Nuprl Lemma : Minkowski-inequality1
∀[n:ℕ]. ∀[x,y:ℝ^n]. (||x + y|| ≤ (||x|| + ||y||))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
real-vec-add: X + Y,
real-vec: ℝ^n,
rleq: x ≤ y,
radd: a + b,
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
cand: A c∧ B,
nat: ℕ,
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
uiff: uiff(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermAdd: left (+) right,
itermVar: vvar,
itermMultiply: left (*) right,
top: Top
Lemmas referenced :
rnexp-rleq-iff,
real-vec-norm_wf,
real-vec-add_wf,
radd_wf,
real-vec-norm-nonneg,
radd-non-neg,
less_than_wf,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
real-vec_wf,
nat_wf,
rnexp_wf,
false_wf,
le_wf,
rmul_wf,
int-to-real_wf,
dot-product_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_transitivity,
rleq_weakening,
radd_functionality_wrt_rleq,
req_inversion,
req_functionality,
req_transitivity,
real-vec-norm-squared,
dot-product-linearity1,
radd_functionality,
req_weakening,
dot-product-comm,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rmul_functionality,
radd_comm,
rmul-assoc,
radd-int,
rmul-identity1,
rmul-distrib2,
radd-assoc,
rmul_comm,
rmul-distrib,
uiff_transitivity,
rnexp2,
req_wf,
Cauchy-Schwarz,
rmul_preserves_rleq2,
rabs_wf,
rleq-int,
rleq_wf,
rleq_functionality,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
equal_wf,
rminus_wf,
rleq-rmax,
rabs-as-rmax
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productElimination,
lambdaEquality,
independent_pairEquality,
applyEquality,
setElimination,
rename,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
lambdaFormation,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
addEquality,
voidEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (||x + y|| \mleq{} (||x|| + ||y||))
Date html generated:
2017_10_03-AM-10_54_58
Last ObjectModification:
2017_07_28-AM-08_20_41
Theory : reals
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