Nuprl Lemma : not-dist-cong
∀g:EuclideanPlane. ∀a,b,c,d:Point. (((¬D(a;b;b;b;c;d)) ∧ (¬D(c;d;d;d;a;b))) ⇒ ab ≅ cd)
Proof
Definitions occuring in Statement :
dist: D(a;b;c;d;e;f),
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
uiff: uiff(P;Q)
Lemmas referenced :
not-dist-lemma,
not_wf,
dist_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
stable__geo-congruent,
false_wf,
or_wf,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
equal_wf,
geo-length-type_wf,
geo-congruent_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-lt-implies-gt,
geo-congruent-symmetry,
geo-congruent-iff-length,
not-lt-or
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
productEquality,
isectElimination,
setElimination,
rename,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
functionEquality,
unionElimination,
voidElimination,
equalitySymmetry
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (((\mneg{}D(a;b;b;b;c;d)) \mwedge{} (\mneg{}D(c;d;d;d;a;b))) {}\mRightarrow{} ab \mcong{} cd)
Date html generated:
2019_10_16-PM-02_51_41
Last ObjectModification:
2018_09_21-AM-09_42_33
Theory : euclidean!plane!geometry
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