Nuprl Lemma : rv-isometry-injective
∀[rv:InnerProductSpace]. ∀f:Point ⟶ Point. (Isometry(f) ⇒ (∀x,y:Point. (f x ≡ f y ⇒ x ≡ y)))
Proof
Definitions occuring in Statement :
rv-isometry: Isometry(f),
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
rv-isometry: Isometry(f),
false: False,
not: ¬A,
ss-eq: x ≡ y,
guard: {T},
prop: ℙ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
req_weakening,
req_functionality,
rmul_wf,
int-to-real_wf,
rleq_wf,
real_wf,
rv-ip_wf,
req_wf,
rv-norm_wf,
ss-sep_wf,
rv-isometry_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
real-vector-space_subtype1,
ss-eq_wf,
rv-sub_wf,
rv-norm-is-zero,
ss-point_wf,
inner-product-space_subtype,
rv-sub-is-zero
Rules used in proof :
natural_numberEquality,
productEquality,
setEquality,
rename,
setElimination,
voidElimination,
dependent_functionElimination,
lambdaEquality,
functionEquality,
instantiate,
functionExtensionality,
because_Cache,
independent_isectElimination,
productElimination,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}f:Point {}\mrightarrow{} Point. (Isometry(f) {}\mRightarrow{} (\mforall{}x,y:Point. (f x \mequiv{} f y {}\mRightarrow{} x \mequiv{} y)))
Date html generated:
2016_11_08-AM-09_18_27
Last ObjectModification:
2016_11_02-PM-11_39_01
Theory : inner!product!spaces
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