Nuprl Lemma : matrix-det-dim1
∀[r:CRng]. ∀[M:Row(1;r)]. (|M| = M[0,0] ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
matrix-det: |M|,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
permutation-sign: permutation-sign(n;f),
let: let,
int-prod: Π(f[x] | x < k),
squash: ↓T,
prop: ℙ,
crng: CRng,
rng: Rng,
infix_ap: x f y,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
false: False,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
select: L[n],
cons: [a / b]
Lemmas referenced :
permutations-list-1,
rng_lsum_cons_lemma,
rng_lsum_nil_lemma,
primrec1_lemma,
primrec0_lemma,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_plus_wf,
rng_prod_unroll_hi,
less_than_wf,
matrix-ap_wf,
select_wf,
int_seg_wf,
cons_wf,
false_wf,
lelt_wf,
nil_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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,
length-singleton,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
rng_zero_wf,
subtype_rel_self,
iff_weakening_equal,
rng_prod_empty_lemma,
matrix_wf,
crng_wf,
rng_times_one,
rng_plus_zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
lambdaEquality,
imageElimination,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
setElimination,
rename,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
lambdaFormation,
independent_isectElimination,
productElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
instantiate,
axiomEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[M:Row(1;r)]. (|M| = M[0,0])
Date html generated:
2018_05_21-PM-09_35_36
Last ObjectModification:
2018_05_19-PM-04_24_42
Theory : matrices
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