Nuprl Lemma : mk_grp
∀[T:Type]. ∀[eq,le:T ⟶ T ⟶ 𝔹]. ∀[op:T ⟶ T ⟶ T]. ∀[id:T]. ∀[inv:T ⟶ T].
(<T, eq, le, op, id, inv> ∈ Group{i}) supposing (Inverse(T;op;id;inv) and Ident(T;op;id) and Assoc(T;op))
Proof
Definitions occuring in Statement :
grp: Group{i},
ident: Ident(T;op;id),
inverse: Inverse(T;op;id;inv),
assoc: Assoc(T;op),
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
pair: <a, b>,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
grp: Group{i},
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
grp_id: e,
grp_inv: ~,
mon: Mon,
grp_sig: GrpSig,
monoid_p: IsMonoid(T;op;id),
and: P ∧ Q
Lemmas referenced :
inverse_wf,
ident_wf,
assoc_wf,
bool_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
grp_inv_wf,
monoid_p_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality,
dependent_set_memberEquality,
setElimination,
rename,
dependent_pairEquality,
productEquality,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq,le:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[op:T {}\mrightarrow{} T {}\mrightarrow{} T]. \mforall{}[id:T]. \mforall{}[inv:T {}\mrightarrow{} T].
(<T, eq, le, op, id, inv> \mmember{} Group\{i\}) supposing
(Inverse(T;op;id;inv) and
Ident(T;op;id) and
Assoc(T;op))
Date html generated:
2016_05_15-PM-00_08_43
Last ObjectModification:
2015_12_26-PM-11_46_50
Theory : groups_1
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