Nuprl Lemma : three-cubes-mod-9
∀a,b,c,k:ℤ. (((a^3 + b^3 + c^3) = k ∈ ℤ) ⇒ ((¬(k ≡ 4 mod 9)) ∧ (¬(k ≡ 5 mod 9))))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
exp: i^n,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
or: P ∨ Q,
and: P ∧ Q,
not: ¬A,
modulus: a mod n,
remainder: n rem m,
true: True,
false: False,
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
absval: |i|
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
istype-int,
cube-mod-9,
modulus-equal-iff-eqmod,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
eqmod_wf,
exp_wf2,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
eqmod_functionality_wrt_eqmod,
add_functionality_wrt_eqmod,
eqmod_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
independent_functionElimination,
Error :equalityIstype,
because_Cache,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
Error :inhabitedIsType,
unionElimination,
independent_pairFormation,
natural_numberEquality,
voidElimination,
productElimination,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
promote_hyp,
addEquality,
minusEquality
Latex:
\mforall{}a,b,c,k:\mBbbZ{}. (((a\^{}3 + b\^{}3 + c\^{}3) = k) {}\mRightarrow{} ((\mneg{}(k \mequiv{} 4 mod 9)) \mwedge{} (\mneg{}(k \mequiv{} 5 mod 9))))
Date html generated:
2019_06_20-PM-02_26_14
Last ObjectModification:
2019_03_18-PM-00_04_05
Theory : num_thy_1
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