Nuprl Lemma : product-eq-0-mod-prime
∀p,a,b:ℤ. (prime(p) ⇒ ((a * b) ≡ 0 mod p) ⇒ ((a ≡ 0 mod p) ∨ (b ≡ 0 mod p)))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
prime: prime(a),
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
decidable: Dec(P),
member: t ∈ T,
exists: ∃x:A. B[x],
divides: b | a,
eqmod: a ≡ b mod m,
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
prime_wf,
eqmod_wf,
int_subtype_base,
equal-wf-base,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
prime_divs_prod
Rules used in proof :
multiplyEquality,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
isectElimination,
unionElimination,
hypothesis,
because_Cache,
dependent_functionElimination,
extract_by_obid,
introduction,
hypothesisEquality,
dependent_pairFormation,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inrFormation,
inlFormation
Latex:
\mforall{}p,a,b:\mBbbZ{}. (prime(p) {}\mRightarrow{} ((a * b) \mequiv{} 0 mod p) {}\mRightarrow{} ((a \mequiv{} 0 mod p) \mvee{} (b \mequiv{} 0 mod p)))
Date html generated:
2018_05_21-PM-00_56_03
Last ObjectModification:
2018_01_01-PM-00_04_06
Theory : num_thy_1
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