Nuprl Lemma : mdivides_cancel
∀g:IAbMonoid. (Cancel(|g|;|g|;*) ⇒ (∀a,b,c:|g|. (((a * b) | (a * c)) ⇒ (b | c))))
Proof
Definitions occuring in Statement :
mdivides: b | a,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: 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,
cancel: Cancel(T;S;op),
member: t ∈ T,
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
mdivides: b | a,
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
mdivides_wf,
grp_op_wf,
grp_car_wf,
cancel_wf,
iabmonoid_wf,
equal_wf,
squash_wf,
true_wf,
mon_assoc,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
isectElimination,
because_Cache,
productElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
dependent_pairFormation
Latex:
\mforall{}g:IAbMonoid. (Cancel(|g|;|g|;*) {}\mRightarrow{} (\mforall{}a,b,c:|g|. (((a * b) | (a * c)) {}\mRightarrow{} (b | c))))
Date html generated:
2017_10_01-AM-09_57_52
Last ObjectModification:
2017_03_03-PM-00_58_57
Theory : factor_1
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