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Theorem signswmnd 32118
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 32113 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 𝑣) = if(𝑣 = 0, 𝑢, 𝑣))
3 ifcl 4469 . . . . 5 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → if(𝑣 = 0, 𝑢, 𝑣) ∈ {-1, 0, 1})
42, 3eqeltrd 2834 . . . 4 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → (𝑢 𝑣) ∈ {-1, 0, 1})
51signspval 32113 . . . . . . . . . . . . 13 (((𝑢 𝑣) ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = if(𝑤 = 0, (𝑢 𝑣), 𝑤))
64, 5stoic3 1783 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 𝑣) 𝑤) = if(𝑤 = 0, (𝑢 𝑣), 𝑤))
7 iftrue 4430 . . . . . . . . . . . 12 (𝑤 = 0 → if(𝑤 = 0, (𝑢 𝑣), 𝑤) = (𝑢 𝑣))
86, 7sylan9eq 2794 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = (𝑢 𝑣))
98adantr 484 . . . . . . . . . 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 4430 . . . . . . . . . . 11 (𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢)
1312adantl 485 . . . . . . . . . 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 2778 . . . . . . . . 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 32113 . . . . . . . . . . . . . 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 4459 . . . . . . . . . . . . 13 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → if(𝑤 = 0, 𝑣, 𝑤) ∈ {-1, 0, 1})
2117, 20eqeltrd 2834 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → (𝑣 𝑤) ∈ {-1, 0, 1})
221signspval 32113 . . . . . . . . . . . 12 ((𝑢 ∈ {-1, 0, 1} ∧ (𝑣 𝑤) ∈ {-1, 0, 1}) → (𝑢 (𝑣 𝑤)) = if((𝑣 𝑤) = 0, 𝑢, (𝑣 𝑤)))
2315, 21, 22syl2anc 587 . . . . . . . . . . 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 4430 . . . . . . . . . . . . 13 (𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣)
2617, 25sylan9eq 2794 . . . . . . . . . . . 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 2794 . . . . . . . . . . 11 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑣 𝑤) = 0)
2928iftrued 4432 . . . . . . . . . 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 2774 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 (𝑣 𝑤)) = 𝑢)
3114, 30eqtr4d 2777 . . . . . . . 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 4433 . . . . . . . . . . . 12 𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣)
3635adantl 485 . . . . . . . . . . 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 2774 . . . . . . . . . 10 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 𝑣) = 𝑣)
3832, 33, 373eqtrd 2778 . . . . . . . . 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 488 . . . . . . . . . . . 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 2774 . . . . . . . . . . . . 13 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 𝑤) = 𝑣)
4443eqeq1d 2741 . . . . . . . . . . . 12 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑣 𝑤) = 0 ↔ 𝑣 = 0))
4540, 44mtbird 328 . . . . . . . . . . 11 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ (𝑣 𝑤) = 0)
4645iffalsed 4435 . . . . . . . . . 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 2778 . . . . . . . . 9 ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 (𝑣 𝑤)) = 𝑣)
4838, 47eqtr4d 2777 . . . . . . . 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 4433 . . . . . . . . 9 𝑤 = 0 → if(𝑤 = 0, (𝑢 𝑣), 𝑤) = 𝑤)
516, 50sylan9eq 2794 . . . . . . . 8 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑢 𝑣) 𝑤) = 𝑤)
5223adantr 484 . . . . . . . . 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 488 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ 𝑤 = 0)
54 iffalse 4433 . . . . . . . . . . . . 13 𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑤)
5517, 54sylan9eq 2794 . . . . . . . . . . . 12 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑣 𝑤) = 𝑤)
5655eqeq1d 2741 . . . . . . . . . . 11 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ((𝑣 𝑤) = 0 ↔ 𝑤 = 0))
5753, 56mtbird 328 . . . . . . . . . 10 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → ¬ (𝑣 𝑤) = 0)
5857iffalsed 4435 . . . . . . . . 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 2778 . . . . . . . 8 (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬ 𝑤 = 0) → (𝑢 (𝑣 𝑤)) = 𝑤)
6051, 59eqtr4d 2777 . . . . . . 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 3097 . . . 4 ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) → ∀𝑤 ∈ {-1, 0, 1} ((𝑢 𝑣) 𝑤) = (𝑢 (𝑣 𝑤)))
644, 63jca 515 . . 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 3116 . 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 10725 . . . 4 0 ∈ V
6766tpid2 4671 . . 3 0 ∈ {-1, 0, 1}
681signsw0glem 32114 . . 3 𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)
69 oveq1 7189 . . . . . 6 (𝑒 = 0 → (𝑒 𝑢) = (0 𝑢))
7069eqeq1d 2741 . . . . 5 (𝑒 = 0 → ((𝑒 𝑢) = 𝑢 ↔ (0 𝑢) = 𝑢))
7170ovanraleqv 7206 . . . 4 (𝑒 = 0 → (∀𝑢 ∈ {-1, 0, 1} ((𝑒 𝑢) = 𝑢 ∧ (𝑢 𝑒) = 𝑢) ↔ ∀𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)))
7271rspcev 3529 . . 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 32115 . . 3 {-1, 0, 1} = (Base‘𝑊)
761, 74signswplusg 32116 . . 3 = (+g𝑊)
7775, 76ismnd 18042 . 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 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wrex 3055  ifcif 4424  {cpr 4528  {ctp 4530  cop 4532  cfv 6349  (class class class)co 7182  cmpo 7184  0cc0 10627  1c1 10628  -cneg 10961  ndxcnx 16595  Basecbs 16598  +gcplusg 16680  Mndcmnd 18039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491  ax-cnex 10683  ax-resscn 10684  ax-1cn 10685  ax-icn 10686  ax-addcl 10687  ax-addrcl 10688  ax-mulcl 10689  ax-mulrcl 10690  ax-mulcom 10691  ax-addass 10692  ax-mulass 10693  ax-distr 10694  ax-i2m1 10695  ax-1ne0 10696  ax-1rid 10697  ax-rnegex 10698  ax-rrecex 10699  ax-cnre 10700  ax-pre-lttri 10701  ax-pre-lttrn 10702  ax-pre-ltadd 10703  ax-pre-mulgt0 10704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-om 7612  df-1st 7726  df-2nd 7727  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-1o 8143  df-er 8332  df-en 8568  df-dom 8569  df-sdom 8570  df-fin 8571  df-pnf 10767  df-mnf 10768  df-xr 10769  df-ltxr 10770  df-le 10771  df-sub 10962  df-neg 10963  df-nn 11729  df-2 11791  df-n0 11989  df-z 12075  df-uz 12337  df-fz 12994  df-struct 16600  df-ndx 16601  df-slot 16602  df-base 16604  df-plusg 16693  df-mgm 17980  df-sgrp 18029  df-mnd 18040
This theorem is referenced by:  signstcl  32126  signstf  32127  signstf0  32129  signstfvn  32130
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