Nuprl Lemma : rv-orthogonal-injective
∀[rv:InnerProductSpace]. ∀f:Point(rv) ⟶ Point(rv). (Orthogonal(f) ⇒ (∀x,y:Point(rv). (f x ≡ f y ⇒ x ≡ y)))
Proof
Definitions occuring in Statement :
rv-orthogonal: Orthogonal(f),
inner-product-space: InnerProductSpace,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
prop: ℙ,
ss-eq: Error :ss-eq,
not: ¬A,
false: False,
rv-orthogonal: Orthogonal(f),
rev_uimplies: rev_uimplies(P;Q),
rv-sub: x - y,
rv-minus: -x
Lemmas referenced :
rv-sub-is-zero,
inner-product-space_subtype,
rv-ip-zero-iff,
rv-sub_wf,
Error :ss-eq_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
Error :ss-point_wf,
rv-orthogonal_wf,
rv-ip_wf,
int-to-real_wf,
req_functionality,
req_weakening,
rv-add_wf,
rv-mul_wf,
rv-0_wf,
Error :ss-eq_functionality,
Error :ss-eq_transitivity,
rv-add_functionality,
Error :ss-eq_weakening,
rv-add-minus2,
rv-ip0,
rv-ip_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
productElimination,
independent_isectElimination,
universeIsType,
instantiate,
inhabitedIsType,
because_Cache,
dependent_functionElimination,
functionIsType,
lambdaEquality_alt,
voidElimination,
functionIsTypeImplies,
natural_numberEquality,
minusEquality,
independent_functionElimination
Latex:
\mforall{}[rv:InnerProductSpace]
\mforall{}f:Point(rv) {}\mrightarrow{} Point(rv). (Orthogonal(f) {}\mRightarrow{} (\mforall{}x,y:Point(rv). (f x \mequiv{} f y {}\mRightarrow{} x \mequiv{} y)))
Date html generated:
2020_05_20-PM-01_11_57
Last ObjectModification:
2019_12_09-PM-11_41_15
Theory : inner!product!spaces
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