Nuprl Lemma : rleq-iff-all-rless
∀[x,y:ℝ]. uiff(x ≤ y;∀e:{e:ℝ| r0 < e} . (x ≤ (y + e)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
guard: {T},
squash: ↓T,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
rsub: x - y
Lemmas referenced :
set_wf,
real_wf,
rless_wf,
int-to-real_wf,
less_than'_wf,
rsub_wf,
radd_wf,
nat_plus_wf,
rleq_wf,
all_wf,
radd-preserves-rleq,
rminus_wf,
rmul_wf,
sq_stable__rleq,
rleq_weakening_rless,
uiff_transitivity,
rleq_functionality,
req_transitivity,
radd_functionality,
req_weakening,
rminus-as-rmul,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-assoc,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both,
rleq_transitivity,
rleq-iff-not-rless,
rdiv_wf,
rless-int,
rmul_preserves_rless,
rless_functionality,
rmul-rdiv-cancel2,
rmul-int,
radd-preserves-rless,
radd-rminus-both,
radd_comm,
radd-ac,
rmul_preserves_rleq2,
rleq-int,
false_wf,
rmul-distrib,
rmul_comm,
rmul-rdiv-cancel,
rmul-one-both,
rless_transitivity1,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
voidElimination,
isect_memberEquality,
independent_isectElimination,
addEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
inrFormation,
multiplyEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}[x,y:\mBbbR{}]. uiff(x \mleq{} y;\mforall{}e:\{e:\mBbbR{}| r0 < e\} . (x \mleq{} (y + e)))
Date html generated:
2016_10_26-AM-09_09_40
Last ObjectModification:
2016_08_15-PM-09_02_25
Theory : reals
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