Nuprl Lemma : real-vec-triangle-inequality
∀[n:ℕ]. ∀[x,y,z:ℝ^n]. (d(x;z) ≤ (d(x;y) + d(y;z)))
Proof
Definitions occuring in Statement :
real-vec-dist: d(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,
real-vec-dist: d(x;y),
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,
prop: ℙ,
real: ℝ,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
req-vec: req-vec(n;x;y),
real-vec-sub: X - Y,
real-vec-add: X + Y,
nat: ℕ,
real-vec: ℝ^n,
rsub: x - y,
uiff: uiff(P;Q)
Lemmas referenced :
less_than'_wf,
rsub_wf,
radd_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
nat_plus_wf,
real-vec_wf,
nat_wf,
real-vec-norm_wf,
real-vec-sub_wf,
real-vec-add_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
Minkowski-inequality1,
real-vec-norm_functionality,
int_seg_wf,
req_wf,
rminus_wf,
req_weakening,
uiff_transitivity,
req_functionality,
req_inversion,
radd-assoc,
radd_functionality,
radd-ac,
radd-rminus-assoc,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
setEquality,
natural_numberEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_isectElimination,
lambdaFormation,
independent_functionElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y,z:\mBbbR{}\^{}n]. (d(x;z) \mleq{} (d(x;y) + d(y;z)))
Date html generated:
2016_10_26-AM-10_27_25
Last ObjectModification:
2016_09_14-PM-06_48_45
Theory : reals
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