Nuprl Lemma : transpose-identity-matrix
∀[r:RngSig]. ∀[n:ℕ]. (I' = I ∈ Matrix(n;n;r))
Proof
Definitions occuring in Statement :
identity-matrix: I,
matrix-transpose: M',
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
rng_sig: RngSig
Definitions unfolded in proof :
true: True,
subtype_rel: A ⊆r B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
squash: ↓T,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
top: Top,
all: ∀x:A. B[x],
matrix-transpose: M',
identity-matrix: I,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
nat_wf,
rng_zero_wf,
int_subtype_base,
equal-wf-base,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
rng_one_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
rng_sig_wf,
rng_car_wf,
int_seg_wf,
true_wf,
squash_wf,
mx_wf,
matrix_ap_mx_lemma
Rules used in proof :
axiomEquality,
baseClosed,
imageMemberEquality,
independent_pairFormation,
int_eqEquality,
approximateComputation,
int_eqReduceFalseSq,
independent_functionElimination,
cumulativity,
instantiate,
promote_hyp,
dependent_pairFormation,
int_eqReduceTrueSq,
independent_isectElimination,
productElimination,
equalityElimination,
unionElimination,
lambdaFormation,
rename,
setElimination,
intEquality,
because_Cache,
natural_numberEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
isectElimination,
imageElimination,
lambdaEquality,
applyEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:RngSig]. \mforall{}[n:\mBbbN{}]. (I' = I)
Date html generated:
2018_05_21-PM-09_38_38
Last ObjectModification:
2017_12_14-PM-00_12_35
Theory : matrices
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