Nuprl Lemma : mprime_imp_matomic
∀g:IAbMonoid. (Cancel(|g|;|g|;*) ⇒ (∀a:|g|. (IsPrime(a) ⇒ Atomic(a))))
Proof
Definitions occuring in Statement :
matomic: Atomic(a),
mprime: IsPrime(a),
all: ∀x:A. B[x],
implies: P ⇒ Q,
iabmonoid: IAbMonoid,
grp_op: *,
grp_car: |g|,
cancel: Cancel(T;S;op)
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
uall: ∀[x:A]. B[x],
matomic: Atomic(a),
and: P ∧ Q,
mprime: IsPrime(a),
not: ¬A,
mreducible: Reducible(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,
or: P ∨ Q,
false: False
Lemmas referenced :
mprime_wf,
grp_car_wf,
cancel_wf,
grp_op_wf,
iabmonoid_wf,
mreducible_wf,
mdivides_wf,
squash_wf,
true_wf,
grp_sig_wf,
iff_weakening_equal,
mdivides_refl,
mproper_div_cond,
equal_wf,
abmonoid_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
because_Cache,
independent_pairFormation,
productElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
unionElimination,
hyp_replacement,
applyLambdaEquality,
voidElimination,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}g:IAbMonoid. (Cancel(|g|;|g|;*) {}\mRightarrow{} (\mforall{}a:|g|. (IsPrime(a) {}\mRightarrow{} Atomic(a))))
Date html generated:
2017_10_01-AM-09_58_24
Last ObjectModification:
2017_03_03-PM-00_59_36
Theory : factor_1
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