Nuprl Lemma : cross-product-equal-0
∀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|))
∨ (∃c1,c2:|r|. ((¬(c1 = 0 ∈ |r|)) ∧ (¬(c2 = 0 ∈ |r|)) ∧ ((c1*a) = (c2*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,
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],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
rng: Rng,
crng: CRng,
integ_dom: IntegDom{i},
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
guard: {T},
or: P ∨ Q,
cand: A c∧ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
vector-mul: (c*a),
zero-vector: 0,
not: ¬A,
integ_dom_p: IsIntegDom(r),
infix_ap: x f y
Lemmas referenced :
integ_dom_wf,
decidable_wf,
all_wf,
vector-mul_wf,
rng_zero_wf,
not_wf,
exists_wf,
or_wf,
zero-vector_wf,
cross-product_wf,
rng_car_wf,
int_seg_wf,
equal_wf,
cross-product-equal-zero,
iff_weakening_equal,
true_wf,
squash_wf,
cross-product-same,
rng_sig_wf,
cross-product-mul1,
cross-product-mul2,
vector-mul-mul,
rng_times_wf,
crng_times_comm
Rules used in proof :
productEquality,
lambdaEquality,
sqequalRule,
because_Cache,
applyEquality,
functionExtensionality,
rename,
setElimination,
natural_numberEquality,
functionEquality,
isectElimination,
independent_pairFormation,
productElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
inrFormation,
equalitySymmetry,
equalityTransitivity,
inlFormation,
unionElimination,
dependent_pairFormation,
independent_isectElimination,
baseClosed,
imageMemberEquality,
levelHypothesis,
equalityUniverse,
universeEquality,
imageElimination,
cumulativity,
applyLambdaEquality
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{}c1,c2:|r|. ((\mneg{}(c1 = 0)) \mwedge{} (\mneg{}(c2 = 0)) \mwedge{} ((c1*a) = (c2*b))))))
Date html generated:
2018_05_21-PM-09_41_48
Last ObjectModification:
2017_12_22-AM-01_35_22
Theory : matrices
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