Nuprl Lemma : rem-exact
∀[g:ℤ-o]. ∀[v:ℤ]. ((g * v rem g) = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
remainder: n rem m,
multiply: n * m,
natural_number: $n,
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,
true: True,
top: Top,
squash: ↓T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
div_rem_sum,
int_nzero_wf,
equal_wf,
mul-commutes,
squash_wf,
true_wf,
add_functionality_wrt_eq,
divide-exact,
iff_weakening_equal,
add-associates,
minus-one-mul,
add-commutes,
add-mul-special,
add-swap,
zero-mul,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
intEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
divideEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
independent_functionElimination,
voidElimination,
natural_numberEquality,
remainderEquality,
addEquality,
voidEquality,
minusEquality,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
independent_isectElimination,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[g:\mBbbZ{}\msupminus{}\msupzero{}]. \mforall{}[v:\mBbbZ{}]. ((g * v rem g) = 0)
Date html generated:
2017_04_14-AM-07_20_11
Last ObjectModification:
2017_02_27-PM-02_53_43
Theory : arithmetic
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