Nuprl Lemma : real-Cramers-rule1
∀[n:ℕ]. ∀[A:ℕn ⟶ ℕn ⟶ ℝ]. ∀[c:{c:ℝ| (c * |A|) = r1} ]. ∀[b:ℝ^n].
(A*c*col(λj.|λx,y. if y=j then b x else (A x y)|)) ≡ col(b)
Proof
Definitions occuring in Statement :
real-det: |M|,
real-matrix-scalar-mul: c*A,
real-matrix-times: (A*B),
reqmatrix: X ≡ Y,
rcolumn: col(b),
real-vec: ℝ^n,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
int_eq: if a=b then c else d,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
matrix: Matrix(n;m;r),
real-ring: real-ring(),
rng_car: |r|,
pi1: fst(t),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
nat: ℕ,
all: ∀x:A. B[x],
rng_times: *,
pi2: snd(t),
rng_one: 1,
infix_ap: x f y,
reqmatrix: X ≡ Y,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
int_seg: {i..j-},
lelt: i ≤ j < k,
rmatrix: ℝ(a × b),
real-vec: ℝ^n,
less_than: a < b,
squash: ↓T,
prop: ℙ,
quotient: x,y:A//B[x; y],
stable: Stable{P},
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
matrix-ap: M[i,j],
mx: matrix(M[x; y]),
rcolumn: col(b),
real-det: |M|,
r-list-sum: r-list-sum(L),
reduce: reduce(f;k;as),
list_ind: list_ind,
map: map(f;as),
permutations-list: permutations-list(n),
list-permutations,
equipollent-iff-list,
equipollent_inversion,
equipollent-factorial,
equipollent_functionality_wrt_equipollent,
equipollent_transitivity,
equipollent_weakening_ext-eq,
equipollent-nsub,
count-combinations,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
matrix-times: (M*N),
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
add_grp_of_rng: r↓+gp,
grp_op: *,
grp_id: e,
rng_plus: +r,
rng_zero: 0,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_stable: SqStable(P)
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n {}\mrightarrow{} \mBbbR{}]. \mforall{}[c:\{c:\mBbbR{}| (c * |A|) = r1\} ]. \mforall{}[b:\mBbbR{}\^{}n].
(A*c*col(\mlambda{}j.|\mlambda{}x,y. if y=j then b x else (A x y)|)) \mequiv{} col(b)
Date html generated:
2020_05_20-PM-00_38_04
Last ObjectModification:
2020_02_07-PM-04_38_40
Theory : reals
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