Nuprl Lemma : diagonal-matrix_wf
∀[r:RngSig]. ∀[n:ℕ]. ∀[F:ℕn ⟶ |r|]. (diagonal-matrix(r;i.F[i]) ∈ Matrix(n;n;r))
Proof
Definitions occuring in Statement :
diagonal-matrix: diagonal-matrix(r;x.F[x]),
matrix: Matrix(n;m;r),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
rng_car: |r|,
rng_sig: RngSig
Definitions unfolded in proof :
so_apply: x[s1;s2],
not: ¬A,
implies: P ⇒ Q,
false: False,
so_apply: x[s],
int_seg: {i..j-},
so_lambda: λ2x y.t[x; y],
nat: ℕ,
diagonal-matrix: diagonal-matrix(r;x.F[x]),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_sig_wf,
nat_wf,
rng_car_wf,
rng_zero_wf,
int_seg_wf,
mx_wf
Rules used in proof :
isect_memberEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
natural_numberEquality,
functionExtensionality,
applyEquality,
int_eqEquality,
lambdaEquality,
hypothesisEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:RngSig]. \mforall{}[n:\mBbbN{}]. \mforall{}[F:\mBbbN{}n {}\mrightarrow{} |r|]. (diagonal-matrix(r;i.F[i]) \mmember{} Matrix(n;n;r))
Date html generated:
2018_05_21-PM-09_38_11
Last ObjectModification:
2018_01_02-PM-02_59_12
Theory : matrices
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