Nuprl Lemma : geo-congruent-preserves-colinear
∀e:BasicGeometry
∀[a,b,c,a',b',c':Point]. (Colinear(a';b';c')) supposing (bc ≅ b'c' and ac ≅ a'c' and ab ≅ a'b' and Colinear(a;b;c))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
and: P ∧ Q,
geo-colinear: Colinear(a;b;c),
false: False,
not: ¬A,
prop: ℙ,
implies: P ⇒ Q,
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uiff: uiff(P;Q)
Lemmas referenced :
geo-congruence-identity,
geo-colinear_functionality,
geo-eq_weakening,
geo-congruent_functionality,
geo-colinear-same,
geo-point_wf,
geo-congruent_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry--subtype,
geo-between_wf,
not_wf,
geo-strict-between_wf,
geo-strict-between-implies-between,
geo-eq_wf,
stable__colinear,
geo-colinear_wf,
basic-geometry-subtype,
geo-colinear-cases,
geo-colinear-iff,
geo-congruent-preserves-between,
geo-between-implies-colinear3,
geo-length-flip,
geo-congruent-iff-length,
geo-congruent-symmetry,
geo-between-implies-colinear2
Rules used in proof :
promote_hyp,
productElimination,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
instantiate,
productEquality,
lambdaEquality,
voidElimination,
independent_isectElimination,
independent_functionElimination,
because_Cache,
rename,
setElimination,
isectElimination,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry
\mforall{}[a,b,c,a',b',c':Point].
(Colinear(a';b';c')) supposing (bc \00D0 b'c' and ac \00D0 a'c' and ab \00D0 a'b' and Colinear(a;b;c))
Date html generated:
2017_10_02-PM-06_15_31
Last ObjectModification:
2017_08_05-PM-04_12_10
Theory : euclidean!plane!geometry
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