Nuprl Lemma : minus_functionality_wrt_le
∀[i,j:ℤ]. (-i) ≤ (-j) supposing i ≥ j
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
le: A ≤ B,
minus: -n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
ge: i ≥ j ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
minus-one-mul,
less_than'_wf,
ge_wf,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
minus-one-mul-top,
one-mul,
add-swap,
add-commutes,
add-associates,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
zero-add,
not-le-2,
mul-associates,
add-zero,
omega-shadow,
less_than_wf,
mul-distributes,
mul-commutes,
le-add-cancel-alt,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
intEquality,
voidElimination,
multiplyEquality,
natural_numberEquality,
independent_isectElimination,
applyEquality,
voidEquality,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination
Latex:
\mforall{}[i,j:\mBbbZ{}]. (-i) \mleq{} (-j) supposing i \mgeq{} j
Date html generated:
2019_06_20-AM-11_26_31
Last ObjectModification:
2018_09_18-PM-00_24_40
Theory : arithmetic
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