Nuprl Lemma : Jacobi-identity
∀[r:CRng]. ∀[a,b,c:ℕ3 ⟶ |r|]. (((a x (b x c)) + ((b x (c x a)) + (c x (a x b)))) = 0 ∈ (ℕ3 ⟶ |r|))
Proof
Definitions occuring in Statement :
cross-product: (a x b),
zero-vector: 0,
vector-add: (a + b),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
zero-vector: 0,
cross-product: (a x b),
vector-add: (a + b),
select: L[n],
cons: [a / b],
subtract: n - m,
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
crng: CRng,
rng: Rng,
true: True,
squash: ↓T,
infix_ap: x f y,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
int_seg_subtype_special,
int_seg_cases,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_seg_wf,
rng_car_wf,
crng_wf,
rng_plus_wf,
infix_ap_wf,
rng_times_wf,
rng_minus_wf,
rng_zero_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rng_times_over_plus,
rng_times_over_minus,
subtype_rel_self,
crng_times_comm,
crng_times_ac_1,
rng_minus_over_plus,
rng_minus_minus,
rng_plus_assoc,
rng_plus_ac_1,
rng_plus_comm,
rng_plus_inv,
rng_plus_inv_assoc,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
functionExtensionality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
applyEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
hypothesis_subsumption,
productElimination,
int_eqEquality,
functionIsType,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[r:CRng]. \mforall{}[a,b,c:\mBbbN{}3 {}\mrightarrow{} |r|]. (((a x (b x c)) + ((b x (c x a)) + (c x (a x b)))) = 0)
Date html generated:
2019_10_16-AM-11_28_44
Last ObjectModification:
2018_12_08-PM-00_16_52
Theory : matrices
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