Nuprl Lemma : real-vec-triangle-equality
∀n:ℕ. ∀x,y,z:ℝ^n. ((r0 < d(y;z)) ⇒ (d(x;z) = (d(x;y) + d(y;z))) ⇒ (real-vec-be(n;x;y;z) ∧ ((r0 < d(x;y)) ⇒ x-y-z)))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec-be: real-vec-be(n;a;b;c),
real-vec-between: a-b-c,
real-vec: ℝ^n,
rless: x < y,
req: x = y,
radd: a + b,
int-to-real: r(n),
nat: ℕ,
all: ∀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,
real-vec-dist: d(x;y),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
req-vec: req-vec(n;x;y),
real-vec-sub: X - Y,
real-vec-add: X + Y,
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
real-vec: ℝ^n,
exists: ∃x:A. B[x],
and: P ∧ Q,
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rge: x ≥ y,
guard: {T},
real-vec-mul: a*X,
rneq: x ≠ y,
or: P ∨ Q,
cand: A c∧ B,
real-vec-be: real-vec-be(n;a;b;c),
top: Top,
real-vec-between: a-b-c,
le: A ≤ B,
false: False,
not: ¬A
Lemmas referenced :
real-vec-norm_functionality,
real-vec-sub_wf,
real-vec-add_wf,
int_seg_wf,
req_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
radd_wf,
rless_wf,
real-vec_wf,
nat_wf,
rminus_wf,
req_weakening,
real-vec-norm_wf,
Minkowski-equality,
uiff_transitivity,
req_functionality,
req_inversion,
radd-assoc,
radd_functionality,
radd-ac,
radd-rminus-assoc,
trivial-rless-radd,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
equal_wf,
rmul_preserves_req,
rmul_wf,
rdiv_wf,
rsub_wf,
req_transitivity,
rmul-distrib,
rmul_functionality,
rmul_over_rminus,
rmul-assoc,
rmul-one-both,
rminus_functionality,
rmul_comm,
rmul-ac,
rmul-rdiv-cancel2,
radd_comm,
radd-preserves-req,
radd-rminus-both,
radd-zero-both,
squash_wf,
true_wf,
iff_weakening_equal,
member_rccint_lemma,
rmul_preserves_rleq,
i-member_wf,
rccint_wf,
req-vec_wf,
real-vec-mul_wf,
member_rooint_lemma,
rmul_preserves_rless,
rooint_wf,
rleq-int,
false_wf,
trivial-rleq-radd,
rleq_functionality,
rmul-zero-both,
rless_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
natural_numberEquality,
setElimination,
rename,
applyEquality,
lambdaEquality,
setEquality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
inrFormation,
imageElimination,
universeEquality,
dependent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
addLevel
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y,z:\mBbbR{}\^{}n.
((r0 < d(y;z))
{}\mRightarrow{} (d(x;z) = (d(x;y) + d(y;z)))
{}\mRightarrow{} (real-vec-be(n;x;y;z) \mwedge{} ((r0 < d(x;y)) {}\mRightarrow{} x-y-z)))
Date html generated:
2017_10_03-AM-11_13_01
Last ObjectModification:
2017_07_28-AM-08_23_52
Theory : reals
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