Nuprl Lemma : matrix-scalar-mul-1
∀[n,m:ℕ]. ∀[r:Rng]. ∀[M:Matrix(n;m;r)]. (1*M = 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],
equal: s = t ∈ T,
rng: Rng,
rng_one: 1
Definitions unfolded in proof :
rng: Rng,
nat: ℕ,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
top: Top,
all: ∀x:A. B[x],
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
matrix-scalar-mul: k*M,
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
nat_wf,
rng_wf,
matrix_wf,
int_seg_wf,
matrix_ap_mx_lemma,
matrix-ap_wf,
rng_car_wf,
equal_wf,
squash_wf,
true_wf,
rng_times_one,
iff_weakening_equal
Rules used in proof :
axiomEquality,
hypothesisEquality,
because_Cache,
rename,
setElimination,
natural_numberEquality,
isectElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
functionExtensionality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[r:Rng]. \mforall{}[M:Matrix(n;m;r)]. (1*M = M)
Date html generated:
2018_05_21-PM-09_38_28
Last ObjectModification:
2017_12_14-PM-01_35_28
Theory : matrices
Home
Index