Nuprl Lemma : hyptrans-perm_wf
∀[rv:InnerProductSpace]. ∀[e:Point]. ∀[t:ℝ]. hyptrans-perm(rv;e;t) ∈ Point supposing e^2 = r1
Proof
Definitions occuring in Statement :
hyptrans-perm: hyptrans-perm(rv;e;t),
hyptrans: hyptrans(rv;e;t;x),
translation-group: translation-group(rv;e;T),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
guard: {T},
hyptrans-perm: hyptrans-perm(rv;e;t),
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
ext-eq: A ≡ B
Lemmas referenced :
translation-group-point,
hyptrans_wf,
ss-point_wf,
real_wf,
hyptrans-is-translation-group-fun,
req_wf,
rv-ip_wf,
int-to-real_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
rminus_wf,
ss-eq_weakening,
all_wf,
ss-eq_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
lambdaEquality,
isectElimination,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
isect_memberEquality,
instantiate,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairEquality,
dependent_pairFormation,
lambdaFormation,
independent_pairFormation,
productEquality,
productElimination,
functionExtensionality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[e:Point]. \mforall{}[t:\mBbbR{}]. hyptrans-perm(rv;e;t) \mmember{} Point supposing e\^{}2 = r1
Date html generated:
2017_10_05-AM-00_28_55
Last ObjectModification:
2017_06_26-AM-10_27_01
Theory : inner!product!spaces
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