Nuprl Lemma : trans-apply-0
∀rv:InnerProductSpace. ∀T:ℝ ⟶ Point ⟶ Point. ∀x:Point. T_r0(x) ≡ x supposing ∃e:Point. translation-group-fun(rv;e;T)
Proof
Definitions occuring in Statement :
trans-apply: T_t(x),
translation-group-fun: translation-group-fun(rv;e;T),
inner-product-space: InnerProductSpace,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
exists: ∃x:A. B[x],
translation-group-fun: translation-group-fun(rv;e;T),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
trans-apply: T_t(x),
stable: Stable{P},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
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-sep_wf,
trans-apply_wf,
real_wf,
int-to-real_wf,
exists_wf,
translation-group-fun_wf,
ss-eq_wf,
rv-add_wf,
rv-mul_wf,
rv-0_wf,
uiff_transitivity,
ss-eq_functionality,
rv-add_functionality,
ss-eq_weakening,
rv-mul0,
rv-0-add,
radd_wf,
trans-apply_functionality,
req_weakening,
stable__ss-eq,
false_wf,
or_wf,
rneq_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
ss-sep_functionality,
not-rneq,
radd-preserves-req,
rminus_wf,
req_functionality,
radd-rminus-assoc,
radd-rminus-both,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
voidElimination,
because_Cache,
functionExtensionality,
natural_numberEquality,
functionEquality,
independent_functionElimination,
dependent_pairFormation,
unionElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point.
\mforall{}x:Point. T\_r0(x) \mequiv{} x supposing \mexists{}e:Point. translation-group-fun(rv;e;T)
Date html generated:
2017_10_05-AM-00_21_33
Last ObjectModification:
2017_07_28-AM-08_55_30
Theory : inner!product!spaces
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