Nuprl Lemma : i-member-diff-bound
∀I:Interval. (icompact(I) ⇒ (∀x,y:{x:ℝ| x ∈ I} . (|x - y| ≤ |I|)))
Proof
Definitions occuring in Statement :
icompact: icompact(I),
i-member: r ∈ I,
i-length: |I|,
interval: Interval,
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rbetween: x≤y≤z,
i-length: |I|,
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
icompact: icompact(I),
rev_implies: P ⇐ Q,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
rsub: x - y,
uiff: uiff(P;Q)
Lemmas referenced :
sq_stable__rleq,
rabs_wf,
rsub_wf,
i-length_wf,
i-member-compact,
left-endpoint_wf,
real_wf,
right-endpoint_wf,
rleq_wf,
equal_wf,
set_wf,
i-member_wf,
icompact_wf,
interval_wf,
rabs-difference-bound-rleq,
radd_wf,
rmul_wf,
int-to-real_wf,
rminus_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
uiff_transitivity,
rleq_functionality,
radd_functionality,
rminus-radd,
req_weakening,
radd-ac,
req_transitivity,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
rminus-as-rmul,
radd-int,
rmul-zero-both,
rminus-rminus,
radd_comm,
radd-zero-both,
radd_functionality_wrt_rleq,
radd-rminus-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
productElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
independent_pairFormation,
minusEquality,
natural_numberEquality,
addEquality
Latex:
\mforall{}I:Interval. (icompact(I) {}\mRightarrow{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . (|x - y| \mleq{} |I|)))
Date html generated:
2017_10_03-AM-09_34_26
Last ObjectModification:
2017_07_28-AM-07_52_16
Theory : reals
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