Nuprl Lemma : real-vec-dist_functionality
∀[n:ℕ]. ∀[x1,x2,y1,y2:ℝ^n]. (d(x1;y1) = d(x2;y2)) supposing (req-vec(n;x1;x2) and req-vec(n;y1;y2))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
req: x = y,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
real-vec-dist: d(x;y),
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req-vec_wf,
real-vec_wf,
nat_wf,
real-vec-norm_wf,
real-vec-sub_wf,
req_weakening,
req_functionality,
real-vec-norm_functionality,
real-vec-sub_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,
because_Cache,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x1,x2,y1,y2:\mBbbR{}\^{}n]. (d(x1;y1) = d(x2;y2)) supposing (req-vec(n;x1;x2) and req-vec(n;y1;y2))
Date html generated:
2016_10_26-AM-10_25_02
Last ObjectModification:
2016_09_24-PM-11_42_41
Theory : reals
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