Nuprl Lemma : grp_hom_formation
∀[g,h:IGroup]. ∀[f:|g| ⟶ |h|]. IsMonHom{g,h}(f) supposing ∀a,b:|g|. ((f (a * b)) = ((f a) * (f b)) ∈ |h|)
Proof
Definitions occuring in Statement :
monoid_hom_p: IsMonHom{M1,M2}(f),
igrp: IGroup,
grp_op: *,
grp_car: |g|,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
monoid_hom_p: IsMonHom{M1,M2}(f),
and: P ∧ Q,
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
squash: ↓T,
prop: ℙ,
igrp: IGroup,
imon: IMonoid,
all: ∀x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
grp_car_wf,
infix_ap_wf,
grp_op_wf,
iff_weakening_equal,
all_wf,
igrp_wf,
grp_eq_op_l,
grp_id_wf,
mon_ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
setElimination,
rename,
dependent_functionElimination,
because_Cache,
functionExtensionality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
independent_pairEquality,
functionEquality
Latex:
\mforall{}[g,h:IGroup]. \mforall{}[f:|g| {}\mrightarrow{} |h|].
IsMonHom\{g,h\}(f) supposing \mforall{}a,b:|g|. ((f (a * b)) = ((f a) * (f b)))
Date html generated:
2017_10_01-AM-08_14_01
Last ObjectModification:
2017_02_28-PM-01_58_17
Theory : groups_1
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