Nuprl Lemma : rv-isometry-inverse
∀[rv:InnerProductSpace]. ∀[f,g:Point(rv) ⟶ Point(rv)].
(Isometry(f)) supposing (Isometry(g) and (∀x:Point(rv). g (f x) ≡ x))
Proof
Definitions occuring in Statement :
rv-isometry: Isometry(f),
inner-product-space: InnerProductSpace,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-isometry: Isometry(f),
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rv-isometry-injective,
rv-isometry-implies-functional,
req_witness,
rv-norm_wf,
rv-sub_wf,
inner-product-space_subtype,
rv-isometry_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
Error :ss-eq_wf,
req_functionality,
req_inversion,
req_weakening,
rv-norm_functionality,
rv-sub_functionality,
Error :ss-eq_weakening,
req-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
because_Cache,
sqequalRule,
isect_memberEquality_alt,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
universeIsType,
functionIsType,
instantiate,
independent_isectElimination,
productElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[f,g:Point(rv) {}\mrightarrow{} Point(rv)].
(Isometry(f)) supposing (Isometry(g) and (\mforall{}x:Point(rv). g (f x) \mequiv{} x))
Date html generated:
2020_05_20-PM-01_12_40
Last ObjectModification:
2020_01_06-PM-00_10_42
Theory : inner!product!spaces
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