Nuprl Lemma : rv-norm-difference-symmetry
∀[rv:InnerProductSpace]. ∀[a,b:Point]. (||a - b|| = ||b - a||)
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
inner-product-space: InnerProductSpace,
req: x = y,
ss-point: Point,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rv-norm-equal-iff,
rv-sub_wf,
req_witness,
rv-norm_wf,
inner-product-space_subtype,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
rv-ip_wf,
ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
req-iff-rsub-is-0,
req_functionality,
req_transitivity,
rv-ip-sub,
rsub_functionality,
rv-ip-sub2,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
productElimination,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
independent_functionElimination,
instantiate,
isect_memberEquality,
dependent_functionElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a,b:Point]. (||a - b|| = ||b - a||)
Date html generated:
2018_05_22-PM-09_29_49
Last ObjectModification:
2018_05_20-PM-10_42_57
Theory : inner!product!spaces
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