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Theorem signswmnd 34572
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 34567 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
3 ifcl 4571 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → if(𝑣 = 0, 𝑢, 𝑣) ∈ {-1, 0, 1})
42, 3eqeltrd 2841 . . . 4 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 𝑣) ∈ {-1, 0, 1})
51signspval 34567 . . . . . . . . . . . . 13 (((𝑢 𝑣) ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = if(𝑤 = 0, (𝑢 𝑣), 𝑤))
64, 5stoic3 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 4531 . . . . . . . . . . . 12 (𝑤 = 0 → if(𝑤 = 0, (𝑢 𝑣), 𝑤) = (𝑢 𝑣))
86, 7sylan9eq 2797 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 𝑣))
98adantr 480 . . . . . . . . . 10 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 𝑣))
1023adant3 1133 . . . . . . . . . . 11 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑢 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
1110ad2antrr 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 4531 . . . . . . . . . . 11 (𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢)
1312adantl 481 . . . . . . . . . 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 2781 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = 𝑢)
15 simp1 1137 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → 𝑢 ∈ {-1, 0, 1})
161signspval 34567 . . . . . . . . . . . . . 14 ((𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 𝑤) = if(𝑤 = 0, 𝑣, 𝑤))
17163adant1 1131 . . . . . . . . . . . . 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})
2018, 19ifclda 4561 . . . . . . . . . . . . 13 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → if(𝑤 = 0, 𝑣, 𝑤) ∈ {-1, 0, 1})
2117, 20eqeltrd 2841 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 𝑤) ∈ {-1, 0, 1})
221signspval 34567 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ (𝑣 𝑤) ∈ {-1, 0, 1}) → (𝑢 (𝑣 𝑤)) = if((𝑣 𝑤) = 0, 𝑢, (𝑣 𝑤)))
2315, 21, 22syl2anc 584 . . . . . . . . . . 11 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑢 (𝑣 𝑤)) = if((𝑣 𝑤) = 0, 𝑢, (𝑣 𝑤)))
2423ad2antrr 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 4531 . . . . . . . . . . . . 13 (𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣)
2617, 25sylan9eq 2797 . . . . . . . . . . . 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 2797 . . . . . . . . . . 11 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑣 𝑤) = 0)
2928iftrued 4533 . . . . . . . . . 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 2777 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 (𝑣 𝑤)) = 𝑢)
3114, 30eqtr4d 2780 . . . . . . . 8 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
326ad2antrr 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))
337ad2antlr 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))
3410ad2antrr 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 4534 . . . . . . . . . . . 12 𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣)
3635adantl 481 . . . . . . . . . . 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 2777 . . . . . . . . . 10 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 𝑣) = 𝑣)
3832, 33, 373eqtrd 2781 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 𝑣) 𝑤) = 𝑣)
3923ad2antrr 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)
4117ad2antrr 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))
4225ad2antlr 727 . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . 13 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 𝑤) = 𝑣)
4443eqeq1d 2739 . . . . . . . . . . . 12 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑣 𝑤) = 0 ↔ 𝑣 = 0))
4540, 44mtbird 325 . . . . . . . . . . 11 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ (𝑣 𝑤) = 0)
4645iffalsed 4536 . . . . . . . . . 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 2781 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 (𝑣 𝑤)) = 𝑣)
4838, 47eqtr4d 2780 . . . . . . . 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 813 . . . . . . 7 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
50 iffalse 4534 . . . . . . . . 9 𝑤 = 0 → if(𝑤 = 0, (𝑢 𝑣), 𝑤) = 𝑤)
516, 50sylan9eq 2797 . . . . . . . 8 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = 𝑤)
5223adantr 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 4534 . . . . . . . . . . . . 13 𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑤)
5517, 54sylan9eq 2797 . . . . . . . . . . . 12 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑣 𝑤) = 𝑤)
5655eqeq1d 2739 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑣 𝑤) = 0 ↔ 𝑤 = 0))
5753, 56mtbird 325 . . . . . . . . . 10 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ (𝑣 𝑤) = 0)
5857iffalsed 4536 . . . . . . . . 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 2781 . . . . . . . 8 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑢 (𝑣 𝑤)) = 𝑤)
6051, 59eqtr4d 2780 . . . . . . 7 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
6149, 60pm2.61dan 813 . . . . . 6 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
62613expa 1119 . . . . 5 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
6362ralrimiva 3146 . . . 4 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → ∀𝑤 ∈ {-1, 0, 1} ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
644, 63jca 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))))
6564rgen2 3199 . 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 11255 . . . 4 0 ∈ V
6766tpid2 4770 . . 3 0 ∈ {-1, 0, 1}
681signsw0glem 34568 . . 3 𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)
69 oveq1 7438 . . . . . 6 (𝑒 = 0 → (𝑒 𝑢) = (0 𝑢))
7069eqeq1d 2739 . . . . 5 (𝑒 = 0 → ((𝑒 𝑢) = 𝑢 ↔ (0 𝑢) = 𝑢))
7170ovanraleqv 7455 . . . 4 (𝑒 = 0 → (∀𝑢 ∈ {-1, 0, 1} ((𝑒 𝑢) = 𝑢 ∧ (𝑢 𝑒) = 𝑢) ↔ ∀𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)))
7271rspcev 3622 . . 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))
7367, 68, 72mp2an 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), ⟩}
751, 74signswbase 34569 . . 3 {-1, 0, 1} = (Base‘𝑊)
761, 74signswplusg 34570 . . 3 = (+g𝑊)
7775, 76ismnd 18750 . 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)))
7865, 73, 77mpbir2an 711 1 𝑊 ∈ Mnd
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ifcif 4525  {cpr 4628  {ctp 4630  cop 4632  cfv 6561  (class class class)co 7431  cmpo 7433  0cc0 11155  1c1 11156  -cneg 11493  ndxcnx 17230  Basecbs 17247  +gcplusg 17297  Mndcmnd 18747
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mgm 18653  df-sgrp 18732  df-mnd 18748
This theorem is referenced by:  signstcl  34580  signstf  34581  signstf0  34583  signstfvn  34584
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