Nuprl Lemma : matrix-mul-col_wf
∀[n,m:ℤ]. ∀[r:RngSig]. ∀[k:|r|]. ∀[i:ℕm]. ∀[M:Matrix(n;m;r)]. (matrix-mul-col(r;k;i;M) ∈ Matrix(n;m;r))
Proof
Definitions occuring in Statement :
matrix-mul-col: matrix-mul-col(r;k;i;M),
matrix: Matrix(n;m;r),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n,
int: ℤ,
rng_car: |r|,
rng_sig: RngSig
Definitions unfolded in proof :
so_apply: x[s1;s2],
not: ¬A,
implies: P ⇒ Q,
false: False,
infix_ap: x f y,
int_seg: {i..j-},
so_lambda: λ2x y.t[x; y],
matrix-mul-col: matrix-mul-col(r;k;i;M),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_sig_wf,
rng_car_wf,
matrix_wf,
int_seg_wf,
matrix-ap_wf,
rng_times_wf,
mx_wf
Rules used in proof :
intEquality,
because_Cache,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
natural_numberEquality,
applyEquality,
hypothesis,
rename,
setElimination,
int_eqEquality,
lambdaEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[r:RngSig]. \mforall{}[k:|r|]. \mforall{}[i:\mBbbN{}m]. \mforall{}[M:Matrix(n;m;r)].
(matrix-mul-col(r;k;i;M) \mmember{} Matrix(n;m;r))
Date html generated:
2018_05_21-PM-09_34_24
Last ObjectModification:
2017_12_11-PM-04_20_51
Theory : matrices
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