Nuprl Lemma : hyptrans_ext
∀rv:InnerProductSpace. ∀e:Point. ∀s,t:ℝ. ∀x,y:Point. (hyptrans(rv;e;s;x) # hyptrans(rv;e;t;y) ⇒ (x # y ∨ s ≠ t))
Proof
Definitions occuring in Statement :
hyptrans: hyptrans(rv;e;t;x),
inner-product-space: InnerProductSpace,
rneq: x ≠ y,
real: ℝ,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
hyptrans: hyptrans(rv;e;t;x),
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
or: P ∨ Q,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rneq: x ≠ y
Lemmas referenced :
rv-add-sep,
inner-product-space_subtype,
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,
rneq_wf,
ss-sep_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
hyptrans_wf,
ss-point_wf,
real_wf,
rv-mul-sep,
ss-sep-irrefl,
rneq-radd,
rneq-rmul,
rsqrt-rneq,
rneq_irreflexivity,
rv-ip-rneq,
rneq-function,
req_functionality,
rsub_functionality,
cosh_functionality,
req_weakening,
req_wf,
rneq-sinh
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
isectElimination,
because_Cache,
natural_numberEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
dependent_set_memberEquality,
unionElimination,
inlFormation,
instantiate,
independent_isectElimination,
promote_hyp,
inrFormation,
voidElimination,
lambdaEquality
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:Point. \mforall{}s,t:\mBbbR{}. \mforall{}x,y:Point.
(hyptrans(rv;e;s;x) \# hyptrans(rv;e;t;y) {}\mRightarrow{} (x \# y \mvee{} s \mneq{} t))
Date html generated:
2017_10_05-AM-00_28_07
Last ObjectModification:
2017_06_26-PM-01_55_33
Theory : inner!product!spaces
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