Nuprl Lemma : same-metric
∀[X:Type]. ∀[d:metric(X)]. ∀[d':X ⟶ X ⟶ ℝ]. d' ∈ metric(X) supposing ∀x,y:X. ((d' x y) = (d x y))
Proof
Definitions occuring in Statement :
metric: metric(X),
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
metric: metric(X),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
prop: ℙ,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rleq_wf,
radd_wf,
req_wf,
int-to-real_wf,
real_wf,
metric_wf,
istype-universe,
rleq_functionality,
radd_functionality,
req_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesisEquality,
productElimination,
lambdaFormation_alt,
inhabitedIsType,
universeIsType,
independent_pairFormation,
hypothesis,
because_Cache,
sqequalRule,
productIsType,
functionIsType,
extract_by_obid,
isectElimination,
applyEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
independent_isectElimination,
dependent_functionElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[d':X {}\mrightarrow{} X {}\mrightarrow{} \mBbbR{}].
d' \mmember{} metric(X) supposing \mforall{}x,y:X. ((d' x y) = (d x y))
Date html generated:
2019_10_29-AM-10_51_33
Last ObjectModification:
2019_10_02-AM-09_33_27
Theory : reals
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