Nuprl Lemma : non-zero-component-exists
∀r:RngSig
((∀x,y:|r|. Dec(x = y ∈ |r|)) ⇒ (∀k:ℕ. {∀a:ℕk ⟶ |r|. ((¬(a = 0 ∈ (ℕk ⟶ |r|))) ⇒ (∃i:ℕk. (¬((a i) = 0 ∈ |r|))))}))
Proof
Definitions occuring in Statement :
zero-vector: 0,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
guard: {T},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
rng_zero: 0,
rng_car: |r|,
rng_sig: RngSig
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
false: False
Lemmas referenced :
non-zero-component_wf,
not_wf,
equal_wf,
int_seg_wf,
rng_car_wf,
zero-vector_wf,
set_wf,
rng_zero_wf,
lelt_wf,
nat_wf,
all_wf,
decidable_wf,
rng_sig_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
rename,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
hypothesis,
functionEquality,
natural_numberEquality,
setElimination,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
voidElimination
Latex:
\mforall{}r:RngSig
((\mforall{}x,y:|r|. Dec(x = y)) {}\mRightarrow{} (\mforall{}k:\mBbbN{}. \{\mforall{}a:\mBbbN{}k {}\mrightarrow{} |r|. ((\mneg{}(a = 0)) {}\mRightarrow{} (\mexists{}i:\mBbbN{}k. (\mneg{}((a i) = 0))))\}))
Date html generated:
2018_05_21-PM-09_42_41
Last ObjectModification:
2018_05_19-PM-04_34_26
Theory : matrices
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