Nuprl Lemma : geo-ge-between-sep
∀g:EuclideanPlane. ∀a,b,c,a1,a2,x,y,t:Point. (a_b_c ⇒ b ≠ c ⇒ ab ≅ a1a2 ⇒ a1a2 ≥ xy ⇒ a_t_c ⇒ at ≅ xy ⇒ t ≠ c)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-ge: cd ≥ ab,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
uiff: uiff(P;Q),
basic-geometry: BasicGeometry,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
euclidean-plane: EuclideanPlane,
squash: ↓T,
geo-eq: a ≡ b,
false: False,
or: P ∨ Q,
not: ¬A,
stable: Stable{P},
geo-le: p ≤ q,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
basic-geometry-: BasicGeometry-,
so_apply: x[s],
so_lambda: λ2x.t[x],
cand: A c∧ B,
geo-ge: cd ≥ ab
Lemmas referenced :
geo-point_wf,
geo-sep_wf,
geo-ge_wf,
geo-between_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-congruent_wf,
geo-congruent-iff-length,
geo-le-iff,
iff_weakening_equal,
subtype_rel_self,
geo-mk-seg_wf,
geo-length_wf,
basic-geometry_wf,
geo-length-type_wf,
true_wf,
squash_wf,
geo-le_wf,
minimal-not-not-excluded-middle,
geo-between_functionality,
geo-eq_weakening,
geo-congruent_functionality,
minimal-double-negation-hyp-elim,
not_wf,
or_wf,
false_wf,
stable__geo-between,
sq_stable__geo-between,
geo-length-equality,
equal_wf,
geo-ge_functionality,
geo-congruent-between-exists,
geo-X_wf,
euclidean-plane-axioms,
geo-sep-sym,
geo-congruent-sep,
geo-le-sep,
geo-le_weakening,
geo-proper-extend-exists,
geo-between-outer-trans,
geo-between-exchange4,
geo-between-inner-trans,
geo-strict-between-implies-between,
geo-O_wf,
geo-strict-between-sep3,
geo-construction-unicity,
geo-between-symmetry,
geo-between-exchange3,
geo-congruence-identity-eq,
geo-eq-self,
geo-zero-length-iff,
geo-le-zero,
exists_wf,
geo-zero-length_wf,
subtype-geo-length-type,
geo-between-trivial,
geo-eq_transitivity,
geo-eq_inversion,
geo-between-sep
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalitySymmetry,
productElimination,
dependent_functionElimination,
independent_functionElimination,
universeEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
rename,
setElimination,
equalityTransitivity,
imageElimination,
lambdaEquality,
voidElimination,
unionElimination,
functionEquality,
setEquality,
productEquality,
independent_pairFormation,
dependent_pairFormation
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,a1,a2,x,y,t:Point.
(a\_b\_c {}\mRightarrow{} b \mneq{} c {}\mRightarrow{} ab \mcong{} a1a2 {}\mRightarrow{} a1a2 \mgeq{} xy {}\mRightarrow{} a\_t\_c {}\mRightarrow{} at \mcong{} xy {}\mRightarrow{} t \mneq{} c)
Date html generated:
2019_10_16-PM-02_52_07
Last ObjectModification:
2018_09_18-PM-03_07_17
Theory : euclidean!plane!geometry
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