Nuprl Lemma : adjugate-transpose
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. (adj(M') = adj(M)' ∈ Matrix(n;n;r))
Proof
Definitions occuring in Statement :
adjugate: adj(M),
matrix-transpose: M',
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
ge: i ≥ j ,
guard: {T},
ifthenelse: if b then t else f fi ,
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-},
true: True,
so_apply: x[s1;s2],
top: Top,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
rng: Rng,
crng: CRng,
nat: ℕ,
prop: ℙ,
squash: ↓T,
adjugate: adj(M),
matrix-transpose: M',
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
ite_rw_false,
ite_rw_true,
iff_weakening_equal,
btrue_neq_bfalse,
not_assert_elim,
and_wf,
assert_elim,
add-commutes,
rng_minus_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
matrix-det_wf,
matrix-minor_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
subtract_wf,
det-transpose,
equal_wf,
eqtt_to_assert,
bool_wf,
isEven_wf,
crng_wf,
nat_wf,
matrix_wf,
matrix_ap_mx_lemma,
rng_sig_wf,
rng_car_wf,
int_seg_wf,
true_wf,
squash_wf,
mx_wf,
transpose-matrix-minor
Rules used in proof :
applyLambdaEquality,
cumulativity,
instantiate,
promote_hyp,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
dependent_set_memberEquality,
independent_isectElimination,
productElimination,
equalityElimination,
unionElimination,
lambdaFormation,
addEquality,
axiomEquality,
baseClosed,
imageMemberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
rename,
setElimination,
intEquality,
because_Cache,
natural_numberEquality,
functionEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. (adj(M') = adj(M)')
Date html generated:
2018_05_21-PM-09_39_05
Last ObjectModification:
2017_12_14-PM-00_28_46
Theory : matrices
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