Nuprl Lemma : le-if-point-be-end
∀e:BasicGeometry. ∀a,b,c,d,x:Point. (a ≠ b ⇒ a_b_x ⇒ ax ≅ cd ⇒ ab≤cd)
Proof
Definitions occuring in Statement :
geo-le-pt: ab≤cd,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-le-pt: ab≤cd,
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
geo-point_wf,
geo-sep_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-congruent-between-exists
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
independent_pairFormation,
dependent_pairFormation,
productElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d,x:Point. (a \mneq{} b {}\mRightarrow{} a\_b\_x {}\mRightarrow{} ax \00D0 cd {}\mRightarrow{} ab\mleq{}cd)
Date html generated:
2017_10_02-PM-06_46_04
Last ObjectModification:
2017_08_05-PM-04_51_03
Theory : euclidean!plane!geometry
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