Nuprl Lemma : rv-between-simple
∀n:ℕ. ∀c,d:ℝ^n. ((r0 < ||d||) ⇒ c - d-c-c + d)
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-norm: ||x||,
real-vec-sub: X - Y,
real-vec-add: X + Y,
real-vec: ℝ^n,
rless: x < y,
int-to-real: r(n),
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
real-vec-between: a-b-c,
exists: ∃x:A. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B,
i-member: r ∈ I,
rooint: (l, u),
nat_plus: ℕ+,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
real-vec-add: 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,
subtype_rel: A ⊆r B,
rv-between: a-b-c,
real-vec-sep: a ≠ b,
real-vec-dist: d(x;y),
rge: x ≥ y
Lemmas referenced :
rless_wf,
int-to-real_wf,
real-vec-norm_wf,
real-vec_wf,
nat_wf,
rdiv_wf,
rless-int,
rless-int-fractions2,
less_than_wf,
rless-int-fractions3,
i-member_wf,
rooint_wf,
req-vec_wf,
real-vec-add_wf,
real-vec-mul_wf,
real-vec-sub_wf,
rsub_wf,
rmul_preserves_req,
req_wf,
rmul_wf,
radd_wf,
rminus_wf,
req_weakening,
uiff_transitivity,
req_functionality,
rmul-rdiv-cancel2,
req_transitivity,
rmul-distrib,
radd_functionality,
rmul_over_rminus,
rmul-one-both,
rminus_functionality,
rmul_comm,
rmul-rdiv-cancel,
uiff_transitivity3,
squash_wf,
true_wf,
real_wf,
rminus-int,
radd-int,
req-vec_functionality,
req-vec_weakening,
real-vec-add_functionality,
real-vec-mul_functionality,
equal_wf,
int_seg_wf,
req_inversion,
radd-assoc,
radd-ac,
radd_comm,
radd-rminus-both,
radd-zero-both,
rmul-distrib2,
rmul_functionality,
rmul-identity1,
rmul-assoc,
iff_weakening_equal,
rmul-ac,
real-vec-dist-between,
real-vec-dist_wf,
rleq_wf,
rless_functionality,
real-vec-dist-symmetry,
rminus-radd,
rminus-as-rmul,
rmul-zero-both,
rminus-rminus,
real-vec-norm_functionality,
real-vec-norm-nonneg,
trivial-rless-radd,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
dependent_pairFormation,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
multiplyEquality,
productEquality,
minusEquality,
addEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
universeEquality,
setEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}c,d:\mBbbR{}\^{}n. ((r0 < ||d||) {}\mRightarrow{} c - d-c-c + d)
Date html generated:
2017_10_03-AM-11_13_22
Last ObjectModification:
2017_07_28-AM-08_24_08
Theory : reals
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