Nuprl Lemma : mk_igrp_wf
∀[T:Type]. ∀[op:T ⟶ T ⟶ T]. ∀[id:T]. ∀[inv:T ⟶ T].
(mk_igrp(T;op;id;inv) ∈ IGroup) supposing (Inverse(T;op;id;inv) and Ident(T;op;id) and Assoc(T;op))
Proof
Definitions occuring in Statement :
mk_igrp: mk_igrp(T;op;id;inv),
igrp: IGroup,
ident: Ident(T;op;id),
inverse: Inverse(T;op;id;inv),
assoc: Assoc(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
mk_igrp: mk_igrp(T;op;id;inv),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
igrp: IGroup,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
grp_id: e,
grp_inv: ~,
imon: IMonoid
Lemmas referenced :
inverse_wf,
ident_wf,
assoc_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
grp_inv_wf,
mk_imon,
btrue_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality,
dependent_set_memberEquality,
setElimination,
rename,
lambdaEquality,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[op:T {}\mrightarrow{} T {}\mrightarrow{} T]. \mforall{}[id:T]. \mforall{}[inv:T {}\mrightarrow{} T].
(mk\_igrp(T;op;id;inv) \mmember{} IGroup) supposing
(Inverse(T;op;id;inv) and
Ident(T;op;id) and
Assoc(T;op))
Date html generated:
2016_05_15-PM-00_08_00
Last ObjectModification:
2015_12_26-PM-11_46_14
Theory : groups_1
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