Nuprl Lemma : rv-orthogonal-compose
∀[rv:InnerProductSpace]. ∀[f,g:Point ⟶ Point]. (Orthogonal(f o g)) supposing (Orthogonal(g) and Orthogonal(f))
Proof
Definitions occuring in Statement :
rv-orthogonal: Orthogonal(f),
inner-product-space: InnerProductSpace,
ss-point: Point,
compose: f o g,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
prop: ℙ,
guard: {T},
false: False,
not: ¬A,
ss-eq: x ≡ y,
rv-orthogonal: Orthogonal(f),
compose: f o g,
cand: A c∧ B,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
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 :
ss-eq_transitivity,
rv-0_wf,
rv-orthogonal_wf,
real_wf,
rv-mul_wf,
rv-ip_wf,
req_witness,
rv-add_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-isometry-compose,
ss-point_wf,
compose_wf,
rv-orthogonal-iff,
rv-orthogonal-implies-functional
Rules used in proof :
voidElimination,
functionEquality,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
instantiate,
lambdaEquality,
independent_pairEquality,
independent_isectElimination,
independent_pairFormation,
functionExtensionality,
sqequalRule,
because_Cache,
applyEquality,
productElimination,
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 o g)) supposing (Orthogonal(g) and Orthogonal(f))
Date html generated:
2016_11_08-AM-09_20_24
Last ObjectModification:
2016_11_02-PM-11_30_30
Theory : inner!product!spaces
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