Nuprl Lemma : mod2-is-zero
∀x:ℤ. ((x mod 2) = 0 ∈ ℤ 
⇐⇒ ∃n:ℤ. (x = (2 * n) ∈ ℤ))
Proof
Definitions occuring in Statement : 
modulus: a mod n
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
multiply: n * m
, 
natural_number: $n
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
squash: ↓T
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
subtract: n - m
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
or: P ∨ Q
, 
decidable: Dec(P)
, 
lelt: i ≤ j < k
, 
int_seg: {i..j-}
, 
subtype_rel: A ⊆r B
, 
absval: |i|
, 
less_than': less_than'(a;b)
, 
le: A ≤ B
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
nat: ℕ
, 
and: P ∧ Q
, 
uiff: uiff(P;Q)
, 
int_nzero: ℤ-o
, 
prop: ℙ
, 
false: False
, 
guard: {T}
, 
sq_type: SQType(T)
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
nequal: a ≠ b ∈ T 
, 
true: True
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
has-value: (a)↓
, 
modulus: a mod n
, 
all: ∀x:A. B[x]
Lemmas referenced : 
rem-exact, 
mul-swap, 
mul-distributes, 
zero-mul, 
mul-distributes-right, 
mul-associates, 
subtract_wf, 
iff_weakening_equal, 
subtype_rel_self, 
squash_wf, 
iff_wf, 
equal_wf, 
less_than_irreflexivity, 
less_than_transitivity1, 
le-add-cancel, 
mul-commutes, 
int_seg_cases, 
le-add-cancel2, 
add-zero, 
subtype_rel_sets, 
exists_wf, 
decidable__int_equal, 
int_seg_wf, 
less_than_wf, 
and_wf, 
le-add-cancel-alt, 
zero-add, 
add-swap, 
add_functionality_wrt_le, 
add-commutes, 
minus-one-mul-top, 
add-associates, 
minus-one-mul, 
minus-add, 
condition-implies-le, 
less-iff-le, 
not-le-2, 
decidable__le, 
absval_pos, 
false_wf, 
le_wf, 
set_subtype_base, 
nat_wf, 
absval_wf, 
absval_strict_ubound, 
rem_bounds_absval, 
nequal_wf, 
div_rem_sum, 
true_wf, 
equal-wf-base, 
int_subtype_base, 
subtype_base_sq, 
int-value-type, 
value-type-has-value
Rules used in proof : 
imageMemberEquality, 
universeEquality, 
imageElimination, 
divideEquality, 
dependent_pairFormation, 
setEquality, 
hypothesis_subsumption, 
promote_hyp, 
levelHypothesis, 
voidEquality, 
isect_memberEquality, 
addEquality, 
unionElimination, 
minusEquality, 
multiplyEquality, 
closedConclusion, 
baseApply, 
rename, 
setElimination, 
applyEquality, 
independent_pairFormation, 
lambdaEquality, 
productElimination, 
because_Cache, 
dependent_set_memberEquality, 
baseClosed, 
voidElimination, 
independent_functionElimination, 
equalitySymmetry, 
equalityTransitivity, 
dependent_functionElimination, 
cumulativity, 
instantiate, 
addLevel, 
natural_numberEquality, 
hypothesisEquality, 
remainderEquality, 
hypothesis, 
independent_isectElimination, 
intEquality, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
callbyvalueReduce, 
sqequalRule, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}x:\mBbbZ{}.  ((x  mod  2)  =  0  \mLeftarrow{}{}\mRightarrow{}  \mexists{}n:\mBbbZ{}.  (x  =  (2  *  n)))
Date html generated:
2018_07_25-PM-01_27_54
Last ObjectModification:
2018_06_27-PM-04_13_38
Theory : arithmetic
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