### 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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