Nuprl Lemma : eu-between-eq-same-side
∀e:EuclideanPlane. ∀[A,B,C,D:Point]. (¬((¬A_C_D) ∧ (¬A_D_C))) supposing ((¬(A = B ∈ Point)) and A_B_C and A_B_D)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
prop: ℙ,
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
cand: A c∧ B,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
eu-point: Point,
record-select: r.x,
sq_type: SQType(T)
Lemmas referenced :
eu-between-eq-same2,
equal_wf,
eu-point_wf,
eu-extend-exists,
not_wf,
eu-between-eq_wf,
euclidean-plane_wf,
eu-congruent_wf,
eu-congruence-identity-sym,
eu-between-eq-symmetry,
eu-between-eq-inner-trans,
eu-between-eq-exchange3,
eu-three-segment,
eu-congruent-iff-length,
eu-length-flip,
and_wf,
eu-construction-unicity,
eu-between-eq-exchange4,
eu-between-eq-outer-trans,
eu-add-length-between,
eu-O_wf,
eu-X_wf,
eu-add-length_wf,
eu-length_wf,
eu-mk-seg_wf,
set_wf,
eu-add-length-assoc,
iff_weakening_equal,
squash_wf,
true_wf,
eu-add-length-comm,
eu-congruent-flip,
eu-five-segment,
not-not-inner-pasch,
exists_wf,
eu-congruent-refl,
eu-inner-five-segment,
eu-congruence-identity3,
eu-congruence-identity2,
eu-between-eq-same,
eu-between-eq-implies-colinear,
eu-congruent-symmetry,
eu-colinear-five-segment,
eu-congruence-identity,
subtype_base_sq,
eu-colinear-equidistant,
eu-between-eq-implies-colinear2,
eu-between-eq-implies-colinear3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
productElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
independent_functionElimination,
voidElimination,
setElimination,
rename,
dependent_functionElimination,
dependent_set_memberEquality,
productEquality,
sqequalRule,
lambdaEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hyp_replacement,
Error :applyLambdaEquality,
independent_pairFormation,
applyEquality,
setEquality,
equalityEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
comment,
instantiate
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[A,B,C,D:Point]. (\mneg{}((\mneg{}A\_C\_D) \mwedge{} (\mneg{}A\_D\_C))) supposing ((\mneg{}(A = B)) and A\_B\_C and A\_B\_D)
Date html generated:
2016_10_26-AM-07_43_14
Last ObjectModification:
2016_07_12-AM-08_12_32
Theory : euclidean!geometry
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