Nuprl Lemma : test-prove-distinct
∀e:EuclideanPlane. ∀A,B,C,X,Y,Z,W,U,V:Point.
((Colinear(A;B;X) ∨ A-X-B ∨ B-X-A)
⇒ (A_B_C ∨ C_B_A)
⇒ (Y_C_A ∨ A_C_Y)
⇒ (ZW=AY ∨ ZW=YA)
⇒ ZW=UV
⇒ (¬(U = V ∈ Point)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-colinear: Colinear(a;b;c),
eu-between: a-b-c,
eu-congruent: ab=cd,
eu-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
or: P ∨ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
equal_wf,
eu-point_wf,
eu-congruent_wf,
or_wf,
eu-between-eq_wf,
eu-colinear_wf,
eu-between_wf,
euclidean-plane_wf,
eu-congruence-identity,
eu-congruence-identity-sym,
eu-between-eq-same,
eu-colinear-def,
eu-between-same2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
because_Cache,
unionElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
independent_isectElimination,
dependent_functionElimination,
productElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}A,B,C,X,Y,Z,W,U,V:Point.
((Colinear(A;B;X) \mvee{} A-X-B \mvee{} B-X-A)
{}\mRightarrow{} (A\_B\_C \mvee{} C\_B\_A)
{}\mRightarrow{} (Y\_C\_A \mvee{} A\_C\_Y)
{}\mRightarrow{} (ZW=AY \mvee{} ZW=YA)
{}\mRightarrow{} ZW=UV
{}\mRightarrow{} (\mneg{}(U = V)))
Date html generated:
2016_10_26-AM-07_44_07
Last ObjectModification:
2016_07_12-AM-08_11_51
Theory : euclidean!geometry
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