Nuprl Lemma : euclid-P3
∀e:EuclideanPlane. ∀A,B,C1,C2:Point. ∃E:Point. (A_E_B ∧ AE=C1C2) supposing (¬(C1 = C2 ∈ Point)) ∧ |C1C2| < |AB|
Proof
Definitions occuring in Statement :
eu-lt: p < q,
eu-length: |s|,
eu-mk-seg: ab,
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-congruent: ab=cd,
eu-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
not: ¬A,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
cand: A c∧ B,
stable: Stable{P},
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eu-lt: p < q
Lemmas referenced :
not_wf,
equal_wf,
eu-point_wf,
eu-lt_wf,
eu-length_wf,
eu-mk-seg_wf,
euclidean-plane_wf,
eu-extend-exists,
eu-lt-null-segment,
eu-congruent_wf,
eu-between-eq_wf,
eu-congruence-identity-sym,
false_wf,
eu-between-eq-same-side2,
eu-between-eq-symmetry,
stable__eu-between-eq,
eu-add-length-between,
eu-congruent-iff-length,
eu-O_wf,
eu-X_wf,
eu-add-length_wf,
squash_wf,
true_wf,
set_wf,
iff_weakening_equal,
eu-le-add1,
eu-lt_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
productEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
dependent_functionElimination,
productElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
independent_isectElimination,
voidElimination,
equalityTransitivity,
equalityEquality,
universeEquality,
dependent_set_memberEquality,
independent_functionElimination,
dependent_pairFormation,
independent_pairFormation,
setEquality,
applyEquality,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:EuclideanPlane. \mforall{}A,B,C1,C2:Point.
\mexists{}E:Point. (A\_E\_B \mwedge{} AE=C1C2) supposing (\mneg{}(C1 = C2)) \mwedge{} |C1C2| < |AB|
Date html generated:
2016_10_26-AM-07_46_08
Last ObjectModification:
2016_08_29-PM-03_31_29
Theory : euclidean!geometry
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