Nuprl Lemma : trans-from-kernel-is-trans
∀rv:InnerProductSpace. ∀e:{e:Point(rv)| e^2 = r1} . ∀f,g:{h:Point(rv)| h ⋅ e = r0} ⟶ ℝ ⟶ ℝ.
(trans-kernel-fun(rv;e;f)
⇒ (∀h:{h:Point(rv)| h ⋅ e = r0} . ∀r:ℝ. ((f h (g h r)) = r))
⇒ translation-group-fun(rv;e;λt,x. trans-from-kernel(rv;e;f;g;t;x)))
Proof
Definitions occuring in Statement :
trans-from-kernel: trans-from-kernel(rv;e;f;g;t;x),
trans-kernel-fun: trans-kernel-fun(rv;e;f),
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),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
and: P ∧ Q,
exists: ∃x:A. B[x],
trans-kernel-fun: trans-kernel-fun(rv;e;f),
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
trans-from-kernel: trans-from-kernel(rv;e;f;g;t;x),
pi1: fst(t),
pi2: snd(t),
rv-decomp: rv-decomp(rv;x;e),
so_lambda: λ2x.t[x],
so_apply: x[s],
rv-sub: x - y,
rv-minus: -x,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
translation-group-fun: translation-group-fun(rv;e;T),
rev_implies: P ⇐ Q,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
true: True,
stable: Stable{P},
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
kernel-fun-properties,
sq_stable__req,
rv-ip_wf,
int-to-real_wf,
rv-sub_wf,
rv-mul_wf,
req_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
real_wf,
trans-kernel-fun_wf,
rsub_wf,
rmul_wf,
itermSubtract_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_functionality,
req_transitivity,
rv-ip-sub,
rsub_functionality,
req_weakening,
rv-ip-mul,
rmul_functionality,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
rv-decomp_wf,
rv-add_wf,
Error :sq_stable__ss-eq,
radd_wf,
pi2_wf,
pi1_wf_top,
Error :ss-eq_weakening,
itermAdd_wf,
Error :ss-eq_wf,
rminus_wf,
itermMinus_wf,
rv-0_wf,
rv-ip-add,
radd_functionality,
real_term_value_add_lemma,
Error :ss-eq_functionality,
Error :ss-eq_transitivity,
Error :ss-eq_inversion,
rv-sub_functionality,
rv-mul_functionality,
uiff_transitivity,
rv-add_functionality,
rv-mul-mul,
rv-mul-add-alt,
rv-add-comm,
rv-mul0,
rv-add-0,
real_term_value_minus_lemma,
req_inversion,
not-rneq,
rneq_wf,
rless_wf,
rneq_functionality,
rneq_irreflexivity,
req-implies-req,
trans-from-kernel-sep,
trans-from-kernel_wf,
trans-from-kernel_functionality,
rleq_wf,
iff_weakening_uiff,
Error :ss-sep_wf,
Error :ss-sep_functionality,
rv-add-sep-iff,
rv-mul-sep-iff,
rneq-by-function,
rv-norm-positive-iff,
rv-norm_wf,
rnexp_wf,
istype-le,
rleq-int,
istype-false,
rless-int,
rv-norm-eq-iff,
rnexp2,
rless_functionality,
stable__rleq,
false_wf,
not_wf,
trivial-rless-radd,
rleq_weakening_rless,
rleq_weakening,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
radd-preserves-req,
rless_transitivity2,
rless_transitivity1,
nat_plus_properties,
full-omega-unsat,
intformless_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rleq-implies-rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
isectElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
dependent_set_memberEquality_alt,
applyEquality,
universeIsType,
functionIsType,
setIsType,
instantiate,
independent_isectElimination,
inhabitedIsType,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
promote_hyp,
applyLambdaEquality,
independent_pairEquality,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
minusEquality,
unionElimination,
inlFormation_alt,
inrFormation_alt,
dependent_pairFormation_alt,
productIsType,
independent_pairFormation,
unionEquality,
functionEquality,
unionIsType
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:\{e:Point(rv)| e\^{}2 = r1\} . \mforall{}f,g:\{h:Point(rv)| h \mcdot{} e = r0\} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}.
(trans-kernel-fun(rv;e;f)
{}\mRightarrow{} (\mforall{}h:\{h:Point(rv)| h \mcdot{} e = r0\} . \mforall{}r:\mBbbR{}. ((f h (g h r)) = r))
{}\mRightarrow{} translation-group-fun(rv;e;\mlambda{}t,x. trans-from-kernel(rv;e;f;g;t;x)))
Date html generated:
2020_05_20-PM-01_17_30
Last ObjectModification:
2019_12_08-PM-07_01_42
Theory : inner!product!spaces
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