Nuprl Lemma : det-swap-cols
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ∀[i,j:ℕn]. |matrix-swap-cols(M;i;j)| = (-r |M|) ∈ |r| supposing ¬(i = j ∈ ℤ)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
matrix-swap-cols: matrix-swap-cols(M;i;j),
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,
crng: CRng,
rng_minus: -r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
matrix-det: |M|,
crng: CRng,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
rng: Rng,
subtype_rel: A ⊆r B,
false: False,
implies: P ⇒ Q,
not: ¬A,
so_apply: x[s],
injection: A →⟶ B,
and: P ∧ Q,
all: ∀x:A. B[x],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
compose: f o g,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
matrix-ap: M[i,j],
matrix-swap-cols: matrix-swap-cols(M;i;j),
mx: matrix(M[x; y]),
flip: (i, j),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
let: let,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
rng_lsum_functionality_wrt_permutation,
int_seg_wf,
inject_wf,
let_wf,
rng_car_wf,
permutation-sign_wf,
equal-wf-base,
int_subtype_base,
rng_minus_wf,
rng_prod_wf,
matrix-ap_wf,
permutations-list_wf,
list_wf,
injection_wf,
no_repeats_wf,
all_wf,
l_member_wf,
map_wf,
compose_wf-injection,
flip-injection,
flip_wf,
flip-permutes-permutations-list2,
equal_wf,
squash_wf,
true_wf,
rng_lsum_map,
compose_wf,
subtype_rel_self,
iff_weakening_equal,
matrix-det_wf,
matrix-swap-cols_wf,
rng_minus_lsum,
subtype_rel_set,
subtype_rel_list,
rng_lsum_wf,
rng_wf,
not_wf,
matrix_wf,
nat_wf,
crng_wf,
rng_sig_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,
permutation-sign-compose,
sign-flip,
set_wf,
absval_cases,
false_wf,
le_wf,
rng_minus_minus,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
setEquality,
functionEquality,
natural_numberEquality,
because_Cache,
sqequalRule,
lambdaEquality,
int_eqEquality,
applyEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
productEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_isectElimination,
dependent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
instantiate,
productElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
lambdaFormation,
unionElimination,
equalityElimination,
int_eqReduceTrueSq,
dependent_pairFormation,
promote_hyp,
cumulativity,
voidElimination,
int_eqReduceFalseSq,
equalityUniverse,
levelHypothesis,
multiplyEquality,
independent_pairFormation,
applyLambdaEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. \mforall{}[i,j:\mBbbN{}n].
|matrix-swap-cols(M;i;j)| = (-r |M|) supposing \mneg{}(i = j)
Date html generated:
2018_05_21-PM-09_36_03
Last ObjectModification:
2018_05_19-PM-04_24_53
Theory : matrices
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