Nuprl Lemma : lcm-is-lcm
∀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_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
divides: b | a,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
squash: ↓T,
true: True,
rev_implies: P ⇐ Q
Lemmas referenced :
lcm-property,
divides_wf,
nat_plus_wf,
equal_wf,
lcm_wf,
coprime_bezout_id,
divides_add,
equal-wf-base-T,
int_subtype_base,
subtype_base_sq,
nat_plus_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
mul_assoc,
iff_weakening_equal,
mul_com,
mul_add_distrib,
itermConstant_wf,
int_term_value_constant_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
independent_pairFormation,
because_Cache,
isectElimination,
intEquality,
dependent_pairFormation,
equalitySymmetry,
multiplyEquality,
independent_functionElimination,
equalityTransitivity,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
instantiate,
cumulativity,
independent_isectElimination,
unionElimination,
natural_numberEquality,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
hyp_replacement,
imageElimination,
imageMemberEquality,
universeEquality
Latex:
\mforall{}n,m:\mBbbN{}\msupplus{}. (((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_37
Last ObjectModification:
2017_02_27-PM-05_41_00
Theory : num_thy_1
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