Nuprl Lemma : mdist-nonneg
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X].  (r0 ≤ mdist(d;x;y))
Proof
Definitions occuring in Statement : 
mdist: mdist(d;x;y)
, 
metric: metric(X)
, 
rleq: x ≤ y
, 
int-to-real: r(n)
, 
uall: ∀[x:A]. B[x]
, 
natural_number: $n
, 
universe: Type
Definitions unfolded in proof : 
mdist: mdist(d;x;y)
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
metric: metric(X)
, 
sq_stable: SqStable(P)
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
squash: ↓T
, 
rleq: x ≤ y
, 
rnonneg: rnonneg(x)
, 
all: ∀x:A. B[x]
, 
le: A ≤ B
, 
uimplies: b supposing a
, 
guard: {T}
Lemmas referenced : 
sq_stable__rleq, 
int-to-real_wf, 
le_witness_for_triv, 
metric_wf, 
istype-universe
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation_alt, 
introduction, 
cut, 
sqequalHypSubstitution, 
setElimination, 
thin, 
rename, 
extract_by_obid, 
isectElimination, 
natural_numberEquality, 
hypothesis, 
applyEquality, 
hypothesisEquality, 
independent_functionElimination, 
productElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
lambdaEquality_alt, 
dependent_functionElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_isectElimination, 
functionIsTypeImplies, 
inhabitedIsType, 
isect_memberEquality_alt, 
isectIsTypeImplies, 
universeIsType, 
instantiate, 
universeEquality
Latex:
\mforall{}[X:Type].  \mforall{}[d:metric(X)].  \mforall{}[x,y:X].    (r0  \mleq{}  mdist(d;x;y))
Date html generated:
2019_10_29-AM-10_57_36
Last ObjectModification:
2019_10_02-AM-09_39_27
Theory : reals
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