Nuprl Lemma : real-vec-sep-iff-dot-product
∀n:ℕ. ∀x,y:ℝ^n. (x ≠ y ⇐⇒ r0 < y - x⋅y - x)
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
dot-product: x⋅y,
real-vec-sub: X - Y,
real-vec: ℝ^n,
rless: x < y,
int-to-real: r(n),
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
rev_implies: P ⇐ Q,
real-vec: ℝ^n,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
nat: ℕ,
prop: ℙ,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
real-vec-sub: X - Y,
rsub: x - y,
radd: a + b,
accelerate: accelerate(k;f),
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
real-vec-sep-iff,
int-to-real_wf,
int_seg_wf,
real-vec-sep_wf,
real-vec-sub_wf,
real-vec-sep-0-iff,
rless_wf,
dot-product_wf,
real-vec_wf,
istype-nat,
rabs_wf,
rsub_wf,
subtype_rel_self,
real_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
rless_functionality,
req_weakening,
rabs-difference-symmetry,
rabs_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_functionElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
lambdaEquality_alt,
isectElimination,
setElimination,
rename,
universeIsType,
natural_numberEquality,
independent_pairFormation,
promote_hyp,
dependent_pairFormation_alt,
applyEquality,
functionEquality,
imageElimination,
productIsType,
independent_isectElimination,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n. (x \mneq{} y \mLeftarrow{}{}\mRightarrow{} r0 < y - x\mcdot{}y - x)
Date html generated:
2019_10_30-AM-08_44_13
Last ObjectModification:
2019_07_29-PM-00_36_48
Theory : reals
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