Nuprl Lemma : subtract-is-int-iff
∀[a,b:Base]. uiff(a - b ∈ ℤ;(a ∈ ℤ) ∧ (b ∈ ℤ))
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
subtract: n - m,
int: ℤ,
base: Base
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtract: n - m,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
minus-is-int-iff,
add-is-int-iff,
iff_weakening_uiff,
iff_transitivity,
uiff_wf,
base_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
lemma_by_obid,
intEquality,
baseApply,
closedConclusion,
baseClosed,
because_Cache,
productEquality,
independent_pairFormation,
instantiate,
cumulativity,
addLevel,
independent_isectElimination,
independent_functionElimination,
lambdaFormation
Latex:
\mforall{}[a,b:Base]. uiff(a - b \mmember{} \mBbbZ{};(a \mmember{} \mBbbZ{}) \mwedge{} (b \mmember{} \mBbbZ{}))
Date html generated:
2016_05_13-PM-03_28_37
Last ObjectModification:
2016_01_14-PM-06_42_07
Theory : arithmetic
Home
Index