Nuprl Lemma : rv-isometry-implies-extensional
∀rv:InnerProductSpace. ∀f:Point ⟶ Point. ∀x,y:Point. (f x # f y ⇒ x # y) supposing Isometry(f)
Proof
Definitions occuring in Statement :
rv-isometry: Isometry(f),
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
rv-isometry: Isometry(f),
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
req_weakening,
rless_functionality,
rv-isometry_wf,
ss-sep_wf,
rv-norm-positive-iff-ext,
rv-sep-iff-ext,
rmul_wf,
int-to-real_wf,
rleq_wf,
real_wf,
rv-ip_wf,
req_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
real-vector-space_subtype1,
ss-point_wf,
inner-product-space_subtype,
rv-sub_wf,
rv-norm_wf,
req_witness
Rules used in proof :
independent_pairFormation,
functionEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
productEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
because_Cache,
independent_isectElimination,
instantiate,
functionExtensionality,
hypothesis,
applyEquality,
extract_by_obid,
hypothesisEquality,
thin,
isectElimination,
isect_memberEquality,
sqequalHypSubstitution,
sqequalRule,
introduction,
cut,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}f:Point {}\mrightarrow{} Point. \mforall{}x,y:Point. (f x \# f y {}\mRightarrow{} x \# y) supposing Isometry(f)
Date html generated:
2016_11_08-AM-09_18_24
Last ObjectModification:
2016_11_02-PM-08_45_12
Theory : inner!product!spaces
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