Nuprl Lemma : assoc_reln
∀a,b:ℤ. ((a | b) ∧ (b | a) ⇐⇒ a = ± b)
Proof
Definitions occuring in Statement :
divides: b | a,
pm_equal: i = ± j,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
pm_equal: i = ± j,
or: P ∨ Q,
divides: b | a,
exists: ∃x:A. B[x],
decidable: Dec(P),
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
subtype_rel: A ⊆r B
Lemmas referenced :
divides_wf,
pm_equal_wf,
istype-int,
divides_anti_sym,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
int_subtype_base,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
Error :productIsType,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
Error :dependent_pairFormation_alt,
natural_numberEquality,
because_Cache,
independent_isectElimination,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :equalityIsType4,
applyEquality,
multiplyEquality,
minusEquality
Latex:
\mforall{}a,b:\mBbbZ{}. ((a | b) \mwedge{} (b | a) \mLeftarrow{}{}\mRightarrow{} a = \mpm{} b)
Date html generated:
2019_06_20-PM-02_20_16
Last ObjectModification:
2018_10_03-AM-00_35_37
Theory : num_thy_1
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