Nuprl Lemma : mreducible_elim
∀g:IAbMonoid. (Cancel(|g|;|g|;*) ⇒ (∀a:|g|. (Reducible(a) ⇐⇒ ∃b:|g|. ((¬(g-unit(b))) ∧ (b p| a)))))
Proof
Definitions occuring in Statement :
mreducible: Reducible(a),
mpdivides: a p| b,
munit: g-unit(u),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: 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,
mreducible: Reducible(a),
member: t ∈ T,
uall: ∀[x:A]. B[x],
iabmonoid: IAbMonoid,
imon: IMonoid,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
infix_ap: x f y,
so_apply: x[s],
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
cand: A c∧ B,
mpdivides: a p| b,
mdivides: b | a,
not: ¬A,
false: False,
munit: g-unit(u),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
grp_car_wf,
cancel_wf,
grp_op_wf,
iabmonoid_wf,
exists_over_and_r,
not_wf,
munit_wf,
equal_wf,
infix_ap_wf,
exists_wf,
iff_wf,
mpdivides_wf,
mdivides_wf,
mproper_div_cond,
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,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
existsFunctionality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
productEquality,
independent_functionElimination,
applyEquality,
existsLevelFunctionality,
dependent_pairFormation,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
voidElimination,
imageElimination,
equalityTransitivity,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}g:IAbMonoid
(Cancel(|g|;|g|;*) {}\mRightarrow{} (\mforall{}a:|g|. (Reducible(a) \mLeftarrow{}{}\mRightarrow{} \mexists{}b:|g|. ((\mneg{}(g-unit(b))) \mwedge{} (b p| a)))))
Date html generated:
2017_10_01-AM-09_58_11
Last ObjectModification:
2017_03_03-PM-00_59_48
Theory : factor_1
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