Nuprl Lemma : mdist-rn-metric-mul
∀[n:ℕ]. ∀[p:ℝ^n]. ∀[c:ℝ]. (mdist(rn-metric(n);c*p;λi.r0) = (|c| * mdist(rn-metric(n);p;λi.r0)))
Proof
Definitions occuring in Statement :
rn-metric: rn-metric(n),
real-vec-mul: a*X,
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rabs: |x|,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
rn-metric: rn-metric(n),
mdist: mdist(d;x;y),
member: t ∈ T,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
real_wf,
real-vec_wf,
istype-nat,
real-vec-dist_wf,
real-vec-mul_wf,
int-to-real_wf,
int_seg_wf,
real-vec-norm_wf,
rmul_wf,
rabs_wf,
real-vec-norm-mul,
req_functionality,
real-vec-dist-from-zero,
rmul_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
lambdaEquality_alt,
setElimination,
rename,
productElimination,
natural_numberEquality,
applyEquality,
independent_isectElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p:\mBbbR{}\^{}n]. \mforall{}[c:\mBbbR{}]. (mdist(rn-metric(n);c*p;\mlambda{}i.r0) = (|c| * mdist(rn-metric(n);p;\mlambda{}i.r0)))
Date html generated:
2019_10_30-AM-08_40_32
Last ObjectModification:
2019_10_02-AM-11_05_06
Theory : reals
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