Nuprl Lemma : three-cubes-6-mod-9
∀a,b,c,k:ℤ.
(((a^3 + b^3 + c^3) = k ∈ ℤ) ⇒ (k ≡ 6 mod 9) ⇒ ((a^3 ≡ (-1) mod 9) ∧ (b^3 ≡ (-1) mod 9) ∧ (c^3 ≡ (-1) mod 9)))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
exp: i^n,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
add: n + m,
minus: -n,
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},
or: P ∨ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
modulus: a mod n,
remainder: n rem m,
true: True,
absval: |i|,
cand: A c∧ B
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
cube-mod-9,
eqmod_wf,
istype-int,
exp_wf2,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
modulus-equal-iff-eqmod,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
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,
unionElimination,
Error :universeIsType,
natural_numberEquality,
Error :equalityIstype,
because_Cache,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
Error :inhabitedIsType,
addEquality,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
voidElimination,
productElimination,
independent_pairFormation,
minusEquality
Latex:
\mforall{}a,b,c,k:\mBbbZ{}.
(((a\^{}3 + b\^{}3 + c\^{}3) = k)
{}\mRightarrow{} (k \mequiv{} 6 mod 9)
{}\mRightarrow{} ((a\^{}3 \mequiv{} (-1) mod 9) \mwedge{} (b\^{}3 \mequiv{} (-1) mod 9) \mwedge{} (c\^{}3 \mequiv{} (-1) mod 9)))
Date html generated:
2019_06_20-PM-02_26_23
Last ObjectModification:
2019_03_18-PM-00_14_26
Theory : num_thy_1
Home
Index