Nuprl Lemma : colinear-cong3
∀e:EuclideanPlane. ∀a,b,c,x,y:Point.
(a ≠ b ⇒ Colinear(a;b;c) ⇒ ab ≅ xy ⇒ (∃z:Point. (Colinear(x;y;z) ∧ Cong3(abc,xyz))))
Proof
Definitions occuring in Statement :
geo-cong-tri: Cong3(abc,a'b'c'),
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
geo-midpoint: a=m=b,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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]),
select: L[n],
cons: [a / b],
subtract: n - m,
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q),
euclidean-plane: EuclideanPlane,
not: ¬A,
false: False,
stable: Stable{P},
squash: ↓T,
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b
Lemmas referenced :
geo-colinear-sep-cases,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear_wf,
geo-sep_wf,
geo-point_wf,
symmetric-point-construction,
geo-congruent-symmetry,
geo-congruent-sep,
geo-extend-exists,
geo-sep-sym,
oriented-colinear-append,
cons_member,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-congruent-iff-length,
geo-length-flip,
geo-cong-tri_wf,
geo-between-symmetry,
geo-between-same-side2,
stable__geo-congruent,
false_wf,
or_wf,
geo-between_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
and_wf,
equal_wf,
geo-length-type_wf,
geo-add-length_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length-between,
geo-add-length-cancel-left,
squash_wf,
true_wf,
basic-geometry_wf,
geo-add-length-assoc,
subtype_rel_self,
iff_weakening_equal,
geo-add-length-implies-eq-zero,
geo-add-length-is-zero,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesisEquality,
independent_functionElimination,
hypothesis,
unionElimination,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
productElimination,
rename,
dependent_pairFormation,
independent_pairFormation,
inrFormation,
inlFormation,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
setElimination,
functionEquality,
dependent_set_memberEquality,
applyLambdaEquality,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y:Point.
(a \mneq{} b {}\mRightarrow{} Colinear(a;b;c) {}\mRightarrow{} ab \mcong{} xy {}\mRightarrow{} (\mexists{}z:Point. (Colinear(x;y;z) \mwedge{} Cong3(abc,xyz))))
Date html generated:
2018_05_22-PM-00_15_42
Last ObjectModification:
2018_04_05-AM-01_03_23
Theory : euclidean!plane!geometry
Home
Index