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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