Nuprl Lemma : r2-upper-dimension
∀[a,b,p,q,r:ℝ^2].
(¬(p ≠ q ∧ q ≠ r ∧ r ≠ p ∧ (¬p-q-r) ∧ (¬q-r-p) ∧ (¬r-p-q))) supposing (ra=rb and qa=qb and pa=pb and a ≠ b)
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec: ℝ^n,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
real-vec-dist: d(x;y),
subtype_rel: A ⊆r B,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real-vec-sep: a ≠ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
true: True,
rneq: x ≠ y,
guard: {T},
rsub: x - y
Lemmas referenced :
r2-equidistant-implies',
vec-midpoint_wf,
r2-perp_wf,
real-vec-sub_wf,
set_wf,
real-vec_wf,
req_wf,
dot-product_wf,
int-to-real_wf,
real-vec-norm_wf,
equal_wf,
real-vec-sep_wf,
false_wf,
le_wf,
not_wf,
rv-between_wf,
rv-congruent_wf,
real-vec-dist_wf,
rless_functionality,
req_weakening,
real-vec-dist-symmetry,
exists_wf,
real_wf,
req-vec_wf,
real-vec-add_wf,
real-vec-mul_wf,
real-vec-sep_functionality,
req-vec_weakening,
rv-between_functionality,
rv-between-translation,
rv-between-vec-mul,
or_wf,
rless_wf,
rless-int,
rmul_wf,
rabs_wf,
rsub_wf,
radd_wf,
rminus_wf,
rabs-positive-iff,
radd-preserves-rless,
real-vec-dist-translation2,
real-vec-dist-vec-mul,
rmul_functionality,
rmul-one-both,
rabs_functionality,
radd_comm,
req_transitivity,
radd_functionality,
rminus-as-rmul,
radd-assoc,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-int,
rmul-zero-both,
radd-zero-both,
radd-ac,
radd-rminus-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
isectElimination,
sqequalRule,
lambdaEquality,
productEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
dependent_set_memberEquality,
independent_pairFormation,
isect_memberEquality,
applyEquality,
independent_isectElimination,
productElimination,
setElimination,
rename,
promote_hyp,
addLevel,
impliesFunctionality,
andLevelFunctionality,
impliesLevelFunctionality,
levelHypothesis,
imageMemberEquality,
baseClosed,
unionElimination,
minusEquality,
addEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[a,b,p,q,r:\mBbbR{}\^{}2].
(\mneg{}(p \mneq{} q \mwedge{} q \mneq{} r \mwedge{} r \mneq{} p \mwedge{} (\mneg{}p-q-r) \mwedge{} (\mneg{}q-r-p) \mwedge{} (\mneg{}r-p-q))) supposing
(ra=rb and
qa=qb and
pa=pb and
a \mneq{} b)
Date html generated:
2017_10_03-PM-00_47_13
Last ObjectModification:
2017_07_28-AM-08_47_17
Theory : reals!model!euclidean!geometry
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