Nuprl Lemma : rleq_transitivity
∀[x,y,z:ℝ]. (x ≤ z) supposing ((y ≤ z) and (x ≤ y))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rleq: x ≤ y,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rsub: x - y
Lemmas referenced :
rnonneg-radd,
rsub_wf,
less_than'_wf,
real_wf,
nat_plus_wf,
rnonneg_wf,
radd_wf,
rminus_wf,
rnonneg_functionality,
radd_comm,
req_inversion,
radd-assoc,
req_transitivity,
radd-ac,
radd_functionality,
req_weakening,
radd-rminus-assoc
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
lambdaEquality,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
addLevel,
independent_isectElimination,
promote_hyp
Latex:
\mforall{}[x,y,z:\mBbbR{}]. (x \mleq{} z) supposing ((y \mleq{} z) and (x \mleq{} y))
Date html generated:
2016_05_18-AM-07_05_47
Last ObjectModification:
2015_12_28-AM-00_36_31
Theory : reals
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