Nuprl Lemma : cross-product-equal-zero
∀r:IntegDom{i}. ∀a,b:ℕ3 ⟶ |r|.
((∀x,y:|r|. Dec(x = y ∈ |r|))
⇒ ((a x b) = 0 ∈ (ℕ3 ⟶ |r|)
⇐⇒ (a = 0 ∈ (ℕ3 ⟶ |r|))
∨ (b = 0 ∈ (ℕ3 ⟶ |r|))
∨ (∃i:ℕ3. ((¬((b i) = 0 ∈ |r|)) ∧ (¬((a i) = 0 ∈ |r|)) ∧ ((b i*a) = (a i*b) ∈ (ℕ3 ⟶ |r|))))))
Proof
Definitions occuring in Statement :
cross-product: (a x b),
zero-vector: 0,
vector-mul: (c*a),
int_seg: {i..j-},
decidable: Dec(P),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
integ_dom: IntegDom{i},
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
integ_dom: IntegDom{i},
crng: CRng,
rng: Rng,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
infix_ap: x f y,
squash: ↓T,
zero-vector: 0,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
cross-product: (a x b),
select: L[n],
cons: [a / b],
subtract: n - m,
sq_stable: SqStable(P),
integ_dom_p: IsIntegDom(r),
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
cand: A c∧ B,
vector-mul: (c*a),
uiff: uiff(P;Q),
ringeq_int_terms: t1 ≡ t2,
nequal: a ≠ b ∈ T
Lemmas referenced :
equal_wf,
int_seg_wf,
rng_car_wf,
cross-product_wf,
zero-vector_wf,
or_wf,
exists_wf,
not_wf,
rng_zero_wf,
vector-mul_wf,
all_wf,
decidable_wf,
integ_dom_wf,
rng_plus_wf,
rng_minus_wf,
trivial-equal,
iff_weakening_equal,
false_wf,
lelt_wf,
infix_ap_wf,
rng_times_wf,
squash_wf,
true_wf,
rng_plus_assoc,
subtype_rel_self,
rng_plus_comm,
rng_plus_inv,
rng_plus_zero,
sq_stable__integ_dom_p,
rng_times_zero,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
int_seg_subtype,
int_seg_cases,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
crng_times_comm,
itermAdd_wf,
itermMultiply_wf,
itermMinus_wf,
ringeq-iff-rsub-is-0,
ring_polynomial_null,
int-to-ring_wf,
ring_term_value_add_lemma,
ring_term_value_mul_lemma,
ring_term_value_var_lemma,
ring_term_value_const_lemma,
int-to-ring-zero,
ring_term_value_minus_lemma,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
cross-product-0,
rng_sig_wf,
cross-product-anti-comm,
rng_one_wf,
mul-zero-vector,
cross-product-same,
cross-product-mul1,
cross-product-mul2,
vector-mul-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
hypothesisEquality,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
applyLambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
dependent_set_memberEquality,
functionExtensionality,
universeEquality,
instantiate,
unionElimination,
hyp_replacement,
inlFormation,
cumulativity,
intEquality,
hypothesis_subsumption,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
inrFormation,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}r:IntegDom\{i\}. \mforall{}a,b:\mBbbN{}3 {}\mrightarrow{} |r|.
((\mforall{}x,y:|r|. Dec(x = y))
{}\mRightarrow{} ((a x b) = 0
\mLeftarrow{}{}\mRightarrow{} (a = 0) \mvee{} (b = 0) \mvee{} (\mexists{}i:\mBbbN{}3. ((\mneg{}((b i) = 0)) \mwedge{} (\mneg{}((a i) = 0)) \mwedge{} ((b i*a) = (a i*b))))))
Date html generated:
2018_05_21-PM-09_41_41
Last ObjectModification:
2018_05_19-PM-04_34_09
Theory : matrices
Home
Index