Nuprl Lemma : and_preserved_by
∀[T:Type]. ∀[P,Q:T ⟶ ℙ]. ∀[R:T ⟶ T ⟶ ℙ]. (R preserves P ⇒ R preserves Q ⇒ R preserves P ∧ Q)
Proof
Definitions occuring in Statement :
prop_and: P ∧ Q,
preserved_by: R preserves P,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
prop_and: P ∧ Q,
preserved_by: R preserves P,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
infix_ap: x f y,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
subtype_rel_self,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
applyEquality,
hypothesisEquality,
productEquality,
instantiate,
introduction,
extract_by_obid,
isectElimination,
universeEquality,
because_Cache,
lambdaEquality,
functionEquality,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P,Q:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (R preserves P {}\mRightarrow{} R preserves Q {}\mRightarrow{} R preserves P \mwedge{} Q)
Date html generated:
2019_06_20-PM-00_31_41
Last ObjectModification:
2018_09_26-AM-11_46_28
Theory : relations
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