Nuprl Lemma : monot_functionality
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ].
∀f:T ⟶ T. ((∀x,y:T. (R[x;y] ⇐⇒ R'[x;y])) ⇒ (monot(T;x,y.R[x;y];f) ⇐⇒ monot(T;x,y.R'[x;y];f)))
Proof
Definitions occuring in Statement :
monot: monot(T;x,y.R[x; y];f),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
monot: monot(T;x,y.R[x; y];f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
allFunctionality,
allLevelFunctionality,
impliesLevelFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}f:T {}\mrightarrow{} T. ((\mforall{}x,y:T. (R[x;y] \mLeftarrow{}{}\mRightarrow{} R'[x;y])) {}\mRightarrow{} (monot(T;x,y.R[x;y];f) \mLeftarrow{}{}\mRightarrow{} monot(T;x,y.R'[x;y];f)))
Date html generated:
2016_05_15-PM-00_03_04
Last ObjectModification:
2015_12_26-PM-11_25_21
Theory : gen_algebra_1
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