Nuprl Lemma : geo-cong-preserves-strict-bet-out
∀e:BasicGeometry. ∀a,b,c,a',b',c':Point. (a-b-c ⇒ ab ≅ a'b' ⇒ ac ≅ a'c' ⇒ out(a' b'c') ⇒ a'-b'-c')
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-strict-between: a-b-c,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
cand: A c∧ B,
geo-out: out(p ab),
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ
Lemmas referenced :
geo-cong-preserves-bet-out,
geo-strict-between-implies-between,
geo-inner-three-segment,
geo-between-symmetry,
geo-congruent-iff-length,
geo-length-flip,
geo-sep-sym,
geo-congruent-symmetry,
geo-congruent-sep,
geo-strict-between-sep3,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-out_wf,
geo-congruent_wf,
geo-strict-between_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
independent_functionElimination,
isectElimination,
independent_isectElimination,
hypothesis,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
instantiate,
sqequalRule,
universeIsType,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',b',c':Point.
(a-b-c {}\mRightarrow{} ab \mcong{} a'b' {}\mRightarrow{} ac \mcong{} a'c' {}\mRightarrow{} out(a' b'c') {}\mRightarrow{} a'-b'-c')
Date html generated:
2019_10_16-PM-01_24_28
Last ObjectModification:
2018_11_07-PM-02_35_33
Theory : euclidean!plane!geometry
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