Nuprl Lemma : real-vec-dist-between-2
∀n:ℕ. ∀a,c:ℝ^n. ∀t:ℝ. (d(t*a + r1 - t*c;c) = (|t| * d(a;c)))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec-mul: a*X,
real-vec-add: X + Y,
real-vec: ℝ^n,
rabs: |x|,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
real-vec-dist: d(x;y),
real-vec-sub: X - Y,
real-vec-mul: a*X,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
nat: ℕ,
real-vec: ℝ^n,
uimplies: b supposing a,
rsub: x - y,
implies: P ⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
real_wf,
real-vec_wf,
nat_wf,
int_seg_wf,
req_wf,
radd_wf,
rmul_wf,
int-to-real_wf,
rminus_wf,
req_weakening,
uiff_transitivity,
req_functionality,
radd_functionality,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rmul-one-both,
rminus_functionality,
rmul_comm,
req_inversion,
radd-assoc,
radd-ac,
radd_comm,
radd-rminus-assoc,
real-vec-norm_wf,
real-vec-sub_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
rabs_wf,
real-vec-norm_functionality,
real-vec-dist_wf,
rleq_wf,
real-vec-norm-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
natural_numberEquality,
setElimination,
rename,
applyEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
productElimination,
lambdaEquality,
setEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,c:\mBbbR{}\^{}n. \mforall{}t:\mBbbR{}. (d(t*a + r1 - t*c;c) = (|t| * d(a;c)))
Date html generated:
2016_10_26-AM-10_34_59
Last ObjectModification:
2016_09_25-AM-00_07_52
Theory : reals
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