Nuprl Lemma : hyptrans-is-translation-group-fun
∀rv:InnerProductSpace. ∀e:Point. ((e^2 = r1) ⇒ translation-group-fun(rv;e;λt,x. hyptrans(rv;e;t;x)))
Proof
Definitions occuring in Statement :
hyptrans: hyptrans(rv;e;t;x),
translation-group-fun: translation-group-fun(rv;e;T),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
req: x = y,
int-to-real: r(n),
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
translation-group-fun: translation-group-fun(rv;e;T),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
cand: A c∧ B,
uiff: uiff(P;Q),
rneq: x ≠ y,
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
rdiv: (x/y),
less_than: a < b,
squash: ↓T,
true: True,
rleq: x ≤ y,
rnonneg: rnonneg(x),
real: ℝ,
sq_stable: SqStable(P)
Lemmas referenced :
hyptrans_ext,
hyptrans_add,
real_wf,
hyptrans_decomp,
set_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rv-ip_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,
rsqrt-1-plus-ip-positive,
rsqrt_wf,
radd-non-neg,
rleq-int,
false_wf,
rv-ip-nonneg,
radd_wf,
rless_wf,
equal_wf,
hyptrans_wf,
rv-add_wf,
rv-mul_wf,
rmul_wf,
sinh_wf,
ss-eq_wf,
ss-sep_wf,
exists_wf,
all_wf,
rneq_wf,
ss-eq_functionality,
hyptrans_lemma,
ss-eq_weakening,
ss-sep_functionality,
hyptrans_functionality,
req_weakening,
rv-add_functionality,
rsub_wf,
inv-sinh_wf,
rdiv_wf,
req_functionality,
rmul_preserves_req,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
rinv_wf2,
radd_comm,
req-implies-req,
rv-ip_functionality,
req_transitivity,
rv-ip-add,
radd_functionality,
rv-ip-mul,
rmul_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
rmul-rinv3,
inv-sinh_functionality,
req_inversion,
inv-sinh-sinh,
rv-mul_functionality,
sinh_functionality,
sinh-inv-sinh,
uiff_transitivity,
rv-mul-add-alt,
rv-add-comm,
rneq-inv-sinh,
rless-implies-rless,
rneq_functionality,
rmul_preserves_rneq_iff2,
rv-0_wf,
rv-ip0,
rless-int,
rv-mul-sep-iff,
rv-ip-rneq,
ss-eq_inversion,
rv-mul-add,
rv-add-sep-iff,
rv-add-assoc,
rmul_preserves_rleq2,
rleq_weakening_rless,
less_than'_wf,
nat_plus_wf,
rminus_wf,
itermMinus_wf,
rleq-implies-rleq,
rleq_functionality,
real_term_value_minus_lemma,
sinh-rleq,
trivial-rleq-radd,
sq_stable__rleq,
rsub_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
because_Cache,
independent_isectElimination,
productElimination,
lambdaEquality,
natural_numberEquality,
applyEquality,
instantiate,
independent_functionElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
setEquality,
productEquality,
functionEquality,
addLevel,
existsFunctionality,
allFunctionality,
impliesFunctionality,
andLevelFunctionality,
allLevelFunctionality,
impliesLevelFunctionality,
levelHypothesis,
existsLevelFunctionality,
inrFormation,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
unionElimination,
inlFormation,
promote_hyp,
imageMemberEquality,
baseClosed,
isect_memberFormation,
independent_pairEquality,
minusEquality,
axiomEquality,
imageElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:Point. ((e\^{}2 = r1) {}\mRightarrow{} translation-group-fun(rv;e;\mlambda{}t,x. hyptrans(rv;e;t;x)\000C))
Date html generated:
2017_10_05-AM-00_28_45
Last ObjectModification:
2017_07_28-AM-08_55_34
Theory : inner!product!spaces
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