) = if((𝑣 𝑤) = 0, 𝑢, (𝑣 𝑤)))
2423ad2antrr 761 . . . . . . . . . 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 4064 . . . . . . . . . . . . 13 (𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣)
2617, 25sylan9eq 2675 . . . . . . . . . . . 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 2675 . . . . . . . . . . 11 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑣 𝑤) = 0)
2928iftrued 4066 . . . . . . . . . 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 2655 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 (𝑣 𝑤)) = 𝑢)
3114, 30eqtr4d 2658 . . . . . . . 8 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
326ad2antrr 761 . . . . . . . . . 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 762 . . . . . . . . . 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 761 . . . . . . . . . . 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 4067 . . . . . . . . . . . 12 𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣)
3635adantl 482 . . . . . . . . . . 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 2655 . . . . . . . . . 10 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 𝑣) = 𝑣)
3832, 33, 373eqtrd 2659 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = 𝑣)
3923ad2antrr 761 . . . . . . . . . 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 477 . . . . . . . . . . . 12 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ 𝑣 = 0)
4117ad2antrr 761 . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . 13 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 𝑤) = 𝑣)
4443eqeq1d 2623 . . . . . . . . . . . 12 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑣 𝑤) = 0 ↔ 𝑣 = 0))
4540, 44mtbird 315 . . . . . . . . . . 11 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ (𝑣 𝑤) = 0)
4645iffalsed 4069 . . . . . . . . . 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 2659 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 (𝑣 𝑤)) = 𝑣)
4838, 47eqtr4d 2658 . . . . . . . 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 831 . . . . . . 7 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
50 iffalse 4067 . . . . . . . . 9 𝑤 = 0 → if(𝑤 = 0, (𝑢 𝑣), 𝑤) = 𝑤)
516, 50sylan9eq 2675 . . . . . . . 8 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = 𝑤)
5223adantr 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 477 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ 𝑤 = 0)
54 iffalse 4067 . . . . . . . . . . . . 13 𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑤)
5517, 54sylan9eq 2675 . . . . . . . . . . . 12 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑣 𝑤) = 𝑤)
5655eqeq1d 2623 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑣 𝑤) = 0 ↔ 𝑤 = 0))
5753, 56mtbird 315 . . . . . . . . . 10 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ (𝑣 𝑤) = 0)
5857iffalsed 4069 . . . . . . . . 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 2659 . . . . . . . 8 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑢 (𝑣 𝑤)) = 𝑤)
6051, 59eqtr4d 2658 . . . . . . 7 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
6149, 60pm2.61dan 831 . . . . . 6 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
62613expa 1262 . . . . 5 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
6362ralrimiva 2960 . . . 4 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → ∀𝑤 ∈ {-1, 0, 1} ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
644, 63jca 554 . . 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 2971 . 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 9978 . . . 4 0 ∈ V
6766tpid2 4274 . . 3 0 ∈ {-1, 0, 1}
681signsw0glem 30410 . . 3 𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)
69 oveq1 6611 . . . . . . 7 (𝑒 = 0 → (𝑒 𝑢) = (0 𝑢))
7069eqeq1d 2623 . . . . . 6 (𝑒 = 0 → ((𝑒 𝑢) = 𝑢 ↔ (0 𝑢) = 𝑢))
71 oveq2 6612 . . . . . . 7 (𝑒 = 0 → (𝑢 𝑒) = (𝑢 0))
7271eqeq1d 2623 . . . . . 6 (𝑒 = 0 → ((𝑢 𝑒) = 𝑢 ↔ (𝑢 0) = 𝑢))
7370, 72anbi12d 746 . . . . 5 (𝑒 = 0 → (((𝑒 𝑢) = 𝑢 ∧ (𝑢 𝑒) = 𝑢) ↔ ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)))
7473ralbidv 2980 . . . 4 (𝑒 = 0 → (∀𝑢 ∈ {-1, 0, 1} ((𝑒 𝑢) = 𝑢 ∧ (𝑢 𝑒) = 𝑢) ↔ ∀𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)))
7574rspcev 3295 . . 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 707 . 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 30411 . . 3 {-1, 0, 1} = (Base‘𝑊)
791, 77signswplusg 30412 . . 3 = (+g𝑊)
8078, 79ismnd 17218 . 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 954 1 𝑊 ∈ Mnd
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ifcif 4058  {cpr 4150  {ctp 4152  cop 4154  cfv 5847  (class class class)co 6604  cmpt2 6606  0cc0 9880  1c1 9881  -cneg 10211  ndxcnx 15778  Basecbs 15781  +gcplusg 15862  Mndcmnd 17215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-plusg 15875  df-mgm 17163  df-sgrp 17205  df-mnd 17216
This theorem is referenced by:  signstcl  30422  signstf  30423  signstf0  30425  signstfvn  30426
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