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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