Nuprl Lemma : geo-congruent-between-implies-equal
∀e:BasicGeometry. ∀[a,b,c,x:Point]. (b ≡ x) supposing (a_b_c and ab ≅ ax and bc ≅ xc)
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
guard: {T},
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
geo-eq: a ≡ b,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
geo-inner-five-segment,
geo-sep-irrefl',
geo-sep_functionality,
geo-congruence-identity,
geo-congruent_functionality,
geo-eq_weakening,
geo-between_functionality,
geo-length-flip,
geo-congruent-iff-length,
geo-congruent-refl,
geo-eq_wf,
geo-point_wf,
geo-congruent_wf,
geo-between_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf
Rules used in proof :
promote_hyp,
independent_functionElimination,
productElimination,
voidElimination,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
independent_isectElimination,
instantiate,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
hypothesis,
because_Cache,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}[a,b,c,x:Point]. (b \mequiv{} x) supposing (a\_b\_c and ab \00D0 ax and bc \00D0 xc)
Date html generated:
2017_10_02-PM-06_15_01
Last ObjectModification:
2017_08_05-PM-04_11_50
Theory : euclidean!plane!geometry
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