Nuprl Lemma : cross-product-equal-0-iff
∀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|)) ∨ (∀l: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],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: 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 :
so_apply: x[s],
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
rng: Rng,
crng: CRng,
integ_dom: IntegDom{i},
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T},
or: P ∨ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
infix_ap: x f y,
squash: ↓T,
true: True,
exists: ∃x:A. B[x],
integ_dom_p: IsIntegDom(r),
decidable: Dec(P),
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
integ_dom_wf,
decidable_wf,
rng_zero_wf,
le_wf,
false_wf,
scalar-product_wf,
iff_wf,
all_wf,
or_wf,
zero-vector_wf,
cross-product_wf,
rng_car_wf,
int_seg_wf,
equal_wf,
cross-product-equal-0,
rng_times_zero,
iff_weakening_equal,
rng_times_wf,
squash_wf,
true_wf,
scalar-product-mul,
crng_times_comm,
cross-product-0,
not_wf,
cross-product-non-zero-implies-ext,
compact-finite,
decidable__equal_compact_domain,
rng_sig_wf,
mul-zero-vector,
rng_one_wf,
rng_minus_wf,
cross-product-anti-comm,
vector-mul_wf
Rules used in proof :
dependent_set_memberEquality,
lambdaEquality,
sqequalRule,
because_Cache,
applyEquality,
functionExtensionality,
hypothesisEquality,
rename,
setElimination,
hypothesis,
natural_numberEquality,
functionEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inrFormation,
equalitySymmetry,
equalityTransitivity,
inlFormation,
unionElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
levelHypothesis,
equalityUniverse,
imageElimination,
applyLambdaEquality,
universeEquality,
productEquality,
cumulativity,
voidElimination
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{} (\mforall{}l:\mBbbN{}3 {}\mrightarrow{} |r|. ((a . l) = 0 \mLeftarrow{}{}\mRightarrow{} (b . l) = 0))))
Date html generated:
2018_05_21-PM-09_44_10
Last ObjectModification:
2018_01_09-PM-02_07_38
Theory : matrices
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