Nuprl Lemma : mod2-add2n
∀k:ℤ. ∀n:ℕ. (((k + (2 * n)) mod 2) = (k mod 2) ∈ ℤ)
Proof
Definitions occuring in Statement :
modulus: a mod n,
nat: ℕ,
all: ∀x:A. B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
squash: ↓T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
mul-commutes,
add-zero,
modulus_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
nat_wf,
mul-distributes,
equal_wf,
squash_wf,
mod2-add2,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
applyEquality,
because_Cache,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
isect_memberEquality,
voidEquality,
minusEquality,
multiplyEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}k:\mBbbZ{}. \mforall{}n:\mBbbN{}. (((k + (2 * n)) mod 2) = (k mod 2))
Date html generated:
2018_07_25-PM-01_27_46
Last ObjectModification:
2018_06_08-PM-06_58_33
Theory : arithmetic
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