Nuprl Lemma : mk_imon
∀[T:Type]. ∀[eq,le:T ⟶ T ⟶ 𝔹]. ∀[op:T ⟶ T ⟶ T]. ∀[id:T]. ∀[inv:T ⟶ T].
  (<T, eq, le, op, id, inv> ∈ IMonoid) supposing (Ident(T;op;id) and Assoc(T;op))
Proof
Definitions occuring in Statement : 
imon: IMonoid
, 
ident: Ident(T;op;id)
, 
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
, 
imon: IMonoid
, 
grp_sig: GrpSig
, 
grp_car: |g|
, 
pi1: fst(t)
, 
grp_op: *
, 
pi2: snd(t)
, 
grp_id: e
, 
monoid_p: IsMonoid(T;op;id)
, 
and: P ∧ Q
Lemmas referenced : 
ident_wf, 
assoc_wf, 
bool_wf, 
monoid_p_wf, 
grp_car_wf, 
grp_op_wf, 
grp_id_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, 
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{}  IMonoid)  supposing  (Ident(T;op;id)  and  Assoc(T;op))
Date html generated:
2016_05_15-PM-00_06_37
Last ObjectModification:
2015_12_26-PM-11_47_45
Theory : groups_1
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