Nuprl Lemma : rmin-rleq
∀[x,y:ℝ]. ((rmin(x;y) ≤ x) ∧ (rmin(x;y) ≤ y))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rmin: rmin(x;y),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
uimplies: b supposing a,
rge: x ≥ y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
less_than'_wf,
rsub_wf,
rmin_wf,
real_wf,
nat_plus_wf,
rminus_wf,
rmax_wf,
rminus_functionality_wrt_rleq,
rleq-rmax,
rleq_functionality,
rmin-req-rminus-rmax,
req_weakening,
req_inversion,
rminus-rminus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
lemma_by_obid,
isectElimination,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_isectElimination
Latex:
\mforall{}[x,y:\mBbbR{}]. ((rmin(x;y) \mleq{} x) \mwedge{} (rmin(x;y) \mleq{} y))
Date html generated:
2016_05_18-AM-07_16_35
Last ObjectModification:
2015_12_28-AM-00_43_19
Theory : reals
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