Nuprl Lemma : rel_plus_functionality_wrt_iff
∀[T:Type]. ∀[R,Q:T ⟶ T ⟶ ℙ]. ((∀x,y:T. (R x y ⇐⇒ Q x y)) ⇒ (∀x,y:T. (R+ x y ⇐⇒ Q+ x y)))
Proof
Definitions occuring in Statement :
rel_plus: R+,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rel_plus: R+,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
infix_ap: x f y,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
rel_exp_wf,
exists_wf,
nat_plus_wf,
nat_plus_subtype_nat,
all_wf,
iff_wf,
rel_exp_functionality_wrt_iff
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
applyEquality,
cut,
lemma_by_obid,
isectElimination,
because_Cache,
hypothesis,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R,Q:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x,y:T. (R x y \mLeftarrow{}{}\mRightarrow{} Q x y)) {}\mRightarrow{} (\mforall{}x,y:T. (R\msupplus{} x y \mLeftarrow{}{}\mRightarrow{} Q\msupplus{} x y)))
Date html generated:
2016_05_14-PM-03_55_25
Last ObjectModification:
2015_12_26-PM-06_55_23
Theory : relations2
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