Nuprl Lemma : eu-congruent-iff-length
∀e:EuclideanPlane. ∀[a,b,c,d:Point]. uiff(ab=cd;|ab| = |cd| ∈ {p:Point| O_X_p} )
Proof
Definitions occuring in Statement :
eu-length: |s|,
eu-mk-seg: ab,
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-X: X,
eu-O: O,
eu-congruent: ab=cd,
eu-point: Point,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
eu-seg-congruent: s1 ≡ s2,
top: Top,
prop: ℙ
Lemmas referenced :
eu-seg-congruent-iff-length,
eu-mk-seg_wf,
eu_seg1_mk_seg_lemma,
eu_seg2_mk_seg_lemma,
eu-congruent_wf,
equal_wf,
eu-point_wf,
eu-between-eq_wf,
eu-O_wf,
eu-X_wf,
eu-length_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
hypothesis,
productElimination,
independent_isectElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
introduction,
axiomEquality,
setEquality,
because_Cache
Latex:
\mforall{}e:EuclideanPlane. \mforall{}[a,b,c,d:Point]. uiff(ab=cd;|ab| = |cd|)
Date html generated:
2016_05_18-AM-06_37_38
Last ObjectModification:
2015_12_28-AM-09_24_37
Theory : euclidean!geometry
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