Nuprl Lemma : matrix-plus-comm
∀[n,m:ℤ]. ∀[r:Rng]. ∀[M,N:Matrix(n;m;r)]. (M + N = N + M ∈ Matrix(n;m;r))
Proof
Definitions occuring in Statement :
matrix-plus: M + N,
matrix: Matrix(n;m;r),
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
true: True,
rng: Rng,
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix-plus: M + N,
matrix: Matrix(n;m;r),
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
prop: ℙ,
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_plus_wf,
infix_ap_wf,
rng_car_wf,
rng_wf,
matrix_wf,
int_seg_wf,
equal_wf,
squash_wf,
true_wf,
rng_plus_comm,
iff_weakening_equal
Rules used in proof :
axiomEquality,
isect_memberEquality,
because_Cache,
setElimination,
hypothesis,
hypothesisEquality,
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,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[r:Rng]. \mforall{}[M,N:Matrix(n;m;r)]. (M + N = N + M)
Date html generated:
2018_05_21-PM-09_34_34
Last ObjectModification:
2017_12_11-PM-00_29_27
Theory : matrices
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