Nuprl Lemma : det-equal-rows
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ∀[i,j:ℕn].
|M| = 0 ∈ |r| supposing (¬(i = j ∈ ℤ)) ∧ (matrix-swap-rows(M;i;j) = M ∈ Matrix(n;n;r))
Proof
Definitions occuring in Statement :
matrix-det: |M|,
matrix-swap-rows: matrix-swap-rows(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,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
crng: CRng,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
matrix-det: |M|,
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
crng: CRng,
so_lambda: λ2x.t[x],
rng: Rng,
false: False,
implies: P ⇒ Q,
not: ¬A,
so_apply: x[s],
all: ∀x:A. B[x],
injection: A →⟶ B,
int_seg: {i..j-},
infix_ap: x f y,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
or: P ∨ Q,
sq_type: SQType(T),
eq_int: (i =z j),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
permutation: permutation(T;L1;L2),
exists: ∃x:A. B[x],
cand: A c∧ B,
compose: f o g,
flip: (i, j),
bool: 𝔹,
unit: Unit,
it: ⋅,
matrix-ap: M[i,j],
matrix-swap-rows: matrix-swap-rows(M;i;j),
mx: matrix(M[x; y]),
assert: ↑b,
let: let
Lemmas referenced :
rng_lsum-split,
int_seg_wf,
eq_int_wf,
permutation-sign_wf,
equal-wf-base,
int_subtype_base,
let_wf,
rng_car_wf,
rng_minus_wf,
rng_prod_wf,
matrix-ap_wf,
permutations-list_wf,
subtype_rel_set,
list_wf,
injection_wf,
no_repeats_wf,
all_wf,
l_member_wf,
subtype_rel_list,
not_wf,
equal_wf,
matrix_wf,
matrix-swap-rows_wf,
nat_wf,
crng_wf,
rng_plus_wf,
rng_lsum_wf,
filter_wf5,
bnot_wf,
rng_zero_wf,
squash_wf,
true_wf,
rng_plus_comm,
subtype_rel_self,
rng_plus_inv,
iff_weakening_equal,
compose_wf,
flip_wf,
rng_minus_lsum,
rng_lsum_map,
rng_lsum_functionality_wrt_permutation,
map_wf,
inject_wf,
compose_wf-injection,
flip-injection,
set_wf,
map-filter,
bool_wf,
permutation-sign-compose,
sign-flip,
absval_cases,
false_wf,
le_wf,
subtype_base_sq,
bfalse_wf,
btrue_wf,
filter_functionality_wrt_permutation,
flip-permutes-permutations-list,
length_wf_nat,
permute_list_wf,
length_wf,
rng_wf,
rng_prod_injection,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
set_subtype_base,
lelt_wf,
rng_minus_minus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
functionEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
lambdaEquality,
hypothesisEquality,
applyEquality,
setEquality,
intEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
int_eqEquality,
productEquality,
independent_isectElimination,
equalityTransitivity,
isect_memberEquality,
axiomEquality,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
instantiate,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
dependent_set_memberEquality,
functionExtensionality,
multiplyEquality,
independent_pairFormation,
unionElimination,
cumulativity,
dependent_pairFormation,
promote_hyp,
minusEquality,
equalityElimination,
int_eqReduceTrueSq,
voidElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. \mforall{}[i,j:\mBbbN{}n].
|M| = 0 supposing (\mneg{}(i = j)) \mwedge{} (matrix-swap-rows(M;i;j) = M)
Date html generated:
2018_05_21-PM-09_36_13
Last ObjectModification:
2018_05_19-PM-04_25_26
Theory : matrices
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