Nuprl Lemma : meq_functionality
∀[X:Type]. ∀[d:metric(X)]. ∀[x1,x2,y1,y2:X]. (uiff(x1 ≡ y1;x2 ≡ y2)) supposing (y1 ≡ y2 and x1 ≡ x2)
Proof
Definitions occuring in Statement :
meq: x ≡ y,
metric: metric(X),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
meq: x ≡ y,
metric: metric(X),
implies: P ⇒ Q,
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
all: ∀x:A. B[x],
guard: {T},
sym: Sym(T;x,y.E[x; y])
Lemmas referenced :
meq-equiv,
req_witness,
int-to-real_wf,
meq_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
sqequalRule,
applyEquality,
setElimination,
rename,
hypothesis,
natural_numberEquality,
independent_functionElimination,
universeIsType,
independent_pairEquality,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x1,x2,y1,y2:X].
(uiff(x1 \mequiv{} y1;x2 \mequiv{} y2)) supposing (y1 \mequiv{} y2 and x1 \mequiv{} x2)
Date html generated:
2019_10_29-AM-10_56_35
Last ObjectModification:
2019_10_02-AM-09_37_46
Theory : reals
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