Nuprl Lemma : real-Cramers-rule
∀[n:ℕ]. ∀[A:ℝ(n × n)]. ∀[b:ℝ^n]. (A*col(λj.(|λx,y. if y=j then b x else (A x y)|/|A|))) ≡ col(b) supposing |A| ≠ r0
Proof
Definitions occuring in Statement :
real-det: |M|,
real-matrix-times: (A*B),
reqmatrix: X ≡ Y,
rcolumn: col(b),
rmatrix: ℝ(a × b),
real-vec: ℝ^n,
rdiv: (x/y),
rneq: x ≠ y,
int-to-real: r(n),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
int_eq: if a=b then c else d,
apply: f a,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rmatrix: ℝ(a × b),
false: False,
implies: P ⇒ Q,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermConstant: "const",
rat_term_ind: rat_term_ind,
pi1: fst(t),
true: True,
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
and: P ∧ Q,
pi2: snd(t),
prop: ℙ,
reqmatrix: X ≡ Y,
all: ∀x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
real-vec: ℝ^n,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rcolumn: col(b),
real-matrix-scalar-mul: c*A
Lemmas referenced :
real-Cramers-rule1,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
real-det_wf,
int-to-real_wf,
istype-int,
rdiv_wf,
req_wf,
rmul_wf,
req_witness,
real-matrix-times_wf,
istype-void,
istype-le,
rcolumn_wf,
subtype_rel_self,
int_seg_wf,
real_wf,
real-vec_wf,
rneq_wf,
rmatrix_wf,
istype-nat,
real-matrix-scalar-mul_wf,
real-matrix-times_functionality,
reqmatrix_functionality,
reqmatrix_weakening,
reqmatrix_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
natural_numberEquality,
hypothesis,
lambdaEquality_alt,
int_eqEquality,
independent_isectElimination,
approximateComputation,
sqequalRule,
independent_pairFormation,
dependent_set_memberEquality_alt,
universeIsType,
dependent_functionElimination,
applyEquality,
lambdaFormation_alt,
voidElimination,
setElimination,
rename,
productElimination,
inhabitedIsType,
functionEquality,
imageElimination,
independent_functionElimination,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
closedConclusion
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[A:\mBbbR{}(n \mtimes{} n)].
\mforall{}[b:\mBbbR{}\^{}n]. (A*col(\mlambda{}j.(|\mlambda{}x,y. if y=j then b x else (A x y)|/|A|))) \mequiv{} col(b) supposing |A| \mneq{} r0
Date html generated:
2019_10_30-AM-08_22_50
Last ObjectModification:
2019_09_19-PM-01_10_24
Theory : reals
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