Nuprl Lemma : real-vec-dist-symmetry
∀[n:ℕ]. ∀[x,y:ℝ^n]. (d(x;y) = d(y;x))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec: ℝ^n,
req: x = y,
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),
real-vec-norm: ||x||,
prop: ℙ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
real-vec-sub: X - Y,
real-vec-mul: a*X,
req-vec: req-vec(n;x;y),
all: ∀x:A. B[x],
nat: ℕ,
real-vec: ℝ^n,
rsub: x - y,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rsqrt_functionality,
dot-product-nonneg,
real-vec-sub_wf,
dot-product_wf,
rleq_wf,
int-to-real_wf,
req_witness,
real-vec-dist_wf,
real_wf,
real-vec_wf,
nat_wf,
int_seg_wf,
req_wf,
rsub_wf,
rmul_wf,
rminus_wf,
radd_wf,
req_weakening,
uiff_transitivity,
req_functionality,
req_inversion,
rminus-as-rmul,
rminus-radd,
radd_comm,
radd_functionality,
rmul_functionality,
req_transitivity,
rminus-rminus,
real-vec-mul_wf,
dot-product_functionality,
dot-product-linearity2,
dot-product-comm,
rmul-assoc,
rmul-int,
rmul-one-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
independent_isectElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
independent_functionElimination,
isect_memberEquality,
because_Cache,
lambdaFormation,
minusEquality,
productElimination,
multiplyEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (d(x;y) = d(y;x))
Date html generated:
2016_10_26-AM-10_25_18
Last ObjectModification:
2016_09_14-PM-06_40_19
Theory : reals
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