Nuprl Lemma : non-zero-component_wf
∀[r:RngSig]. ∀[eq:∀x,y:|r|. Dec(x = y ∈ |r|)]. ∀[k:ℕ]. ∀[a:{a:ℕk ⟶ |r|| ¬(a = 0 ∈ (ℕk ⟶ |r|))} ].
(non-zero-component(r;eq;k;a) ∈ {i:ℕk| ¬((a i) = 0 ∈ |r|)} )
Proof
Definitions occuring in Statement :
non-zero-component: non-zero-component(r;eq;k;a),
zero-vector: 0,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
member: t ∈ T,
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 :
pi1: fst(t),
uimplies: b supposing a,
sq_exists: ∃x:A [B[x]],
nat: ℕ,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
non-zero-vector-implies-ext,
non-zero-component: non-zero-component(r;eq;k;a),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
set_wf,
subtype_rel_function,
rng_zero_wf,
sq_exists_wf,
zero-vector_wf,
not_wf,
int_seg_wf,
nat_wf,
equal_wf,
decidable_wf,
rng_car_wf,
all_wf,
rng_sig_wf,
subtype_rel_self,
pi1-axiom,
non-zero-vector-implies-ext
Rules used in proof :
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_isectElimination,
because_Cache,
rename,
setElimination,
natural_numberEquality,
setEquality,
lambdaEquality,
hypothesisEquality,
cumulativity,
functionEquality,
isectElimination,
sqequalHypSubstitution,
instantiate,
applyEquality,
thin,
hypothesis,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:RngSig]. \mforall{}[eq:\mforall{}x,y:|r|. Dec(x = y)]. \mforall{}[k:\mBbbN{}]. \mforall{}[a:\{a:\mBbbN{}k {}\mrightarrow{} |r|| \mneg{}(a = 0)\} ].
(non-zero-component(r;eq;k;a) \mmember{} \{i:\mBbbN{}k| \mneg{}((a i) = 0)\} )
Date html generated:
2018_05_21-PM-09_42_37
Last ObjectModification:
2018_05_21-AM-07_07_22
Theory : matrices
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