Nuprl Lemma : eu-add-length-between-iff
∀e:EuclideanPlane. ∀[a,b,c:Point]. uiff(a_b_c;|ac| = |ab| + |bc| ∈ {p:Point| O_X_p} )
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,
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,
prop: ℙ,
euclidean-plane: EuclideanPlane,
stable: Stable{P},
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B
Lemmas referenced :
eu-between-eq_wf,
equal_wf,
eu-point_wf,
eu-O_wf,
eu-X_wf,
eu-length_wf,
eu-mk-seg_wf,
eu-add-length_wf,
euclidean-plane_wf,
eu-add-length-between,
stable__eu-between-eq,
eu-between-eq-trivial-left,
not_wf,
eu-le-add1,
eu-le_wf,
eu-le-null-segment,
eu-congruence-identity,
eu-congruent-iff-length,
eu-extend-exists,
eu-congruent_wf,
eu-congruence-identity-sym,
false_wf,
squash_wf,
true_wf,
euclidean-structure_wf,
iff_weakening_equal,
eu-between-eq-trivial-right,
set_wf,
eu-add-length-zero2,
eu-add-length-assoc,
eu-add-length-cancel-left,
eu-between-eq-same-side2,
eu-between-eq-symmetry,
eu-congruent-between-implies-equal,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
independent_pairFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
axiomEquality,
setEquality,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
sqequalRule,
voidElimination,
dependent_set_memberEquality,
productElimination,
equalityTransitivity,
equalityEquality,
universeEquality,
applyEquality,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:EuclideanPlane. \mforall{}[a,b,c:Point]. uiff(a\_b\_c;|ac| = |ab| + |bc|)
Date html generated:
2016_10_26-AM-07_44_23
Last ObjectModification:
2016_07_12-AM-08_12_02
Theory : euclidean!geometry
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