Nuprl Lemma : hyptrans_functionality
∀[rv:InnerProductSpace]. ∀[e1,x1:Point]. ∀[t1:ℝ]. ∀[e2,x2:Point]. ∀[t2:ℝ].
(hyptrans(rv;e1;t1;x1) ≡ hyptrans(rv;e2;t2;x2)) supposing (x1 ≡ x2 and e1 ≡ e2 and (t1 = t2))
Proof
Definitions occuring in Statement :
hyptrans: hyptrans(rv;e;t;x),
inner-product-space: InnerProductSpace,
req: x = y,
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
hyptrans: hyptrans(rv;e;t;x),
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
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,
ss-eq_wf,
req_wf,
real_wf,
ss-point_wf,
rv-add_wf,
rv-mul_wf,
radd_wf,
rmul_wf,
rv-ip_wf,
rsub_wf,
cosh_wf,
int-to-real_wf,
rsqrt_wf,
radd-non-neg,
rleq-int,
false_wf,
rv-ip-nonneg,
rleq_wf,
sinh_wf,
ss-eq_weakening,
ss-eq_functionality,
rv-add_functionality,
rv-mul_functionality,
radd_functionality,
rmul_functionality,
rsqrt_functionality,
rv-ip_functionality,
req_weakening,
sinh_functionality,
rsub_functionality,
cosh_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
natural_numberEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
setElimination,
rename,
setEquality,
productEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[e1,x1:Point]. \mforall{}[t1:\mBbbR{}]. \mforall{}[e2,x2:Point]. \mforall{}[t2:\mBbbR{}].
(hyptrans(rv;e1;t1;x1) \mequiv{} hyptrans(rv;e2;t2;x2)) supposing (x1 \mequiv{} x2 and e1 \mequiv{} e2 and (t1 = t2))
Date html generated:
2017_10_05-AM-00_27_23
Last ObjectModification:
2017_06_21-PM-01_02_44
Theory : inner!product!spaces
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