Nuprl Lemma : rv-be-dist
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. (a_b_c ⇒ (d(a;c) = (d(a;b) + d(b;c))))
Proof
Definitions occuring in Statement :
rv-be: a_b_c,
real-vec-dist: d(x;y),
real-vec: ℝ^n,
req: x = y,
radd: a + b,
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
cand: A c∧ B,
rv-be: a_b_c,
not: ¬A,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-between_wf,
not_wf,
real-vec-sep_wf,
rv-T-iff,
nat_wf,
real-vec_wf,
radd_wf,
int-to-real_wf,
rleq_wf,
real_wf,
real-vec-dist_wf,
req_witness,
rv-be_wf,
rv-T-dist
Rules used in proof :
productEquality,
independent_pairFormation,
productElimination,
isect_memberEquality,
because_Cache,
natural_numberEquality,
setEquality,
rename,
setElimination,
applyEquality,
lambdaEquality,
sqequalRule,
isectElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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_28-AM-07_38_13
Last ObjectModification:
2016_10_27-PM-02_10_16
Theory : reals
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