Nuprl Lemma : det-fun-row-op
∀[r:Rng]. ∀[n:ℕ]. ∀[d:det-fun(r;n)]. ∀[M:Matrix(n;n;r)]. ∀[a,b:ℕn]. ∀[k:|r|].
(d row-op(r;a;b;k;M)) = (d M) ∈ |r| supposing ¬(a = b ∈ ℤ)
Proof
Definitions occuring in Statement :
row-op: row-op(r;a;b;k;M),
det-fun: det-fun(r;n),
matrix: Matrix(n;m;r),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
det-fun: det-fun(r;n),
and: P ∧ Q,
all: ∀x:A. B[x],
infix_ap: x f y,
rng: Rng,
nat: ℕ,
prop: ℙ,
squash: ↓T,
true: True,
int_seg: {i..j-},
matrix: Matrix(n;m;r),
matrix-ap: M[i,j],
mx: matrix(M[x; y]),
row-op: row-op(r;a;b;k;M),
false: False,
implies: P ⇒ Q,
not: ¬A,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
matrix-mul-row: matrix-mul-row(r;k;i;M),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
matrix-swap-rows: matrix-swap-rows(M;i;j),
lelt: i ≤ j < k,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtype_rel: A ⊆r B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rng_times_wf,
matrix-ap_wf,
int_seg_wf,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
not_wf,
matrix_wf,
det-fun_wf,
nat_wf,
rng_wf,
rng_plus_comm,
mx_wf,
rng_plus_wf,
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,
rng_sig_wf,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
rng_zero_wf,
rng_times_zero,
subtype_rel_self,
rng_plus_zero,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
extract_by_obid,
isectElimination,
because_Cache,
natural_numberEquality,
sqequalRule,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
universeEquality,
imageMemberEquality,
baseClosed,
intEquality,
isect_memberEquality,
axiomEquality,
functionExtensionality,
int_eqEquality,
applyLambdaEquality,
lambdaFormation,
unionElimination,
equalityElimination,
independent_isectElimination,
int_eqReduceTrueSq,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
int_eqReduceFalseSq,
functionEquality,
approximateComputation,
voidEquality,
independent_pairFormation,
dependent_set_memberEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[n:\mBbbN{}]. \mforall{}[d:det-fun(r;n)]. \mforall{}[M:Matrix(n;n;r)]. \mforall{}[a,b:\mBbbN{}n]. \mforall{}[k:|r|].
(d row-op(r;a;b;k;M)) = (d M) supposing \mneg{}(a = b)
Date html generated:
2018_05_21-PM-09_37_01
Last ObjectModification:
2018_05_19-PM-04_28_42
Theory : matrices
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