Nuprl Lemma : rem_to_div
∀[a:ℤ]. ∀[n:ℤ-o]. ((a rem n) = (a - (a ÷ n) * n) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
remainder: n rem m,
divide: n ÷ m,
multiply: n * m,
subtract: n - m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
subtract: n - m
Lemmas referenced :
div_rem_sum,
equal_wf,
minus-one-mul,
mul-commutes,
int_nzero_wf,
add-associates,
add-commutes,
minus-one-mul-top,
add-swap,
add-mul-special,
zero-mul,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
hypothesis,
minusEquality,
multiplyEquality,
divideEquality,
setElimination,
rename,
lambdaFormation,
independent_functionElimination,
voidElimination,
intEquality,
natural_numberEquality,
because_Cache,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
remainderEquality,
equalitySymmetry,
axiomEquality
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}]. ((a rem n) = (a - (a \mdiv{} n) * n))
Date html generated:
2016_05_13-PM-03_35_22
Last ObjectModification:
2015_12_26-AM-09_43_10
Theory : arithmetic
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