Nuprl Lemma : Minkowski-inequality2
∀[n:ℕ]. ∀[x,y:ℝ^n]. (||x - y|| ≤ (||x|| + ||y||))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
real-vec-sub: X - Y,
real-vec: ℝ^n,
rleq: x ≤ y,
radd: a + b,
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
true: True,
absval: |i|,
uimplies: b supposing a,
uiff: uiff(P;Q),
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
real-vec-mul: a*X,
real-vec-add: X + Y,
real-vec-sub: X - Y,
req-vec: req-vec(n;x;y),
nat: ℕ,
real-vec: ℝ^n,
rsub: x - y
Lemmas referenced :
rminus-as-rmul,
req_inversion,
req_functionality,
rminus_wf,
int_seg_wf,
real-vec-norm_functionality,
rleq_weakening,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rmul-one-both,
iff_weakening_equal,
rabs-int,
true_wf,
squash_wf,
rleq_wf,
real-vec-norm-mul,
radd_functionality,
req_weakening,
rleq_functionality,
rabs_wf,
rmul_wf,
real-vec-add_wf,
nat_wf,
real-vec_wf,
nat_plus_wf,
real_wf,
real-vec-sub_wf,
real-vec-norm_wf,
radd_wf,
rsub_wf,
less_than'_wf,
int-to-real_wf,
real-vec-mul_wf,
Minkowski-inequality1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
minusEquality,
natural_numberEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
setElimination,
rename,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
callbyvalueReduce,
sqleReflexivity,
independent_isectElimination,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
lambdaFormation
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (||x - y|| \mleq{} (||x|| + ||y||))
Date html generated:
2016_05_18-AM-09_50_29
Last ObjectModification:
2016_01_17-AM-02_51_51
Theory : reals
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