Nuprl Lemma : translation-group_wf
∀rv:InnerProductSpace. ∀e:Point. ∀T:ℝ ⟶ Point ⟶ Point.
translation-group(rv;e;T) ∈ s-Group supposing translation-group-fun(rv;e;T)
Proof
Definitions occuring in Statement :
translation-group: translation-group(rv;e;T),
translation-group-fun: translation-group-fun(rv;e;T),
inner-product-space: InnerProductSpace,
real: ℝ,
s-group: s-Group,
ss-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
translation-group: translation-group(rv;e;T),
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
guard: {T},
top: Top,
so_apply: x[s],
sg-subgroup: sg-subgroup(sg;x.P[x]),
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
prop: ℙ,
pi1: fst(t),
exists: ∃x:A. B[x],
trans-apply: T_t(x),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
translation-group-fun: translation-group-fun(rv;e;T),
sq_stable: SqStable(P),
squash: ↓T,
ss-eq: x ≡ y,
not: ¬A,
false: False,
compose: f o g
Lemmas referenced :
mk-s-subgroup_wf,
rv-permutation-group_wf,
exists_wf,
real_wf,
all_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-eq_wf,
rv-perm-point,
pi1_wf_top,
subtype_rel_product,
top_wf,
s-group-structure_subtype1,
s-group_subtype1,
s-group_wf,
s-group-structure_wf,
translation-group-fun_wf,
rv-perm-id,
int-to-real_wf,
trans-apply_wf,
ss-eq_weakening,
ss-eq_functionality,
trans-apply-0,
rv-perm-inv,
rminus_wf,
sq_stable__ss-eq,
radd_wf,
radd-rminus,
ss-eq_inversion,
trans-apply_functionality,
ss-sep_wf,
rv-perm-op,
compose_wf,
ss-eq_transitivity,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
setElimination,
rename,
functionExtensionality,
independent_pairFormation,
productElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
dependent_pairFormation,
natural_numberEquality,
independent_functionElimination,
promote_hyp,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:Point. \mforall{}T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point.
translation-group(rv;e;T) \mmember{} s-Group supposing translation-group-fun(rv;e;T)
Date html generated:
2017_10_05-AM-00_22_04
Last ObjectModification:
2017_06_24-PM-04_53_20
Theory : inner!product!spaces
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