Nuprl Lemma : eu-lt_transitivity
∀e:EuclideanPlane. ∀[p,q,r:{p:Point| O_X_p} ]. (p < r) supposing (q < r and p ≤ q)
Proof
Definitions occuring in Statement :
eu-lt: p < q,
eu-le: p ≤ q,
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-X: X,
eu-O: O,
eu-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
eu-lt: p < q,
eu-le: p ≤ q,
member: t ∈ T,
prop: ℙ,
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
sq_stable: SqStable(P),
false: False,
squash: ↓T
Lemmas referenced :
eu-lt_wf,
eu-between-eq_wf,
eu-O_wf,
eu-X_wf,
eu-le_wf,
set_wf,
eu-point_wf,
euclidean-plane_wf,
not_wf,
equal_wf,
squash_wf,
eu-between-eq-symmetry,
eu-between-eq-inner-trans,
eu-between-eq-exchange3,
eu-between-eq-exchange4,
sq_stable__and,
sq_stable__eu-between-eq,
sq_stable__not,
eu-between-eq-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
sqequalRule,
lambdaEquality,
isect_memberEquality,
productElimination,
equalityEquality,
independent_isectElimination,
independent_pairFormation,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
independent_functionElimination,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity
Latex:
\mforall{}e:EuclideanPlane. \mforall{}[p,q,r:\{p:Point| O\_X\_p\} ]. (p < r) supposing (q < r and p \mleq{} q)
Date html generated:
2016_10_26-AM-07_41_41
Last ObjectModification:
2016_07_12-AM-08_07_55
Theory : euclidean!geometry
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