Nuprl Lemma : rel_plus_field
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ].
((∀x,y:T. ((R x y) ⇒ ((P x) ∧ (P y)))) ⇒ (∀x,y:T. ((R+ x y) ⇒ ((P x) ∧ (P y)))))
Proof
Definitions occuring in Statement :
rel_plus: R+,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
rel_implies: R1 => R2,
infix_ap: x f y,
trans: Trans(T;x,y.E[x; y]),
cand: A c∧ B
Lemmas referenced :
istype-universe,
subtype_rel_self,
rel_plus_minimal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalRule,
Error :functionIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
Error :universeIsType,
applyEquality,
instantiate,
universeEquality,
Error :productIsType,
because_Cache,
Error :lambdaEquality_alt,
productEquality,
independent_functionElimination,
productElimination,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. ((R x y) {}\mRightarrow{} ((P x) \mwedge{} (P y)))) {}\mRightarrow{} (\mforall{}x,y:T. ((R\msupplus{} x y) {}\mRightarrow{} ((P x) \mwedge{} (P y)))))
Date html generated:
2019_06_20-PM-02_02_27
Last ObjectModification:
2018_10_06-AM-11_23_50
Theory : relations2
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