Nuprl Lemma : rmin_lb
∀[x,y,z:ℝ].  rmin(x;y) ≤ z supposing (x ≤ z) ∨ (y ≤ z)
Proof
Definitions occuring in Statement : 
rleq: x ≤ y
, 
rmin: rmin(x;y)
, 
real: ℝ
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
or: P ∨ Q
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
or: P ∨ Q
, 
and: P ∧ Q
, 
guard: {T}
, 
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
Lemmas referenced : 
rmin-rleq, 
rleq_transitivity, 
rmin_wf, 
less_than'_wf, 
rsub_wf, 
real_wf, 
nat_plus_wf, 
or_wf, 
rleq_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
unionElimination, 
productElimination, 
hypothesis, 
independent_isectElimination, 
sqequalRule, 
lambdaEquality, 
dependent_functionElimination, 
independent_pairEquality, 
because_Cache, 
applyEquality, 
setElimination, 
rename, 
minusEquality, 
natural_numberEquality, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
isect_memberEquality, 
voidElimination
Latex:
\mforall{}[x,y,z:\mBbbR{}].    rmin(x;y)  \mleq{}  z  supposing  (x  \mleq{}  z)  \mvee{}  (y  \mleq{}  z)
Date html generated:
2016_05_18-AM-07_16_55
Last ObjectModification:
2015_12_28-AM-00_44_03
Theory : reals
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