Nuprl Lemma : right-angles-not-acute2
∀e:BasicGeometry. ∀a,b,c,a':Point. (b ≠ c ⇒ Rabc ⇒ Ra'bc ⇒ a_a'_c ⇒ a ≡ a')
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
right-angle: Rabc,
geo-eq: a ≡ b,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
false: False,
geo-eq: a ≡ b,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-
Lemmas referenced :
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
right-angle_wf,
geo-sep_wf,
geo-point_wf,
stable__not,
false_wf,
or_wf,
not_wf,
geo-eq_inversion,
minimal-double-negation-hyp-elim,
right-angle_functionality,
geo-eq_weakening,
geo-sep_functionality,
geo-sep-irrefl',
minimal-not-not-excluded-middle,
geo-between_functionality,
adjacent-right-angles,
colinear-right-angle,
oriented-colinear-append,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
exists_wf,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf,
geo-eq_transitivity,
subtype_rel_self,
basic-geometry-_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
functionEquality,
voidElimination,
independent_functionElimination,
unionElimination,
dependent_functionElimination,
productElimination,
dependent_pairFormation,
independent_pairFormation,
inrFormation,
inlFormation,
productEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a':Point. (b \mneq{} c {}\mRightarrow{} Rabc {}\mRightarrow{} Ra'bc {}\mRightarrow{} a\_a'\_c {}\mRightarrow{} a \mequiv{} a')
Date html generated:
2018_05_22-PM-00_03_18
Last ObjectModification:
2018_04_02-AM-11_30_54
Theory : euclidean!plane!geometry
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