Nuprl Lemma : compose_wf_for_mon_hom
∀[A,B,C:IMonoid]. ∀[f:MonHom(A,B)]. ∀[g:MonHom(B,C)]. (g o f ∈ MonHom(A,C))
Proof
Definitions occuring in Statement :
monoid_hom: MonHom(M1,M2),
imon: IMonoid,
compose: f o g,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
imon: IMonoid,
monoid_hom: MonHom(M1,M2),
prop: ℙ,
monoid_hom_p: IsMonHom{M1,M2}(f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
and: P ∧ Q,
cand: A c∧ B,
compose: f o g,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
monoid_hom_wf,
imon_wf,
compose_wf,
grp_car_wf,
monoid_hom_p_wf,
monoid_hom_properties,
and_wf,
equal_wf,
squash_wf,
true_wf,
infix_ap_wf,
grp_op_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality,
because_Cache,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
productElimination,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
lambdaEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[A,B,C:IMonoid]. \mforall{}[f:MonHom(A,B)]. \mforall{}[g:MonHom(B,C)]. (g o f \mmember{} MonHom(A,C))
Date html generated:
2017_10_01-AM-08_14_17
Last ObjectModification:
2017_02_28-PM-01_58_41
Theory : groups_1
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