Nuprl Lemma : lcm-is-lcm-nat
∀n,m:ℕ. (((n | lcm(n;m)) ∧ (m | lcm(n;m))) ∧ (∀v:ℤ. ((n | v) ⇒ (m | v) ⇒ (lcm(n;m) | v))))
Proof
Definitions occuring in Statement :
lcm: lcm(a;b),
divides: b | a,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
lcm: lcm(a;b),
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
subtract: n - m
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
nat_wf,
any_divs_zero,
divides_wf,
value-type-has-value,
int-value-type,
set-value-type,
le_wf,
gcd_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lcm-is-lcm,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
productElimination,
sqequalRule,
callbyvalueReduce,
lambdaEquality,
equalityElimination,
voidElimination,
dependent_pairFormation,
promote_hyp,
dependent_set_memberEquality,
addEquality,
applyEquality,
isect_memberEquality,
voidEquality,
minusEquality
Latex:
\mforall{}n,m:\mBbbN{}. (((n | lcm(n;m)) \mwedge{} (m | lcm(n;m))) \mwedge{} (\mforall{}v:\mBbbZ{}. ((n | v) {}\mRightarrow{} (m | v) {}\mRightarrow{} (lcm(n;m) | v))))
Date html generated:
2017_04_17-AM-09_46_43
Last ObjectModification:
2017_02_27-PM-05_40_50
Theory : num_thy_1
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