Nuprl Lemma : rleq-iff-rleq2
∀x,y:ℝ. (x ≤ y ⇐⇒ rleq2(x;y))
Proof
Definitions occuring in Statement :
rleq2: rleq2(x;y),
rleq: x ≤ y,
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
rleq2: rleq2(x;y),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
real: ℝ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
int_upper: {i...},
le: A ≤ B,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
exists: ∃x:A. B[x]
Lemmas referenced :
rleq2_wf,
rleq-iff,
rleq_wf,
iff_wf,
all_wf,
nat_plus_wf,
exists_wf,
int_upper_wf,
le_wf,
subtract_wf,
less_than_transitivity1,
less_than_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
lambdaEquality,
multiplyEquality,
minusEquality,
natural_numberEquality,
applyEquality,
dependent_set_memberEquality,
independent_isectElimination
Latex:
\mforall{}x,y:\mBbbR{}. (x \mleq{} y \mLeftarrow{}{}\mRightarrow{} rleq2(x;y))
Date html generated:
2016_05_18-AM-07_15_27
Last ObjectModification:
2015_12_28-AM-00_42_27
Theory : reals
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