Nuprl Lemma : hyptrans_lemma
∀[rv:InnerProductSpace]. ∀[e,h:Point].
∀tau,t:ℝ. hyptrans(rv;e;t;h + sinh(tau) * rsqrt(r1 + h^2)*e) ≡ h + sinh(tau + t) * rsqrt(r1 + h^2)*e
supposing (e^2 = r1) ∧ (h ⋅ e = r0)
Proof
Definitions occuring in Statement :
hyptrans: hyptrans(rv;e;t;x),
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,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
hyptrans: hyptrans(rv;e;t;x),
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
prop: ℙ,
nat: ℕ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
rleq: x ≤ y,
rnonneg: rnonneg(x),
rge: x ≥ y,
or: P ∨ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
real_wf,
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
hyptrans_wf,
rv-add_wf,
rv-mul_wf,
rmul_wf,
sinh_wf,
rsqrt_wf,
radd-non-neg,
int-to-real_wf,
rv-ip_wf,
rleq-int,
false_wf,
rv-ip-nonneg,
radd_wf,
rleq_wf,
req_wf,
ss-point_wf,
rnexp_wf,
le_wf,
req_functionality,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rv-ip-symmetry,
req_weakening,
req_transitivity,
rv-ip-add,
radd_functionality,
rv-ip-add2,
rv-ip-mul,
rmul_functionality,
rv-ip-mul2,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
rmul-assoc,
rsqrt_squared,
rnexp2,
cosh2-sinh2,
radd-preserves-req,
rsub_wf,
cosh_wf,
radd_comm,
trivial-rleq-radd,
rmul_preserves_rleq2,
rnexp2-nonneg,
less_than'_wf,
nat_plus_wf,
rleq_functionality,
rmul-zero-both,
rleq-implies-rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
uiff_transitivity,
rsqrt-unique,
rmul-nonneg,
rsqrt_nonneg,
cosh-ge-1,
set_wf,
equal_wf,
ss-eq_functionality,
rv-add_functionality,
ss-eq_weakening,
rv-mul_functionality,
req_inversion,
sinh-radd,
ss-eq_inversion,
rv-mul-add,
rminus_wf,
itermMinus_wf,
real_term_value_minus_lemma,
ss-eq_wf,
ss-eq_transitivity,
rv-add-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
natural_numberEquality,
independent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality,
setElimination,
rename,
setEquality,
independent_pairEquality,
minusEquality,
axiomEquality,
inlFormation
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[e,h:Point].
\mforall{}tau,t:\mBbbR{}.
hyptrans(rv;e;t;h + sinh(tau) * rsqrt(r1 + h\^{}2)*e) \mequiv{} h + sinh(tau + t) * rsqrt(r1 + h\^{}2)*e
supposing (e\^{}2 = r1) \mwedge{} (h \mcdot{} e = r0)
Date html generated:
2017_10_05-AM-00_27_46
Last ObjectModification:
2017_06_21-PM-02_21_45
Theory : inner!product!spaces
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