Nuprl Lemma : rv-orthogonal-inverse
∀[rv:InnerProductSpace]. ∀[f,g:Point ⟶ Point]. (Orthogonal(f)) supposing (Orthogonal(g) and (∀x:Point. g (f x) ≡ x))
Proof
Definitions occuring in Statement :
rv-orthogonal: Orthogonal(f),
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
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 :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
guard: {T},
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
ss-eq: x ≡ y,
rv-orthogonal: Orthogonal(f),
cand: A c∧ B,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-isometry-inverse,
ss-eq_weakening,
ss-eq_functionality,
ss-eq_inversion,
rv-0_wf,
ss-eq_wf,
all_wf,
rv-orthogonal_wf,
real_wf,
rv-mul_wf,
rv-ip_wf,
req_witness,
rv-add_wf,
ss-point_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-sep_wf,
rv-orthogonal-iff,
rv-orthogonal-injective
Rules used in proof :
voidElimination,
functionEquality,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
functionExtensionality,
independent_isectElimination,
instantiate,
applyEquality,
lambdaEquality,
independent_pairEquality,
sqequalRule,
independent_pairFormation,
productElimination,
because_Cache,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[f,g:Point {}\mrightarrow{} Point].
(Orthogonal(f)) supposing (Orthogonal(g) and (\mforall{}x:Point. g (f x) \mequiv{} x))
Date html generated:
2016_11_08-AM-09_20_32
Last ObjectModification:
2016_11_02-PM-11_43_38
Theory : inner!product!spaces
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