Nuprl Lemma : real-vec-dist-vec-mul
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[a,b:ℝ]. (d(a*x;b*x) = (|a - b| * ||x||))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
real-vec-mul: a*X,
real-vec: ℝ^n,
rabs: |x|,
rsub: x - y,
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-mul: a*X,
real-vec-sub: X - Y,
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,
rsub_wf,
real-vec-norm_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-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:\mBbbR{}\^{}n]. \mforall{}[a,b:\mBbbR{}]. (d(a*x;b*x) = (|a - b| * ||x||))
Date html generated:
2016_10_26-AM-10_27_10
Last ObjectModification:
2016_09_29-AM-01_22_00
Theory : reals
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