Nuprl Lemma : inverse-unique
∀[r:CRng]. ∀[n:ℕ]. ∀[A,B,C:Matrix(n;n;r)].
((((A*B) = I ∈ Matrix(n;n;r)) ∨ ((B*A) = I ∈ Matrix(n;n;r)))
⇒ (((A*C) = I ∈ Matrix(n;n;r)) ∨ ((C*A) = I ∈ Matrix(n;n;r)))
⇒ (B = C ∈ Matrix(n;n;r)))
Proof
Definitions occuring in Statement :
identity-matrix: I,
matrix-times: (M*N),
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
rng: Rng,
crng: CRng,
nat: ℕ,
prop: ℙ,
exists: ∃x:A. B[x],
invertible-matrix: invertible-matrix(r;n;A),
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q
Lemmas referenced :
crng_wf,
nat_wf,
iff_weakening_equal,
rng_wf,
true_wf,
squash_wf,
invertible-matrix-iff-left,
identity-matrix_wf,
matrix-times_wf,
matrix_wf,
equal_wf,
matrix-times-assoc,
matrix-times-id-left,
matrix-times-id-right,
or_wf
Rules used in proof :
isect_memberEquality,
axiomEquality,
independent_isectElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
intEquality,
levelHypothesis,
equalityUniverse,
universeEquality,
imageElimination,
lambdaEquality,
applyEquality,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
productElimination,
dependent_functionElimination,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
hypothesisEquality,
dependent_pairFormation,
lambdaFormation,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[A,B,C:Matrix(n;n;r)].
((((A*B) = I) \mvee{} ((B*A) = I)) {}\mRightarrow{} (((A*C) = I) \mvee{} ((C*A) = I)) {}\mRightarrow{} (B = C))
Date html generated:
2018_05_21-PM-09_40_07
Last ObjectModification:
2017_12_14-PM-03_14_36
Theory : matrices
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