### Nuprl Lemma : multiply_nat_plus_iff

`∀[i:ℕ+]. ∀[x:ℤ].  (i * x ∈ ℕ `⇐⇒` x ∈ ℕ)`

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` nat: `ℕ` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` member: `t ∈ T` multiply: `n * m` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` rev_implies: `P `` Q` nat: `ℕ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` exists: `∃x:A. B[x]` top: `Top` ge: `i ≥ j ` subtract: `n - m` uiff: `uiff(P;Q)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` le: `A ≤ B` not: `¬A` false: `False`
Lemmas referenced :  istype-nat int_subtype_base set_subtype_base less_than_wf mul_bounds_1a nat_plus_subtype_nat istype-le istype-int nat_plus_wf nat_plus_properties nat_properties decidable__le mul_bounds_1b istype-less_than minus-one-mul subtract_wf subtype_base_sq istype-sqequal le_weakening2 mul-swap istype-void add_functionality_wrt_le le_reflexive minus-one-mul-top zero-add one-mul add-mul-special add-associates two-mul add-commutes mul-distributes-right zero-mul less-iff-le mul-associates add-swap add-zero omega-shadow not-lt-2 minus-zero not-le-2 istype-false decidable__lt
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut independent_pairFormation Error :lambdaFormation_alt,  sqequalHypSubstitution hypothesis sqequalRule Error :equalityIstype,  extract_by_obid baseApply closedConclusion baseClosed hypothesisEquality applyEquality isectElimination thin intEquality Error :lambdaEquality_alt,  natural_numberEquality Error :inhabitedIsType,  independent_isectElimination because_Cache sqequalBase equalitySymmetry Error :dependent_set_memberEquality_alt,  multiplyEquality setElimination rename equalityTransitivity productElimination independent_pairEquality dependent_functionElimination axiomEquality Error :functionIsTypeImplies,  Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  Error :universeIsType,  applyLambdaEquality unionElimination minusEquality addEquality instantiate cumulativity independent_functionElimination Error :dependent_pairFormation_alt,  promote_hyp voidElimination imageMemberEquality

Latex:
\mforall{}[i:\mBbbN{}\msupplus{}].  \mforall{}[x:\mBbbZ{}].    (i  *  x  \mmember{}  \mBbbN{}  \mLeftarrow{}{}\mRightarrow{}  x  \mmember{}  \mBbbN{})

Date html generated: 2019_06_20-AM-11_26_47
Last ObjectModification: 2018_12_11-PM-01_01_39

Theory : arithmetic

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