Nuprl Lemma : det-add-row
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ∀[i:ℕn]. ∀[row:ℕn ⟶ |r|].
(|matrix(if x=i then (row y) +r M[x,y] else M[x,y])| = (|matrix(if x=i then row y else M[x,y])| +r |M|) ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
int_eq: if a=b then c else d,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
matrix-det: |M|,
squash: ↓T,
prop: ℙ,
crng: CRng,
rng: Rng,
nat: ℕ,
so_lambda: λ2x.t[x],
injection: A →⟶ B,
subtype_rel: A ⊆r B,
false: False,
implies: P ⇒ Q,
not: ¬A,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
int_seg: {i..j-},
infix_ap: x f y,
so_apply: x[s1;s2],
and: P ∧ Q,
all: ∀x:A. B[x],
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
let: let,
top: Top,
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),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
ringeq_int_terms: t1 ≡ t2,
rng_prod: rng_prod,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
rng_car: |r|,
pi1: fst(t),
grp_car: |g|,
mul_mon_of_rng: r↓xmn,
grp_op: *,
pi2: snd(t)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_lsum_wf,
injection_wf,
int_seg_wf,
let_wf,
permutation-sign_wf,
equal-wf-base,
int_subtype_base,
rng_minus_wf,
rng_prod_wf,
matrix-ap_wf,
mx_wf,
rng_plus_wf,
permutations-list_wf,
list_wf,
no_repeats_wf,
all_wf,
l_member_wf,
rng_lsum_plus,
subtype_rel_self,
iff_weakening_equal,
rng_wf,
matrix_ap_mx_lemma,
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,
matrix_wf,
nat_wf,
crng_wf,
itermAdd_wf,
itermMinus_wf,
itermVar_wf,
ringeq-iff-rsub-is-0,
ring_polynomial_null,
int-to-ring_wf,
ring_term_value_add_lemma,
ring_term_value_minus_lemma,
ring_term_value_var_lemma,
ring_term_value_const_lemma,
int-to-ring-zero,
mul_mon_of_rng_wf_c,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
grp_car_wf,
mul_mon_of_rng_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
lelt_wf,
infix_ap_wf,
rng_times_wf,
int_term_value_add_lemma,
mon_itop_split_el,
rng_times_over_plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
sqequalRule,
int_eqEquality,
setEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
productEquality,
imageMemberEquality,
instantiate,
independent_isectElimination,
productElimination,
independent_functionElimination,
functionEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
unionElimination,
equalityElimination,
int_eqReduceTrueSq,
dependent_pairFormation,
promote_hyp,
cumulativity,
int_eqReduceFalseSq,
axiomEquality,
approximateComputation,
independent_pairFormation,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
functionExtensionality,
addEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. \mforall{}[i:\mBbbN{}n]. \mforall{}[row:\mBbbN{}n {}\mrightarrow{} |r|].
(|matrix(if x=i then (row y) +r M[x,y] else M[x,y])|
= (|matrix(if x=i then row y else M[x,y])| +r |M|))
Date html generated:
2018_05_21-PM-09_36_40
Last ObjectModification:
2018_05_19-PM-04_28_23
Theory : matrices
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