Nuprl Lemma : le_transitivity
∀[x,y,z:ℤ]. (x ≤ z) supposing ((y ≤ z) and (x ≤ y))
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
prop: ℙ,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
squash: ↓T,
subtract: n - m,
top: Top,
true: True
Lemmas referenced :
le-iff-nonneg,
add-nonneg,
subtract_wf,
less_than'_wf,
le_wf,
true_wf,
squash_wf,
zero-add,
zero-mul,
add-mul-special,
add-swap,
add-associates,
add-commutes,
minus-one-mul
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
dependent_functionElimination,
because_Cache,
intEquality,
isect_memberFormation,
sqequalRule,
independent_pairEquality,
lambdaEquality,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
baseClosed,
imageMemberEquality,
imageElimination,
applyEquality,
hyp_replacement,
natural_numberEquality,
minusEquality,
multiplyEquality,
addEquality,
voidEquality
Latex:
\mforall{}[x,y,z:\mBbbZ{}]. (x \mleq{} z) supposing ((y \mleq{} z) and (x \mleq{} y))
Date html generated:
2019_06_20-AM-11_22_59
Last ObjectModification:
2018_08_01-PM-04_17_48
Theory : arithmetic
Home
Index