Nuprl Lemma : real-vec-dist-minus
∀[n:ℕ]. ∀[x,y:ℝ^n]. (d(r(-1)*x;r(-1)*y) = d(x;y))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y)
,
real-vec-mul: a*X
,
real-vec: ℝ^n
,
req: x = y
,
int-to-real: r(n)
,
nat: ℕ
,
uall: ∀[x:A]. B[x]
,
minus: -n
,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
subtype_rel: A ⊆r B
,
prop: ℙ
,
implies: P
⇒ Q
,
uimplies: b supposing a
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
rev_uimplies: rev_uimplies(P;Q)
,
true: True
,
nat: ℕ
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
absval: |i|
,
false: False
,
not: ¬A
,
sq_type: SQType(T)
,
all: ∀x:A. B[x]
,
guard: {T}
,
squash: ↓T
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
Lemmas referenced :
req_witness,
real-vec-dist_wf,
real-vec-mul_wf,
int-to-real_wf,
real_wf,
rleq_wf,
real-vec_wf,
nat_wf,
rmul_wf,
rabs_wf,
req_functionality,
real-vec-dist-dilation,
req_weakening,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
false_wf,
absval_wf,
rmul-identity1,
req_wf,
squash_wf,
true_wf,
rabs-int,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
minusEquality,
natural_numberEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
independent_functionElimination,
isect_memberEquality,
because_Cache,
independent_isectElimination,
productElimination,
instantiate,
cumulativity,
intEquality,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (d(r(-1)*x;r(-1)*y) = d(x;y))
Date html generated:
2017_10_03-AM-10_56_19
Last ObjectModification:
2017_06_18-PM-02_51_12
Theory : reals
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