Nuprl Lemma : mon_hom_p_comp
∀[g,h,k:GrpSig]. ∀[r:|g| ⟶ |h|]. ∀[s:|h| ⟶ |k|].
(IsMonHom{g,k}(s o r)) supposing (IsMonHom{h,k}(s) and IsMonHom{g,h}(r))
Proof
Definitions occuring in Statement :
monoid_hom_p: IsMonHom{M1,M2}(f),
grp_car: |g|,
grp_sig: GrpSig,
compose: f o g,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
monoid_hom_p: IsMonHom{M1,M2}(f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
compose: f o g,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
infix_ap: x f y,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
grp_car_wf,
uall_wf,
equal_wf,
infix_ap_wf,
grp_op_wf,
grp_id_wf,
grp_sig_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
extract_by_obid,
isectElimination,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
independent_pairEquality,
productEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[g,h,k:GrpSig]. \mforall{}[r:|g| {}\mrightarrow{} |h|]. \mforall{}[s:|h| {}\mrightarrow{} |k|].
(IsMonHom\{g,k\}(s o r)) supposing (IsMonHom\{h,k\}(s) and IsMonHom\{g,h\}(r))
Date html generated:
2017_10_01-AM-08_14_13
Last ObjectModification:
2017_02_28-PM-01_58_51
Theory : groups_1
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