Nuprl Lemma : eu-seg-congruent-iff-length
∀e:EuclideanPlane. ∀[s,t:Segment]. uiff(s ≡ t;|s| = |t| ∈ {p:Point| O_X_p} )
Proof
Definitions occuring in Statement :
eu-length: |s|,
eu-seg-congruent: s1 ≡ s2,
eu-segment: Segment,
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-X: X,
eu-O: O,
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,
eu-length: |s|,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
prop: ℙ,
eu-seg-congruent: s1 ≡ s2,
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T
Lemmas referenced :
sq_stable__eu-congruent,
eu-extend-equal-iff-congruent,
eu-length_wf,
eu-seg-congruent_wf,
eu-between-eq_wf,
eu-seg2_wf,
eu-seg1_wf,
eu-point_wf,
equal_wf,
not_wf,
eu-X_wf,
eu-not-colinear-OXY,
eu-O_wf,
eu-extend-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
independent_pairFormation,
cut,
dependent_set_memberEquality,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
introduction,
axiomEquality,
setEquality,
independent_isectElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}[s,t:Segment]. uiff(s \mequiv{} t;|s| = |t|)
Date html generated:
2016_05_18-AM-06_37_35
Last ObjectModification:
2016_01_16-PM-10_31_11
Theory : euclidean!geometry
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