Nuprl Lemma : equiv_rel_and
∀[T:Type]. ∀[E1,E2:T ⟶ T ⟶ ℙ].
(EquivRel(T;x,y.E2[x;y]) ⇒ EquivRel(T;x,y.E1[x;y]) ⇒ EquivRel(T;x,y.E1[x;y] ∧ E2[x;y]))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
guard: {T}
Lemmas referenced :
and_wf,
equiv_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
hypothesisEquality,
lemma_by_obid,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[E1,E2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(EquivRel(T;x,y.E2[x;y]) {}\mRightarrow{} EquivRel(T;x,y.E1[x;y]) {}\mRightarrow{} EquivRel(T;x,y.E1[x;y] \mwedge{} E2[x;y]))
Date html generated:
2016_05_13-PM-04_14_57
Last ObjectModification:
2015_12_26-AM-11_30_09
Theory : rel_1
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