Nuprl Lemma : real-vec-dist-dilation
∀[n:ℕ]. ∀[x,y:ℝ^n]. ∀[a:ℝ].  (d(a*x;a*y) = (|a| * d(x;y)))
Proof
Definitions occuring in Statement : 
real-vec-dist: d(x;y)
, 
real-vec-mul: a*X
, 
real-vec: ℝ^n
, 
rabs: |x|
, 
req: x = y
, 
rmul: a * b
, 
real: ℝ
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
real-vec-dist: d(x;y)
, 
subtype_rel: A ⊆r B
, 
prop: ℙ
, 
implies: P 
⇒ Q
, 
real-vec-sub: X - Y
, 
real-vec-mul: a*X
, 
req-vec: req-vec(n;x;y)
, 
all: ∀x:A. B[x]
, 
nat: ℕ
, 
real-vec: ℝ^n
, 
uimplies: b supposing a
, 
rsub: x - y
, 
and: P ∧ Q
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced : 
req_witness, 
real-vec-dist_wf, 
real-vec-mul_wf, 
real_wf, 
rleq_wf, 
int-to-real_wf, 
rmul_wf, 
rabs_wf, 
real-vec_wf, 
nat_wf, 
int_seg_wf, 
req_wf, 
radd_wf, 
rminus_wf, 
req_weakening, 
uiff_transitivity, 
req_functionality, 
req_transitivity, 
rmul-distrib, 
radd_functionality, 
rmul_over_rminus, 
rminus_functionality, 
rmul_comm, 
real-vec-norm_wf, 
real-vec-sub_wf, 
real-vec-norm-mul, 
real-vec-norm_functionality
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
applyEquality, 
lambdaEquality, 
setElimination, 
rename, 
setEquality, 
natural_numberEquality, 
sqequalRule, 
independent_functionElimination, 
isect_memberEquality, 
because_Cache, 
lambdaFormation, 
independent_isectElimination, 
productElimination
Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[x,y:\mBbbR{}\^{}n].  \mforall{}[a:\mBbbR{}].    (d(a*x;a*y)  =  (|a|  *  d(x;y)))
Date html generated:
2016_10_26-AM-10_26_55
Last ObjectModification:
2016_09_28-PM-00_36_44
Theory : reals
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