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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