Nuprl Lemma : matrix-times-distrib-left
∀[n,k,m:ℕ]. ∀[r:Rng]. ∀[M:Matrix(n;k;r)]. ∀[N,K:Matrix(k;m;r)]. ((M*N + K) = (M*N) + (M*K) ∈ Matrix(n;m;r))
Proof
Definitions occuring in Statement :
matrix-times: (M*N),
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 :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
true: True,
infix_ap: x f y,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
rng: Rng,
prop: ℙ,
squash: ↓T,
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix-plus: M + N,
matrix-times: (M*N),
matrix: Matrix(n;m;r),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
nat_wf,
rng_wf,
matrix_wf,
iff_weakening_equal,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
rng_sum_plus,
int_seg_wf,
rng_plus_wf,
matrix-ap_wf,
rng_times_wf,
infix_ap_wf,
rng_sum_wf,
rng_car_wf,
true_wf,
squash_wf,
equal_wf,
rng_times_over_plus,
rng_plus_comm
Rules used in proof :
axiomEquality,
baseClosed,
imageMemberEquality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
unionElimination,
dependent_functionElimination,
productElimination,
independent_isectElimination,
because_Cache,
natural_numberEquality,
setElimination,
universeEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
sqequalRule,
rename,
functionExtensionality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n,k,m:\mBbbN{}]. \mforall{}[r:Rng]. \mforall{}[M:Matrix(n;k;r)]. \mforall{}[N,K:Matrix(k;m;r)]. ((M*N + K) = (M*N) + (M*K))
Date html generated:
2018_05_21-PM-09_34_42
Last ObjectModification:
2017_12_11-PM-00_29_30
Theory : matrices
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