Nuprl Lemma : munit_of_op
∀g:IAbMonoid. ∀a,b:|g|. ((g-unit(a * b)) ⇒ ((g-unit(a)) ∧ (g-unit(b))))
Proof
Definitions occuring in Statement :
munit: g-unit(u),
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iabmonoid: IAbMonoid,
grp_op: *,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
munit: g-unit(u),
mdivides: b | a,
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
munit_wf,
grp_op_wf,
grp_car_wf,
iabmonoid_wf,
equal_wf,
squash_wf,
true_wf,
abmonoid_comm,
iff_weakening_equal,
mon_assoc,
grp_id_wf,
abmonoid_ac_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
isectElimination,
because_Cache,
productElimination,
dependent_pairFormation,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}g:IAbMonoid. \mforall{}a,b:|g|. ((g-unit(a * b)) {}\mRightarrow{} ((g-unit(a)) \mwedge{} (g-unit(b))))
Date html generated:
2017_10_01-AM-09_57_54
Last ObjectModification:
2017_03_03-PM-00_58_59
Theory : factor_1
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