Nuprl Lemma : geo-krippen-lemma
∀e:BasicGeometry. ∀a1,a2,b1,b2,c,m1,m2:Point.
(a1_c_a2 ⇒ b1_c_b2 ⇒ ca1 ≅ cb1 ⇒ ca2 ≅ cb2 ⇒ a1=m1=b1 ⇒ a2=m2=b2 ⇒ m1_c_m2)
Proof
Definitions occuring in Statement :
geo-midpoint: a=m=b,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
stable: Stable{P},
false: False,
not: ¬A,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q
Lemmas referenced :
double-negation-hyp-elim,
not_wf,
geo-le_wf,
or_wf,
stable__geo-between,
geo-point_wf,
geo-between_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-midpoint_wf,
geo-mk-seg_wf,
geo-length_wf,
geo-le-cases,
geo-krippen-aux,
geo-between-symmetry
Rules used in proof :
voidElimination,
independent_functionElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
rename,
setElimination,
hypothesisEquality,
isectElimination,
because_Cache,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}a1,a2,b1,b2,c,m1,m2:Point.
(a1\_c\_a2 {}\mRightarrow{} b1\_c\_b2 {}\mRightarrow{} ca1 \00D0 cb1 {}\mRightarrow{} ca2 \00D0 cb2 {}\mRightarrow{} a1=m1=b1 {}\mRightarrow{} a2=m2=b2 {}\mRightarrow{} m1\_c\_m2)
Date html generated:
2017_10_02-PM-06_35_46
Last ObjectModification:
2017_08_05-PM-04_44_53
Theory : euclidean!plane!geometry
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