Nuprl Lemma : Dcong-iff-cong
∀g:EuclideanPlane. ∀a,b,c,d:Point. (Dcong(g;a;b;c;d) ⇐⇒ ab ≅ cd)
Proof
Definitions occuring in Statement :
dist-cong: Dcong(g;a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
cand: A c∧ B,
not: ¬A,
euclidean-plane: EuclideanPlane,
dist-cong: Dcong(g;a;b;c;d),
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
false: False
Lemmas referenced :
dist-cong_wf,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
not-dist-cong,
dist_wf,
dist-lemma-lt-2,
not-lt-and-eq,
geo-length_wf,
geo-mk-seg_wf,
geo-congruent-iff-length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
productElimination,
voidElimination,
equalitySymmetry
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (Dcong(g;a;b;c;d) \mLeftarrow{}{}\mRightarrow{} ab \mcong{} cd)
Date html generated:
2019_10_16-PM-02_55_38
Last ObjectModification:
2019_06_05-PM-03_13_09
Theory : euclidean!plane!geometry
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