Nuprl Lemma : hyptrans_decomp
∀rv:InnerProductSpace. ∀e,x:Point.
∃h:Point. ∃tau:ℝ. ((h ⋅ e = r0) ∧ x ≡ h + sinh(tau) * rsqrt(r1 + h^2)*e) supposing e^2 = r1
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
sinh: sinh(x),
rsqrt: rsqrt(x),
req: x = y,
rmul: a * b,
radd: a + b,
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],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
exists: ∃x:A. B[x],
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
req_witness,
rv-ip_wf,
int-to-real_wf,
req_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,
rv-decomp_wf,
rsqrt-1-plus-ip-positive,
equal_wf,
rless_wf,
rsqrt_wf,
radd-non-neg,
rleq-int,
false_wf,
rv-ip-nonneg,
radd_wf,
pi1_wf_top,
subtype_rel_product,
real_wf,
top_wf,
rleq_wf,
inv-sinh_wf,
rdiv_wf,
ss-eq_wf,
rv-add_wf,
rv-mul_wf,
rmul_wf,
sinh_wf,
exists_wf,
set_wf,
sq_stable__req,
sq_stable__ss-eq,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
ss-eq_functionality,
ss-eq_weakening,
rv-add_functionality,
rv-mul_functionality,
rmul_functionality,
sinh-inv-sinh,
req_weakening,
req_transitivity,
rmul-rinv3,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_functionElimination,
rename,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inrFormation,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_pairFormation,
dependent_set_memberEquality,
setElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
productEquality,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addLevel,
approximateComputation,
int_eqEquality,
intEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e,x:Point.
\mexists{}h:Point. \mexists{}tau:\mBbbR{}. ((h \mcdot{} e = r0) \mwedge{} x \mequiv{} h + sinh(tau) * rsqrt(r1 + h\^{}2)*e) supposing e\^{}2 = r1
Date html generated:
2017_10_05-AM-00_27_53
Last ObjectModification:
2017_06_21-PM-02_26_41
Theory : inner!product!spaces
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