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
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