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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