Nuprl Lemma : non_munit_diff_imp_mpdivides
∀g:IAbMonoid. (Cancel(|g|;|g|;*) ⇒ (∀a,b,c:|g|. ((¬(g-unit(b))) ⇒ ((a * b) = c ∈ |g|) ⇒ (a p| c))))
Proof
Definitions occuring in Statement :
mpdivides: a p| b,
munit: g-unit(u),
infix_ap: x f y,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T,
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,
uall: ∀[x:A]. B[x],
iabmonoid: IAbMonoid,
imon: IMonoid,
infix_ap: x f y,
mpdivides: a p| b,
and: P ∧ Q,
mdivides: b | a,
exists: ∃x:A. B[x],
not: ¬A,
false: False,
munit: g-unit(u),
cancel: Cancel(T;S;op),
uimplies: b supposing a,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
grp_car_wf,
grp_op_wf,
not_wf,
munit_wf,
cancel_wf,
iabmonoid_wf,
mdivides_wf,
grp_id_wf,
squash_wf,
true_wf,
mon_ident,
infix_ap_wf,
iff_weakening_equal,
mon_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
dependent_functionElimination,
independent_pairFormation,
dependent_pairFormation,
equalitySymmetry,
independent_functionElimination,
voidElimination,
productElimination,
independent_isectElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed
Latex:
\mforall{}g:IAbMonoid. (Cancel(|g|;|g|;*) {}\mRightarrow{} (\mforall{}a,b,c:|g|. ((\mneg{}(g-unit(b))) {}\mRightarrow{} ((a * b) = c) {}\mRightarrow{} (a p| c))))
Date html generated:
2017_10_01-AM-09_57_57
Last ObjectModification:
2017_03_03-PM-00_59_08
Theory : factor_1
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