Nuprl Lemma : triple-cross-product-zero
∀r:CRng. ∀a,p,q:ℕ3 ⟶ |r|. (((p . a) = 0 ∈ |r|) ⇒ ((q . a) = 0 ∈ |r|) ⇒ ((a x (p x q)) = 0 ∈ (ℕ3 ⟶ |r|)))
Proof
Definitions occuring in Statement :
scalar-product: (a . b),
cross-product: (a x b),
zero-vector: 0,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
zero-vector: 0,
cross-product: (a x b),
select: L[n],
cons: [a / b],
subtract: n - m,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
crng: CRng,
rng: Rng,
nat: ℕ,
less_than: a < b,
squash: ↓T,
true: True,
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,
int_seg_subtype,
false_wf,
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,
int_seg_wf,
equal_wf,
rng_car_wf,
scalar-product_wf,
le_wf,
rng_zero_wf,
crng_wf,
rng_plus_wf,
lelt_wf,
infix_ap_wf,
rng_times_wf,
rng_minus_wf,
squash_wf,
true_wf,
rng_times_over_plus,
rng_times_over_minus,
subtype_rel_self,
rng_minus_over_plus,
rng_minus_minus,
rng_plus_assoc,
iff_weakening_equal,
rng_plus_ac_1,
rng_plus_comm,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
rng_times_zero,
rng_minus_zero,
rng_plus_zero,
scalar-product-3,
crng_times_comm,
crng_times_ac_1,
rng_plus_inv_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
functionExtensionality,
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,
hypothesis_subsumption,
addEquality,
independent_pairFormation,
productElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
applyEquality,
functionEquality,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
baseClosed,
imageElimination,
universeEquality
Latex:
\mforall{}r:CRng. \mforall{}a,p,q:\mBbbN{}3 {}\mrightarrow{} |r|. (((p . a) = 0) {}\mRightarrow{} ((q . a) = 0) {}\mRightarrow{} ((a x (p x q)) = 0))
Date html generated:
2018_05_21-PM-09_44_17
Last ObjectModification:
2018_05_19-PM-04_35_16
Theory : matrices
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