Nuprl Lemma : mdist-triangle-inequality1
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y,z:X]. (mdist(d;x;z) ≤ (mdist(d;x;y) + mdist(d;z;y)))
Proof
Definitions occuring in Statement :
mdist: mdist(d;x;y),
metric: metric(X),
rleq: x ≤ y,
radd: a + b,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
mdist: mdist(d;x;y),
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,
radd_wf,
le_witness_for_triv,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
productElimination,
sqequalRule,
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,z:X]. (mdist(d;x;z) \mleq{} (mdist(d;x;y) + mdist(d;z;y)))
Date html generated:
2019_10_29-AM-10_58_16
Last ObjectModification:
2019_10_02-AM-09_40_02
Theory : reals
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