Nuprl Lemma : rv-orthogonal-implies-extensional-ext
∀rv:InnerProductSpace. ∀f:Point(rv) ⟶ Point(rv). ∀x,y:Point(rv). (f x # f y ⇒ x # y) supposing Orthogonal(f)
Proof
Definitions occuring in Statement :
rv-orthogonal: Orthogonal(f),
inner-product-space: InnerProductSpace,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
member: t ∈ T,
rv-orthogonal-implies-extensional,
rv-isometry-implies-extensional,
rv-sep-iff-ext,
rv-norm-positive-iff-ext,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
rv-orthogonal-implies-extensional,
lifting-strict-spread,
has-value_wf_base,
base_wf,
is-exception_wf,
strict4-spread,
rv-isometry-implies-extensional,
rv-sep-iff-ext,
rv-norm-positive-iff-ext
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueApply,
baseApply,
closedConclusion,
hypothesisEquality,
applyExceptionCases,
inrFormation,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}f:Point(rv) {}\mrightarrow{} Point(rv).
\mforall{}x,y:Point(rv). (f x \# f y {}\mRightarrow{} x \# y) supposing Orthogonal(f)
Date html generated:
2020_05_20-PM-01_12_01
Last ObjectModification:
2020_05_01-AM-07_35_31
Theory : inner!product!spaces
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