Nuprl Lemma : orthogonal-group_wf
∀[rv:InnerProductSpace]. (O(rv) ∈ s-Group)
Proof
Definitions occuring in Statement :
orthogonal-group: O(rv),
inner-product-space: InnerProductSpace,
s-group: s-Group,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
squash: ↓T,
sq_stable: SqStable(P),
prop: ℙ,
implies: P ⇒ Q,
cand: A c∧ B,
pi1: fst(t),
and: P ∧ Q,
sg-subgroup: sg-subgroup(sg;x.P[x]),
so_apply: x[s],
top: Top,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
orthogonal-group: O(rv),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-orthogonal-inverse,
sq_stable__rv-orthogonal,
rv-orthogonal-compose,
rv-perm-op,
rv-perm-inv,
rv-orthogonal-id,
rv-perm-id,
s-group_subtype1,
top_wf,
subtype_rel_product,
rv-perm-point,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-point_wf,
pi1_wf_top,
rv-orthogonal_wf,
rv-permutation-group_wf,
mk-s-subgroup_wf
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
productElimination,
independent_pairFormation,
lambdaFormation,
rename,
setElimination,
voidEquality,
voidElimination,
isect_memberEquality,
independent_isectElimination,
instantiate,
applyEquality,
functionEquality,
dependent_functionElimination,
because_Cache,
lambdaEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. (O(rv) \mmember{} s-Group)
Date html generated:
2016_11_08-AM-09_21_19
Last ObjectModification:
2016_11_03-PM-01_43_57
Theory : inner!product!spaces
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