Nuprl Lemma : non-zero-vector-implies
∀r:RngSig
((∀x,y:|r|. Dec(x = y ∈ |r|)) ⇒ (∀k:ℕ. ∀a:{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),
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
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,
member: t ∈ T,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
false: False,
not: ¬A,
zero-vector: 0,
exists: ∃x:A. B[x],
prop: ℙ,
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
decidable__exists_int_seg,
not_wf,
equal_wf,
rng_car_wf,
rng_zero_wf,
int_seg_wf,
decidable__not,
set_wf,
zero-vector_wf,
nat_wf,
all_wf,
decidable_wf,
rng_sig_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
independent_functionElimination,
unionElimination,
functionExtensionality,
dependent_pairFormation,
voidElimination,
productElimination,
dependent_set_memberEquality,
functionEquality
Latex:
\mforall{}r:RngSig
((\mforall{}x,y:|r|. Dec(x = y)) {}\mRightarrow{} (\mforall{}k:\mBbbN{}. \mforall{}a:\{a:\mBbbN{}k {}\mrightarrow{} |r|| \mneg{}(a = 0)\} . (\mexists{}i:\mBbbN{}k [(\mneg{}((a i) = 0))])))
Date html generated:
2018_05_21-PM-09_42_29
Last ObjectModification:
2018_05_19-PM-04_34_13
Theory : matrices
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