Nuprl Lemma : metric-leq-meq
∀[X:Type]. ∀[d1,d2:metric(X)]. (d1 ≤ d2 ⇒ (∀x,y:X. (x ≡ y ⇒ x ≡ y)))
Proof
Definitions occuring in Statement :
metric-leq: d1 ≤ d2,
meq: x ≡ y,
metric: metric(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
meq: x ≡ y,
mdist: mdist(d;x;y),
metric-leq: d1 ≤ d2,
prop: ℙ,
metric: metric(X),
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
meq_wf,
metric-leq_wf,
req_witness,
int-to-real_wf,
metric_wf,
istype-universe,
mdist-nonneg,
rleq_antisymmetry,
rleq_transitivity,
mdist_wf,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
universeIsType,
extract_by_obid,
isectElimination,
inhabitedIsType,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setElimination,
rename,
natural_numberEquality,
independent_functionElimination,
functionIsTypeImplies,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
instantiate,
universeEquality,
independent_isectElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d1,d2:metric(X)]. (d1 \mleq{} d2 {}\mRightarrow{} (\mforall{}x,y:X. (x \mequiv{} y {}\mRightarrow{} x \mequiv{} y)))
Date html generated:
2019_10_29-AM-11_07_32
Last ObjectModification:
2019_10_02-AM-09_48_56
Theory : reals
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