Nuprl Lemma : geo-seg-congruent-equiv
∀e:BasicGeometry. EquivRel(geo-segment(e);s,t.geo-seg-congruent(e; s; t))
Proof
Definitions occuring in Statement :
geo-seg-congruent: geo-seg-congruent(e; s1; s2),
geo-segment: geo-segment(e),
basic-geometry: BasicGeometry,
equiv_rel: EquivRel(T;x,y.E[x; y]),
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
geo-seg-congruent: geo-seg-congruent(e; s1; s2)
Lemmas referenced :
geo-segment_wf,
geo-seg-congruent_symmetry,
geo-seg-congruent_wf,
basic-geometry_wf,
geo-congruent-refl,
geo-seg1_wf,
geo-seg2_wf,
geo-seg-congruent_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
independent_isectElimination,
because_Cache
Latex:
\mforall{}e:BasicGeometry. EquivRel(geo-segment(e);s,t.geo-seg-congruent(e; s; t))
Date html generated:
2018_05_22-AM-11_54_14
Last ObjectModification:
2018_05_11-PM-06_43_13
Theory : euclidean!plane!geometry
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