Nuprl Lemma : icompact-endpoints-rleq
∀[I:Interval]. left-endpoint(I) ≤ right-endpoint(I) supposing icompact(I)
Proof
Definitions occuring in Statement :
icompact: icompact(I),
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
interval: Interval,
rleq: x ≤ y,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
i-length: |I|,
uiff: uiff(P;Q),
and: P ∧ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
icompact: icompact(I),
rsub: x - y
Lemmas referenced :
icompact-length-nonneg,
radd-preserves-rleq,
int-to-real_wf,
rsub_wf,
right-endpoint_wf,
left-endpoint_wf,
icompact_wf,
less_than'_wf,
nat_plus_wf,
interval_wf,
rleq_wf,
radd_wf,
rminus_wf,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd_functionality,
req_weakening,
radd-rminus-assoc,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
natural_numberEquality,
because_Cache,
productElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
applyEquality,
minusEquality,
independent_functionElimination
Latex:
\mforall{}[I:Interval]. left-endpoint(I) \mleq{} right-endpoint(I) supposing icompact(I)
Date html generated:
2016_05_18-AM-08_47_09
Last ObjectModification:
2015_12_27-PM-11_47_48
Theory : reals
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