Nuprl Lemma : rv-between-vec-mul
∀n:ℕ. ∀a,b,c:ℝ. ∀z:ℝ^n. ((r0 < ||z||) ⇒ (a*z-b*z-c*z ⇐⇒ ((a < b) ∧ (b < c)) ∨ ((c < b) ∧ (b < a))))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-norm: ||x||,
real-vec-mul: a*X,
real-vec: ℝ^n,
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
rv-between: a-b-c,
real-vec-between: a-b-c,
exists: ∃x:A. B[x],
real-vec-sep: a ≠ b,
real-vec-dist: d(x;y),
real-vec-mul: a*X,
real-vec-sub: X - Y,
req-vec: req-vec(n;x;y),
nat: ℕ,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
uiff: uiff(P;Q),
uimplies: b supposing a,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
real-vec-add: X + Y,
rev_uimplies: rev_uimplies(P;Q),
rneq: x ≠ y,
cand: A c∧ B,
guard: {T},
i-member: r ∈ I,
rooint: (l, u),
rdiv: (x/y),
rat_term_to_real: rat_term_to_real(f;t),
rtermAdd: left "+" right,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
rtermSubtract: left "-" right,
rtermConstant: "const",
pi1: fst(t),
true: True,
pi2: snd(t)
Lemmas referenced :
rv-between_wf,
real-vec-mul_wf,
rless_wf,
int-to-real_wf,
real-vec-norm_wf,
real-vec_wf,
real_wf,
istype-nat,
int_seg_wf,
rsub_wf,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
real-vec-sub_wf,
rabs_wf,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rless_functionality,
req_weakening,
real-vec-norm_functionality,
real-vec-norm-mul,
real-vec-norm-positive-iff,
rmul_preserves_req,
radd_wf,
itermAdd_wf,
itermConstant_wf,
itermMinus_wf,
rminus_wf,
req-implies-req,
req_functionality,
real_term_value_add_lemma,
real_term_value_minus_lemma,
rmul-is-positive,
rabs-positive-iff,
radd-preserves-rless,
rless_transitivity2,
rleq_weakening_rless,
rless_irreflexivity,
rmul_preserves_rless,
rdiv_wf,
i-member_wf,
rooint_wf,
req-vec_wf,
real-vec-add_wf,
member_rooint_lemma,
rinv_wf2,
rless-implies-rless,
req_transitivity,
radd_functionality,
rmul-rinv3,
rminus_functionality,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermVar_wf,
rtermAdd_wf,
rtermDivide_wf,
rtermSubtract_wf,
rtermConstant_wf,
real-vec-between-symmetry,
radd-preserves-rleq,
rabs-difference-symmetry,
rabs-of-nonneg,
rleq_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
unionIsType,
productIsType,
natural_numberEquality,
inhabitedIsType,
productElimination,
setElimination,
rename,
applyEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_functionElimination,
unionElimination,
inrFormation_alt,
inlFormation_alt,
promote_hyp,
dependent_pairFormation_alt
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}. \mforall{}z:\mBbbR{}\^{}n.
((r0 < ||z||) {}\mRightarrow{} (a*z-b*z-c*z \mLeftarrow{}{}\mRightarrow{} ((a < b) \mwedge{} (b < c)) \mvee{} ((c < b) \mwedge{} (b < a))))
Date html generated:
2019_10_30-AM-08_49_02
Last ObjectModification:
2019_04_02-PM-04_35_54
Theory : reals
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