Nuprl Lemma : massoc_imp_unit_diff
∀g:IAbMonoid. (Cancel(|g|;|g|;*) ⇒ (∀a,b:|g|. ((a ~ b) ⇒ (∃u:|g|. ((g-unit(u)) c∧ (a = (u * b) ∈ |g|))))))
Proof
Definitions occuring in Statement :
massoc: a ~ b,
munit: g-unit(u),
cand: A c∧ B,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
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,
iabmonoid: IAbMonoid,
imon: IMonoid,
uall: ∀[x:A]. B[x],
cancel: Cancel(T;S;op),
massoc: a ~ b,
symmetrize: Symmetrize(x,y.R[x; y];a;b),
and: P ∧ Q,
mdivides: b | a,
exists: ∃x:A. B[x],
infix_ap: x f y,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
munit: g-unit(u)
Lemmas referenced :
massoc_wf,
grp_car_wf,
cancel_wf,
grp_op_wf,
iabmonoid_wf,
equal_wf,
squash_wf,
true_wf,
mon_assoc,
iff_weakening_equal,
mon_ident,
grp_id_wf,
munit_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,
productElimination,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
dependent_pairFormation,
independent_pairFormation,
productEquality
Latex:
\mforall{}g:IAbMonoid
(Cancel(|g|;|g|;*) {}\mRightarrow{} (\mforall{}a,b:|g|. ((a \msim{} b) {}\mRightarrow{} (\mexists{}u:|g|. ((g-unit(u)) c\mwedge{} (a = (u * b)))))))
Date html generated:
2017_10_01-AM-09_58_03
Last ObjectModification:
2017_03_03-PM-00_59_33
Theory : factor_1
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