Nuprl Lemma : real-vec-dist-equal-iff
∀[n:ℕ]. ∀[x,y,a,b:ℝ^n]. uiff(d(x;y) = d(a;b);x - y⋅x - y = a - b⋅a - b)
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
dot-product: x⋅y,
real-vec-sub: X - Y,
real-vec: ℝ^n,
req: x = y,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
guard: {T}
Lemmas referenced :
dot-product-nonneg,
real-vec-sub_wf,
dot-product_wf,
real_wf,
req_witness,
req_wf,
rsqrt_wf,
rleq_wf,
int-to-real_wf,
rmul_wf,
req_functionality,
rsqrt_functionality,
req_weakening,
equal_wf,
real-vec-dist_wf,
real-vec_wf,
nat_wf,
rsqrt_squared,
req_inversion,
req_transitivity,
rmul_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaFormation,
because_Cache,
independent_pairFormation,
isect_memberFormation,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
sqequalRule,
independent_isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_pairEquality,
isect_memberEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y,a,b:\mBbbR{}\^{}n]. uiff(d(x;y) = d(a;b);x - y\mcdot{}x - y = a - b\mcdot{}a - b)
Date html generated:
2017_10_03-AM-10_56_04
Last ObjectModification:
2017_07_28-AM-08_21_28
Theory : reals
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