Nuprl Lemma : Euclid-Prop5_2
∀e:EuclideanPlane. ∀a,b,c,x,y:Point. ((ISOΔ(a;b;c) ∧ a-b-x ∧ a-c-y) ⇒ xbc ≅a ycb)
Proof
Definitions occuring in Statement :
geo-isosceles: ISOΔ(a;b;c),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
basic-geometry-: BasicGeometry-,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
cand: A c∧ B,
geo-isosceles: ISOΔ(a;b;c),
uiff: uiff(P;Q),
geo-tri: Triangle(a;b;c),
geo-cong-tri: Cong3(abc,a'b'c'),
iff: P ⇐⇒ Q
Lemmas referenced :
segment-density-strict,
geo-strict-between-sep3,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
sq_stable__geo-strict-between,
geo-proper-extend-exists,
geo-strict-between-sep2,
geo-isosceles_wf,
geo-strict-between_wf,
geo-point_wf,
geo-strict-between-sym,
geo-strict-between-trans,
geo-strict-between-trans2,
colinear-lsep,
geo-sep-sym,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
colinear-lsep-cycle,
lsep-all-sym,
geo-strict-between-sep1,
geo-congruent-iff-length,
geo-add-length-between,
geo-strict-between-implies-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
Euclid-Prop4,
lsep-implies-sep,
euclidean-plane-axioms,
geo-congruent-flip,
geo-between-out,
geo-out_weakening,
geo-eq_weakening,
cong-angle-out-aux2_1,
geo-congruent-symmetry,
geo-length-flip,
geo-out-iff-between1,
geo-between-outer-trans,
geo-between-symmetry,
geo-out_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
independent_functionElimination,
dependent_set_memberEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
productEquality,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
independent_pairFormation,
lambdaEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y:Point. ((ISO\mDelta{}(a;b;c) \mwedge{} a-b-x \mwedge{} a-c-y) {}\mRightarrow{} xbc \mcong{}\msuba{} ycb)
Date html generated:
2018_05_22-PM-00_08_20
Last ObjectModification:
2018_05_11-PM-01_17_42
Theory : euclidean!plane!geometry
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