Nuprl Lemma : real-vec-dist-between
∀n:ℕ. ∀a,b,c:ℝ^n. (a-b-c ⇒ (d(a;c) = (d(a;b) + d(b;c))))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec-between: a-b-c,
real-vec: ℝ^n,
req: x = y,
radd: a + b,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
real-vec-between: a-b-c,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
top: Top,
guard: {T},
rsub: x - y
Lemmas referenced :
real-vec-between_wf,
real-vec_wf,
nat_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
radd_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
rmul_wf,
rabs_wf,
req_functionality,
req_weakening,
radd_functionality,
real-vec-dist_functionality,
req-vec_weakening,
real-vec-dist-between-2,
real-vec-dist-between-1,
member_rooint_lemma,
rleq_weakening_rless,
radd-preserves-rleq,
rminus_wf,
rmul_functionality,
rabs-of-nonneg,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd-ac,
radd-rminus-both,
radd-zero-both,
req_wf,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rmul-one-both,
rmul_comm,
rminus_functionality,
req_inversion,
radd-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
natural_numberEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n. (a-b-c {}\mRightarrow{} (d(a;c) = (d(a;b) + d(b;c))))
Date html generated:
2016_10_26-AM-10_35_13
Last ObjectModification:
2016_09_25-AM-00_15_35
Theory : reals
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