Nuprl Lemma : mcomp_imp_not_unit
∀g:IAbMonoid. ∀a:|g|. (Reducible(a) ⇒ (¬(g-unit(a))))
Proof
Definitions occuring in Statement :
mreducible: Reducible(a),
munit: g-unit(u),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
iabmonoid: IAbMonoid,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
uall: ∀[x:A]. B[x],
mreducible: Reducible(a),
exists: ∃x:A. B[x],
and: P ∧ Q,
munit: g-unit(u),
mdivides: b | a,
uimplies: b supposing a,
squash: ↓T,
infix_ap: x f y,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
munit_wf,
mreducible_wf,
grp_car_wf,
iabmonoid_wf,
grp_op_l,
equal_wf,
squash_wf,
true_wf,
abmonoid_comm,
abmonoid_ac_1,
grp_op_wf,
iff_weakening_equal,
grp_id_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation
Latex:
\mforall{}g:IAbMonoid. \mforall{}a:|g|. (Reducible(a) {}\mRightarrow{} (\mneg{}(g-unit(a))))
Date html generated:
2017_10_01-AM-09_58_19
Last ObjectModification:
2017_03_03-PM-00_59_30
Theory : factor_1
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