Nuprl Lemma : geo-parallel-right-comm
∀e:EuclideanPlane. ∀a,b,c,d:Point. (geo-parallel(e;a;b;c;d) ⇒ geo-parallel(e;a;b;d;c))
Proof
Definitions occuring in Statement :
geo-parallel: geo-parallel(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-parallel: geo-parallel(e;a;b;c;d),
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
guard: {T},
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
geo-sep-sym,
lsep-all-sym,
geo-colinear_wf,
geo-parallel_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
because_Cache,
isectElimination,
applyEquality,
sqequalRule,
instantiate,
independent_isectElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (geo-parallel(e;a;b;c;d) {}\mRightarrow{} geo-parallel(e;a;b;d;c))
Date html generated:
2018_05_22-PM-00_13_40
Last ObjectModification:
2017_10_12-AM-11_26_49
Theory : euclidean!plane!geometry
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