Nuprl Lemma : geo-le-pt-functionality
∀e:BasicGeometry. ∀a,b,c,d,a',b',c',d':Point. (a ≠ b ⇒ ab≤cd ⇒ ab ≅ a'b' ⇒ cd ≅ c'd' ⇒ a'b'≤c'd')
Proof
Definitions occuring in Statement :
geo-le-pt: ab≤cd,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
basic-geometry: BasicGeometry,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
geo-le-pt: ab≤cd,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
cand: A c∧ B,
geo-cong-tri: Cong3(abc,a'b'c')
Lemmas referenced :
geo-point_wf,
geo-sep_wf,
geo-le-pt_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-sep,
geo-congruent-symmetry,
geo-congruent-between-exists,
geo-length-flip,
geo-congruent-iff-length,
geo-cong-tri_wf,
geo-between_wf
Rules used in proof :
rename,
setElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
productEquality,
dependent_pairFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d,a',b',c',d':Point.
(a \mneq{} b {}\mRightarrow{} ab\mleq{}cd {}\mRightarrow{} ab \00D0 a'b' {}\mRightarrow{} cd \00D0 c'd' {}\mRightarrow{} a'b'\mleq{}c'd')
Date html generated:
2017_10_02-PM-06_46_11
Last ObjectModification:
2017_08_05-PM-04_51_10
Theory : euclidean!plane!geometry
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