Nuprl Lemma : mdist-max-metric-mul
∀[n:ℕ]. ∀[p:ℝ^n]. ∀[c:ℝ]. (mdist(max-metric(n);c*p;λi.r0) = (|c| * mdist(max-metric(n);p;λi.r0)))
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
real-vec-mul: a*X,
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rabs: |x|,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
real-vec: ℝ^n,
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
req-vec: req-vec(n;x;y),
all: ∀x:A. B[x],
real-vec-mul: a*X,
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top
Lemmas referenced :
int-to-real_wf,
int_seg_wf,
real_wf,
real-vec_wf,
istype-nat,
mdist_wf,
max-metric_wf,
real-vec-mul_wf,
rmul_wf,
rabs_wf,
meq-max-metric,
meq-same,
req_functionality,
req_weakening,
req_inversion,
mdist-max-metric-mul2,
mdist_functionality,
itermSubtract_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
lambdaEquality_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
universeIsType,
natural_numberEquality,
hypothesisEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaFormation_alt,
dependent_functionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p:\mBbbR{}\^{}n]. \mforall{}[c:\mBbbR{}]. (mdist(max-metric(n);c*p;\mlambda{}i.r0) = (|c| * mdist(max-metric(n);p;\mlambda{}i.r0)))
Date html generated:
2019_10_30-AM-08_41_53
Last ObjectModification:
2019_10_02-AM-11_06_02
Theory : reals
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