Nuprl Lemma : geo-congruence-identity-eq
∀e:BasicGeometry. ∀a,b,c:Point. ∀[d:Point]. (c ≡ d ⇒ ab ≅ cd ⇒ a ≡ b)
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-congruent: ab ≅ cd,
geo-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
and: P ∧ Q,
iff: P ⇐⇒ Q,
false: False,
not: ¬A,
geo-eq: a ≡ b,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
geo-eq_weakening,
geo-congruent_functionality,
geo-point_wf,
geo-sep_wf,
geo-eq_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-congruence-identity-sym
Rules used in proof :
productElimination,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
lambdaEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
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:Point. \mforall{}[d:Point]. (c \mequiv{} d {}\mRightarrow{} ab \00D0 cd {}\mRightarrow{} a \mequiv{} b)
Date html generated:
2017_10_02-PM-04_43_02
Last ObjectModification:
2017_08_05-AM-08_42_28
Theory : euclidean!plane!geometry
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