Nuprl Lemma : scalar-triple-product-non-zero
∀[r:IntegDom{i}]. ∀[a,b,c:ℕ3 ⟶ |r|].
∀[u:ℕ3 ⟶ |r|]. (((a . u) = 0 ∈ |r|) ⇒ ((b . u) = 0 ∈ |r|) ⇒ ((c . u) = 0 ∈ |r|) ⇒ (u = 0 ∈ (ℕ3 ⟶ |r|)))
supposing ¬(|a,b,c| = 0 ∈ |r|)
Proof
Definitions occuring in Statement :
scalar-triple-product: |a,b,c|,
scalar-product: (a . b),
zero-vector: 0,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
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 :
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
rng: Rng,
crng: CRng,
prop: ℙ,
implies: P ⇒ Q,
uimplies: b supposing a,
integ_dom: IntegDom{i},
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
matrix: Matrix(n;m;r),
matrix-ap: M[i,j],
matrix-times: (M*N),
zero-matrix: 0,
subtract: n - m,
scalar-product: (a . b),
cons: [a / b],
select: L[n],
sq_type: SQType(T),
less_than: a < b,
zero-vector: 0
Lemmas referenced :
integ_dom_wf,
scalar-triple-product_wf,
not_wf,
rng_zero_wf,
int_seg_wf,
le_wf,
false_wf,
scalar-product_wf,
rng_car_wf,
equal_wf,
scalar-triple-product-as-det,
mx_wf,
iff_weakening_equal,
true_wf,
squash_wf,
null-space-unique,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
intformless_wf,
decidable__lt,
length_of_nil_lemma,
length_of_cons_lemma,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_seg_properties,
nil_wf,
cons_wf,
select_wf,
matrix_ap_mx_lemma,
rng_sig_wf,
int_seg_cases,
int_seg_subtype,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
lelt_wf,
matrix-ap_wf
Rules used in proof :
isect_memberEquality,
functionEquality,
axiomEquality,
dependent_functionElimination,
lambdaEquality,
equalitySymmetry,
equalityTransitivity,
applyEquality,
functionExtensionality,
independent_pairFormation,
sqequalRule,
natural_numberEquality,
dependent_set_memberEquality,
because_Cache,
lambdaFormation,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
baseClosed,
imageMemberEquality,
levelHypothesis,
equalityUniverse,
universeEquality,
imageElimination,
addEquality,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
unionElimination,
productElimination,
independent_isectElimination,
hypothesis_subsumption,
cumulativity,
instantiate,
applyLambdaEquality
Latex:
\mforall{}[r:IntegDom\{i\}]. \mforall{}[a,b,c:\mBbbN{}3 {}\mrightarrow{} |r|].
\mforall{}[u:\mBbbN{}3 {}\mrightarrow{} |r|]. (((a . u) = 0) {}\mRightarrow{} ((b . u) = 0) {}\mRightarrow{} ((c . u) = 0) {}\mRightarrow{} (u = 0))
supposing \mneg{}(|a,b,c| = 0)
Date html generated:
2018_05_21-PM-09_45_21
Last ObjectModification:
2017_12_20-PM-06_25_31
Theory : matrices
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