Nuprl Lemma : linorder_functionality_wrt_ext-eq
∀[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. Linorder(A;x,y.R[x;y]) ⇐⇒ Linorder(B;x,y.R[x;y]) supposing A ≡ B
Proof
Definitions occuring in Statement :
linorder: Linorder(T;x,y.R[x; y]),
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
linorder: Linorder(T;x,y.R[x; y]),
order: Order(T;x,y.R[x; y]),
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
guard: {T},
trans: Trans(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
prop: ℙ,
so_apply: x[s1;s2],
connex: Connex(T;x,y.R[x; y]),
so_lambda: λ2x y.t[x; y],
rev_implies: P ⇐ Q
Lemmas referenced :
ext-eq_inversion,
subtype_rel_weakening,
equal_wf,
linorder_wf,
ext-eq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
rename,
independent_pairFormation,
lambdaFormation,
promote_hyp,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
extract_by_obid,
isectElimination,
independent_isectElimination,
because_Cache,
independent_functionElimination,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
cumulativity,
functionExtensionality,
lambdaEquality,
universeEquality,
functionEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. Linorder(A;x,y.R[x;y]) \mLeftarrow{}{}\mRightarrow{} Linorder(B;x,y.R[x;y]) supposing A \mequiv{} B
Date html generated:
2016_10_21-AM-09_42_29
Last ObjectModification:
2016_07_12-AM-05_03_54
Theory : rel_1
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