Nuprl Lemma : rmin_ub
∀x,y,z:ℝ. ((z ≤ x) ∧ (z ≤ y) ⇐⇒ z ≤ rmin(x;y))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rmin: rmin(x;y),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rsub: x - y,
uimplies: b supposing a,
cand: A c∧ B
Lemmas referenced :
and_wf,
rleq_wf,
rmin_wf,
real_wf,
rnonneg_functionality,
radd_wf,
rminus_wf,
radd_comm,
radd-rmin,
rmin_functionality,
rmin-nonneg,
rmin-rleq,
rleq_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
because_Cache
Latex:
\mforall{}x,y,z:\mBbbR{}. ((z \mleq{} x) \mwedge{} (z \mleq{} y) \mLeftarrow{}{}\mRightarrow{} z \mleq{} rmin(x;y))
Date html generated:
2016_05_18-AM-07_16_49
Last ObjectModification:
2015_12_28-AM-00_44_15
Theory : reals
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