Nuprl Lemma : eu-colinear-five-segment
∀e:EuclideanPlane
∀[a,b,c,d,A,B,C,D:Point]. (cd=CD) supposing (bd=BD and ad=AD and bc=BC and ab=AB and ac=AC and Colinear(a;b;c))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-colinear: Colinear(a;b;c),
eu-congruent: ab=cd,
eu-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
not: ¬A,
false: False,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q)
Lemmas referenced :
sq_stable__eu-congruent,
eu-colinear-cases,
eu-congruent_wf,
stable__eu-congruent,
not_wf,
equal_wf,
eu-point_wf,
eu-between_wf,
eu-colinear_wf,
euclidean-plane_wf,
eu-congruence-identity-sym,
eu-between-eq-def,
eu-congruent-preserves-between,
eu-congruent-iff-length,
eu-length-flip,
eu-five-segment,
eu-between-eq-symmetry,
eu-inner-five-segment
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
isectElimination,
hypothesisEquality,
independent_functionElimination,
productElimination,
productEquality,
equalityEquality,
voidElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
independent_isectElimination,
promote_hyp,
equalityTransitivity
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[a,b,c,d,A,B,C,D:Point].
(cd=CD) supposing (bd=BD and ad=AD and bc=BC and ab=AB and ac=AC and Colinear(a;b;c))
Date html generated:
2016_10_26-AM-07_42_42
Last ObjectModification:
2016_07_12-AM-08_09_03
Theory : euclidean!geometry
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