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))


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: 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: supposing a prop: 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

\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 

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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