Nuprl Lemma : preserved_by_monotone
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[R1,R2:T ⟶ T ⟶ ℙ].  ((∀x,y:T.  ((x R1 y) 
⇒ (x R2 y))) 
⇒ R2 preserves P 
⇒ R1 preserves P)
Proof
Definitions occuring in Statement : 
preserved_by: R preserves P
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
infix_ap: x f y
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
preserved_by: R preserves P
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
prop: ℙ
, 
infix_ap: x f y
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
guard: {T}
Lemmas referenced : 
all_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
Error :isect_memberFormation_alt, 
lambdaFormation, 
applyEquality, 
hypothesisEquality, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
lambdaEquality, 
functionEquality, 
hypothesis, 
Error :inhabitedIsType, 
Error :functionIsType, 
Error :universeIsType, 
universeEquality, 
dependent_functionElimination, 
independent_functionElimination
Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[R1,R2:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    ((\mforall{}x,y:T.    ((x  R1  y)  {}\mRightarrow{}  (x  R2  y)))  {}\mRightarrow{}  R2  preserves  P  {}\mRightarrow{}  R1  preserves  P)
Date html generated:
2019_06_20-PM-00_30_29
Last ObjectModification:
2018_09_26-PM-00_40_01
Theory : relations
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