Nuprl Lemma : matrix-times-id-left
∀[k,m:ℕ]. ∀[r:Rng]. ∀[N:Matrix(k;m;r)]. ((I*N) = N ∈ Matrix(k;m;r))
Proof
Definitions occuring in Statement :
identity-matrix: I,
matrix-times: (M*N),
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
matrix: Matrix(n;m;r),
identity-matrix: I,
matrix-times: (M*N),
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
matrix-ap: M[i,j],
mx: matrix(M[x; y]),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
rng: Rng,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
infix_ap: x f y,
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
label: ...$L... t,
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m
Lemmas referenced :
matrix_ap_mx_lemma,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
matrix_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
rng_wf,
nat_wf,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_sum_unroll_hi,
rng_times_wf,
rng_one_wf,
rng_zero_wf,
matrix-ap_wf,
subtype_rel_self,
iff_weakening_equal,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
infix_ap_wf,
rng_plus_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
lelt_wf,
rng_sum_wf,
rng_times_zero,
rng_sum_0,
rng_sig_wf,
decidable__equal_int,
rng_times_one,
rng_plus_comm,
rng_plus_zero,
subtype_rel_function,
int_seg_subtype,
false_wf,
not-le-2,
not-equal-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
rename,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
hypothesisEquality,
setElimination,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
axiomEquality,
because_Cache,
productElimination,
unionElimination,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
instantiate,
equalityElimination,
int_eqReduceTrueSq,
promote_hyp,
cumulativity,
int_eqReduceFalseSq,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
functionEquality,
addEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}[k,m:\mBbbN{}]. \mforall{}[r:Rng]. \mforall{}[N:Matrix(k;m;r)]. ((I*N) = N)
Date html generated:
2018_05_21-PM-09_35_04
Last ObjectModification:
2018_05_19-PM-04_23_53
Theory : matrices
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