Nuprl Lemma : matrix-scalar-mul-mul
∀[n,m:ℕ]. ∀[r:Rng]. ∀[k1,k2:|r|]. ∀[M:Matrix(n;m;r)]. (k1*k2*M = k1 * k2*M ∈ Matrix(n;m;r))
Proof
Definitions occuring in Statement :
matrix-scalar-mul: k*M,
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
equal: s = t ∈ T,
rng: Rng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
true: True,
rng: Rng,
nat: ℕ,
prop: ℙ,
squash: ↓T,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
top: Top,
all: ∀x:A. B[x],
matrix-scalar-mul: k*M,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
matrix-ap_wf,
rng_times_wf,
infix_ap_wf,
nat_wf,
rng_wf,
matrix_wf,
rng_sig_wf,
rng_car_wf,
int_seg_wf,
true_wf,
squash_wf,
mx_wf,
matrix_ap_mx_lemma,
equal_wf,
rng_times_assoc,
iff_weakening_equal
Rules used in proof :
axiomEquality,
baseClosed,
imageMemberEquality,
rename,
setElimination,
intEquality,
because_Cache,
natural_numberEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
isectElimination,
imageElimination,
lambdaEquality,
applyEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[r:Rng]. \mforall{}[k1,k2:|r|]. \mforall{}[M:Matrix(n;m;r)]. (k1*k2*M = k1 * k2*M)
Date html generated:
2018_05_21-PM-09_38_25
Last ObjectModification:
2017_12_14-PM-01_34_57
Theory : matrices
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