Nuprl Lemma : cross-product-non-zero-implies
∀r:IntegDom{i}
((∀x,y:|r|. Dec(x = y ∈ |r|))
⇒ (∀a:{a:ℕ3 ⟶ |r|| ¬(a = 0 ∈ (ℕ3 ⟶ |r|))} . ∀b:{b:ℕ3 ⟶ |r||
(¬(b = 0 ∈ (ℕ3 ⟶ |r|))) ∧ (¬((a x b) = 0 ∈ (ℕ3 ⟶ |r|)))} .
(∃l:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} [(((a . l) = 0 ∈ |r|) ∧ (¬((b . l) = 0 ∈ |r|)))])))
Proof
Definitions occuring in Statement :
scalar-product: (a . b),
cross-product: (a x b),
zero-vector: 0,
int_seg: {i..j-},
decidable: Dec(P),
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
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,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
member: t ∈ T,
integ_dom: IntegDom{i},
crng: CRng,
rng: Rng,
sq_stable: SqStable(P),
squash: ↓T,
and: P ∧ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
prop: ℙ,
decidable: Dec(P),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
nat: ℕ,
less_than': less_than'(a;b),
cand: A c∧ B,
subtype_rel: A ⊆r B,
infix_ap: x f y,
true: True,
uimplies: b supposing a,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
sq_type: SQType(T),
vector-mul: (c*a),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
sq_exists: ∃x:A [B[x]],
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
bnot: ¬bb,
zero-vector: 0,
uiff: uiff(P;Q),
eq_int: (i =z j),
integ_dom_p: IsIntegDom(r),
nequal: a ≠ b ∈ T
Lemmas referenced :
sq_stable__and,
not_wf,
equal_wf,
int_seg_wf,
rng_car_wf,
zero-vector_wf,
cross-product_wf,
istype-void,
sq_stable__not,
cross-product-equal-zero,
rng_zero_wf,
vector-mul_wf,
decidable_wf,
integ_dom_wf,
decidable__equal_compact_domain,
compact-finite,
istype-le,
non-zero-component_wf,
rng_plus_wf,
rng_minus_wf,
iff_weakening_equal,
rng_plus_inv,
infix_ap_wf,
squash_wf,
true_wf,
istype-universe,
rng_plus_assoc,
subtype_rel_self,
rng_plus_comm,
rng_plus_zero,
decidable__not,
decidable__cand,
and_wf,
decidable__exists_int_seg,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
int-value-type,
istype-int,
lelt_wf,
set-value-type,
rng_times_wf,
crng_times_comm,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
int_seg_properties,
scalar-product_wf,
ifthenelse_wf,
eq_int_wf,
btrue_neq_bfalse,
assert_elim,
bool_subtype_base,
bfalse_wf,
btrue_wf,
eq_int_eq_true,
bool_wf,
istype-assert,
equal-wf-base,
bnot_wf,
assert_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformeq_wf,
intformand_wf,
bool_cases,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
rng_minus_minus,
rng_minus_zero,
int_seg_cases,
int_seg_subtype_special,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
decidable__equal_int,
scalar-product-3,
rng_times_over_minus,
rng_times_zero,
rng_plus_ac_1,
rng_plus_inv_assoc,
rng_one_wf,
rng_times_one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
setElimination,
thin,
rename,
cut,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
because_Cache,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
natural_numberEquality,
hypothesisEquality,
isect_memberEquality_alt,
sqequalRule,
functionIsType,
equalityIstype,
inhabitedIsType,
lambdaEquality_alt,
dependent_functionElimination,
functionIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
unionIsType,
productIsType,
applyEquality,
setIsType,
universeIsType,
unionElimination,
inlFormation_alt,
equalityTransitivity,
equalitySymmetry,
inrFormation_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
dependent_pairFormation_alt,
independent_isectElimination,
applyLambdaEquality,
instantiate,
universeEquality,
cumulativity,
promote_hyp,
intEquality,
cutEval,
functionExtensionality,
sqequalBase,
approximateComputation,
dependent_set_memberFormation_alt,
equalityElimination,
closedConclusion,
baseApply,
int_eqEquality,
hypothesis_subsumption
Latex:
\mforall{}r:IntegDom\{i\}
((\mforall{}x,y:|r|. Dec(x = y))
{}\mRightarrow{} (\mforall{}a:\{a:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(a = 0)\} . \mforall{}b:\{b:\mBbbN{}3 {}\mrightarrow{} |r|| (\mneg{}(b = 0)) \mwedge{} (\mneg{}((a x b) = 0))\} .
(\mexists{}l:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} [(((a . l) = 0) \mwedge{} (\mneg{}((b . l) = 0)))])))
Date html generated:
2020_05_20-AM-09_03_55
Last ObjectModification:
2019_12_26-PM-04_06_30
Theory : matrices
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