Nuprl Lemma : rmax_lb
∀[x,y,z:ℝ]. uiff((x ≤ z) ∧ (y ≤ z);rmax(x;y) ≤ z)
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rmax: rmax(x;y),
real: ℝ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
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,
guard: {T},
rsub: x - y,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
less_than'_wf,
rsub_wf,
rmax_wf,
real_wf,
nat_plus_wf,
and_wf,
rleq_wf,
rleq-rmax,
rleq_transitivity,
rminus_wf,
rmin_wf,
req_weakening,
req_functionality,
req_transitivity,
rmin-req-rminus-rmax,
rminus_functionality,
rmax_functionality,
rminus-rminus,
radd_wf,
rnonneg_functionality,
radd_functionality,
radd-rmin,
rmin-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
independent_pairEquality,
because_Cache,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
voidElimination,
isect_memberEquality,
independent_functionElimination
Latex:
\mforall{}[x,y,z:\mBbbR{}]. uiff((x \mleq{} z) \mwedge{} (y \mleq{} z);rmax(x;y) \mleq{} z)
Date html generated:
2016_05_18-AM-07_16_23
Last ObjectModification:
2015_12_28-AM-00_43_36
Theory : reals
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