Nuprl Lemma : mdivisor_of_atom_is_assoc2
∀g:IAbMonoid. ((∀a,b:|g|. Stable{a | b}) ⇒ (∀a,b:|g|. ((¬(g-unit(a))) ⇒ Atomic(b) ⇒ (a | b) ⇒ (a ~ b))))
Proof
Definitions occuring in Statement :
matomic: Atomic(a),
massoc: a ~ b,
munit: g-unit(u),
mdivides: b | a,
stable: Stable{P},
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,
member: t ∈ T,
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
matomic: Atomic(a),
and: P ∧ Q,
massoc: a ~ b,
symmetrize: Symmetrize(x,y.R[x; y];a;b),
stable: Stable{P},
uimplies: b supposing a,
not: ¬A,
mreducible: Reducible(a),
false: False,
mdivides: b | a,
exists: ∃x:A. B[x],
cand: A c∧ B,
infix_ap: x f y,
munit: g-unit(u),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
mdivides_wf,
matomic_wf,
not_wf,
munit_wf,
grp_car_wf,
all_wf,
stable_wf,
iabmonoid_wf,
equal_wf,
grp_op_wf,
exists_wf,
squash_wf,
true_wf,
mon_assoc,
iff_weakening_equal,
mon_ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
because_Cache,
productElimination,
independent_pairFormation,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_pairFormation,
productEquality,
applyEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}g:IAbMonoid
((\mforall{}a,b:|g|. Stable\{a | b\}) {}\mRightarrow{} (\mforall{}a,b:|g|. ((\mneg{}(g-unit(a))) {}\mRightarrow{} Atomic(b) {}\mRightarrow{} (a | b) {}\mRightarrow{} (a \msim{} b))))
Date html generated:
2017_10_01-AM-09_58_16
Last ObjectModification:
2017_03_03-PM-00_59_44
Theory : factor_1
Home
Index