Nuprl Lemma : matrix-times-0-left
∀[k,m,n:ℕ]. ∀[r:Rng]. ∀[N:Matrix(k;m;r)]. ((0*N) = 0 ∈ Matrix(n;m;r))
Proof
Definitions occuring in Statement :
zero-matrix: 0,
matrix-times: (M*N),
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
matrix: Matrix(n;m;r),
zero-matrix: 0,
matrix-times: (M*N),
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
nat: ℕ,
rng: Rng,
true: True,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
matrix_ap_mx_lemma,
int_seg_wf,
matrix_wf,
rng_wf,
nat_wf,
rng_car_wf,
matrix-ap_wf,
rng_zero_wf,
int_seg_properties,
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,
equal_wf,
squash_wf,
true_wf,
rng_sum_wf,
rng_times_zero,
subtype_rel_self,
iff_weakening_equal,
rng_sum_0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
rename,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
natural_numberEquality,
setElimination,
because_Cache,
hypothesisEquality,
axiomEquality,
productElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionEquality,
imageMemberEquality,
baseClosed,
instantiate
Latex:
\mforall{}[k,m,n:\mBbbN{}]. \mforall{}[r:Rng]. \mforall{}[N:Matrix(k;m;r)]. ((0*N) = 0)
Date html generated:
2018_05_21-PM-09_35_18
Last ObjectModification:
2018_05_19-PM-04_24_35
Theory : matrices
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