Nuprl Lemma : Minkowski-equality
∀n:ℕ. ∀x,y:ℝ^n.
((r0 < ||y||) ⇒ (||x + y|| = (||x|| + ||y||)) ⇒ (∃t:ℝ. ((r0 ≤ t) ∧ req-vec(n;x;t*y) ∧ ((r0 < ||x||) ⇒ (r0 < t)))))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
real-vec-mul: a*X,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rleq: x ≤ y,
rless: x < y,
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
exists: ∃x:A. B[x],
cand: A c∧ B,
rdiv: (x/y)
Lemmas referenced :
rnexp-positive,
real-vec-norm_wf,
false_wf,
le_wf,
req_wf,
real-vec-add_wf,
radd_wf,
rless_wf,
int-to-real_wf,
real-vec_wf,
nat_wf,
rnexp_wf,
dot-product_wf,
rmul_wf,
dot-product-comm,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
req-iff-rsub-is-0,
req_functionality,
req_transitivity,
real-vec-norm-squared,
dot-product-linearity1,
radd_functionality,
req_weakening,
rmul_functionality,
radd-preserves-req,
rminus_wf,
rnexp_functionality,
rnexp2,
itermMinus_wf,
real_term_value_minus_lemma,
rmul_preserves_req,
rless-int,
req_inversion,
rabs_wf,
rmul-nonneg-case1,
real-vec-norm-nonneg,
rabs-of-nonneg,
rleq_functionality,
rdiv_wf,
rmul_preserves_rleq,
rmul_preserves_rless,
rleq_wf,
req-vec_wf,
real-vec-mul_wf,
rinv_wf2,
rmul-rinv,
rless_functionality,
rmul-is-positive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
because_Cache,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_isectElimination,
inrFormation,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
productEquality,
functionEquality,
inlFormation
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n.
((r0 < ||y||)
{}\mRightarrow{} (||x + y|| = (||x|| + ||y||))
{}\mRightarrow{} (\mexists{}t:\mBbbR{}. ((r0 \mleq{} t) \mwedge{} req-vec(n;x;t*y) \mwedge{} ((r0 < ||x||) {}\mRightarrow{} (r0 < t)))))
Date html generated:
2017_10_03-AM-10_55_22
Last ObjectModification:
2017_07_28-AM-08_20_58
Theory : reals
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