Nuprl Lemma : massoc_cancel
∀g:IAbMonoid. (Cancel(|g|;|g|;*) ⇒ (∀a,b,c:|g|. (((a * b) ~ (a * c)) ⇒ (b ~ c))))
Proof
Definitions occuring in Statement :
massoc: a ~ b,
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 :
massoc: a ~ b,
symmetrize: Symmetrize(x,y.R[x; y];a;b),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
uall: ∀[x:A]. B[x],
infix_ap: x f y
Lemmas referenced :
mdivides_wf,
infix_ap_wf,
grp_car_wf,
grp_op_wf,
cancel_wf,
iabmonoid_wf,
mdivides_cancel
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
productEquality,
lemma_by_obid,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
isectElimination,
because_Cache,
independent_functionElimination
Latex:
\mforall{}g:IAbMonoid. (Cancel(|g|;|g|;*) {}\mRightarrow{} (\mforall{}a,b,c:|g|. (((a * b) \msim{} (a * c)) {}\mRightarrow{} (b \msim{} c))))
Date html generated:
2016_05_16-AM-07_43_43
Last ObjectModification:
2015_12_28-PM-05_54_31
Theory : factor_1
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