Nuprl Lemma : matrix-plus-zero-left
∀[k,m:ℕ]. ∀[r:Rng]. ∀[N:Matrix(k;m;r)]. (0 + N = N ∈ Matrix(k;m;r))
Proof
Definitions occuring in Statement :
zero-matrix: 0,
matrix-plus: M + N,
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
true: True,
rng: Rng,
nat: ℕ,
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix-plus: M + N,
zero-matrix: 0,
matrix: Matrix(n;m;r),
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
matrix-ap_wf,
rng_zero_wf,
rng_car_wf,
nat_wf,
rng_wf,
matrix_wf,
int_seg_wf,
equal_wf,
squash_wf,
true_wf,
rng_plus_comm,
rng_plus_zero,
iff_weakening_equal
Rules used in proof :
axiomEquality,
isect_memberEquality,
hypothesisEquality,
hypothesis,
because_Cache,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
rename,
functionExtensionality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[k,m:\mBbbN{}]. \mforall{}[r:Rng]. \mforall{}[N:Matrix(k;m;r)]. (0 + N = N)
Date html generated:
2018_05_21-PM-09_35_09
Last ObjectModification:
2017_12_11-PM-00_29_37
Theory : matrices
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