Nuprl Lemma : real-vec-dist-from-zero
∀[n:ℕ]. ∀[p:ℝ^n]. (d(p;λi.r0) = ||p||)
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
real-vec: ℝ^n,
req: x = y,
int-to-real: r(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
real-vec-dist: d(x;y),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
req-vec: req-vec(n;x;y),
all: ∀x:A. B[x],
real-vec-sub: X - Y,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
real-vec-norm_functionality,
real-vec-sub_wf,
int-to-real_wf,
int_seg_wf,
req_witness,
real-vec-dist_wf,
real-vec-norm_wf,
real-vec_wf,
istype-nat,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
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,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
productElimination,
hypothesis,
universeIsType,
natural_numberEquality,
independent_isectElimination,
applyEquality,
because_Cache,
independent_functionElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
lambdaFormation_alt,
dependent_functionElimination,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p:\mBbbR{}\^{}n]. (d(p;\mlambda{}i.r0) = ||p||)
Date html generated:
2019_10_30-AM-08_29_06
Last ObjectModification:
2019_07_02-AM-11_00_41
Theory : reals
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