Nuprl Lemma : mod2-2n
∀n:ℤ. (((2 * n) mod 2) = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
modulus: a mod n,
all: ∀x:A. B[x],
multiply: 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,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
modulus: a mod n,
remainder: n rem m,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
top: Top,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
less_than: a < b,
sq_type: SQType(T)
Lemmas referenced :
zero-add,
equal_wf,
squash_wf,
true_wf,
istype-universe,
mod2-add2n,
subtype_rel_self,
iff_weakening_equal,
istype-nat,
decidable__le,
istype-int,
istype-le,
le_weakening2,
minus-one-mul,
not-lt-2,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
minus-one-mul-top,
istype-void,
minus-zero,
add-zero,
one-mul,
add-commutes,
add-mul-special,
zero-mul,
not-le-2,
istype-false,
add-associates,
add-swap,
omega-shadow,
istype-less_than,
decidable__lt,
subtype_base_sq,
int_subtype_base,
mul-swap,
mul-associates,
mul-distributes-right
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
Error :inhabitedIsType,
instantiate,
universeEquality,
intEquality,
dependent_functionElimination,
because_Cache,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
unionElimination,
Error :dependent_set_memberEquality_alt,
minusEquality,
Error :isect_memberEquality_alt,
voidElimination,
addEquality,
independent_pairFormation,
cumulativity
Latex:
\mforall{}n:\mBbbZ{}. (((2 * n) mod 2) = 0)
Date html generated:
2019_06_20-AM-11_26_14
Last ObjectModification:
2019_03_15-PM-05_58_07
Theory : arithmetic
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