Nuprl Lemma : scalar-product-0
∀[r:Rng]. ∀[n:ℕ]. ∀[a:ℕn ⟶ |r|]. ((0 . a) = 0 ∈ |r|)
Proof
Definitions occuring in Statement :
scalar-product: (a . b),
zero-vector: 0,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
rng: Rng,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
scalar-product: (a . b),
nat: ℕ,
uimplies: b supposing a,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
rng: Rng,
so_lambda: λ2x.t[x],
zero-vector: 0,
true: True,
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rng_sum_0,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rng_sum_wf,
squash_wf,
true_wf,
int_seg_wf,
rng_car_wf,
nat_wf,
rng_wf,
equal_wf,
rng_times_zero,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
imageMemberEquality,
baseClosed,
axiomEquality,
universeEquality,
productElimination,
instantiate
Latex:
\mforall{}[r:Rng]. \mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n {}\mrightarrow{} |r|]. ((0 . a) = 0)
Date html generated:
2018_05_21-PM-09_42_00
Last ObjectModification:
2018_05_19-PM-04_33_41
Theory : matrices
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