Nuprl Lemma : int_mod_isect_int_mod
∀[n,m:ℕ+]. ℤ_n ⋂ ℤ_m ≡ ℤ_lcm(n;m)
Proof
Definitions occuring in Statement :
int_mod: ℤ_n,
lcm: lcm(a;b),
isect2: T1 ⋂ T2,
nat_plus: ℕ+,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_mod: ℤ_n,
so_lambda: λ2x y.t[x; y],
nat_plus: ℕ+,
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
and: P ∧ Q,
implies: P ⇒ Q,
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
quotient: x,y:A//B[x; y],
cand: A c∧ B,
divides: b | a,
exists: ∃x:A. B[x],
eqmod: a ≡ b mod m,
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
Lemmas referenced :
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
nat_plus_properties,
int_subtype_base,
subtype_base_sq,
equal_wf,
lcm-property,
subtract_wf,
lcm-is-lcm,
and_wf,
equal-wf-base,
quotient-member-eq,
nat_plus_wf,
lcm_wf,
equiv_rel_and,
quotient_wf,
isect2_wf,
ext-eq_transitivity,
eqmod_equiv_rel,
eqmod_wf,
isect2_quotient
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
because_Cache,
productEquality,
independent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
independent_pairFormation,
pointwiseFunctionalityForEquality,
pertypeElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
multiplyEquality,
promote_hyp,
instantiate,
cumulativity,
unionElimination,
natural_numberEquality,
int_eqEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[n,m:\mBbbN{}\msupplus{}]. \mBbbZ{}\_n \mcap{} \mBbbZ{}\_m \mequiv{} \mBbbZ{}\_lcm(n;m)
Date html generated:
2016_05_14-PM-09_27_35
Last ObjectModification:
2016_01_14-PM-11_32_08
Theory : num_thy_1
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