Nuprl Lemma : eu-add-length-between
∀[e:EuclideanPlane]. ∀[a,b,c:Point]. |ac| = |ab| + |bc| ∈ {p:Point| O_X_p} supposing a_b_c
Proof
Definitions occuring in Statement :
eu-add-length: p + q,
eu-length: |s|,
eu-mk-seg: ab,
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],
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
eu-length: |s|,
all: ∀x:A. B[x],
euclidean-plane: EuclideanPlane,
and: P ∧ Q,
prop: ℙ,
eu-add-length: p + q,
top: Top,
implies: P ⇒ Q,
cand: A c∧ B,
not: ¬A,
false: False,
stable: Stable{P},
uiff: uiff(P;Q)
Lemmas referenced :
eu-extend-property,
eu-O_wf,
eu-not-colinear-OXY,
eu-X_wf,
not_wf,
equal_wf,
eu-point_wf,
eu-seg1_wf,
eu-mk-seg_wf,
eu-seg2_wf,
eu-between-eq_wf,
euclidean-plane_wf,
eu_seg1_mk_seg_lemma,
eu_seg2_mk_seg_lemma,
eu-extend_wf,
eu-between-eq-same2,
eu-construction-unicity,
eu-between-eq-symmetry,
eu-between-eq-inner-trans,
eu-between-eq-exchange3,
eu-between-eq-exchange4,
eu-congruent_wf,
stable__eu-congruent,
eu-congruence-identity,
eu-congruent-iff-length,
eu-three-segment
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
voidEquality,
lambdaFormation,
independent_isectElimination,
independent_functionElimination,
productEquality,
equalityEquality,
hyp_replacement,
Error :applyLambdaEquality,
promote_hyp
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[a,b,c:Point]. |ac| = |ab| + |bc| supposing a\_b\_c
Date html generated:
2016_10_26-AM-07_42_11
Last ObjectModification:
2016_07_12-AM-08_08_39
Theory : euclidean!geometry
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