Nuprl Lemma : geo-le-pt-switch-congruence
∀e:BasicGeometry. ∀a,b,c,d:Point. (a ≠ b ⇒ ab≤cd ⇒ cd≤ab ⇒ ab ≅ cd)
Proof
Definitions occuring in Statement :
geo-le-pt: ab≤cd,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
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,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
exists: ∃x:A. B[x],
geo-le-pt: ab≤cd,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T
Lemmas referenced :
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
geo-le-pt_wf,
geo-le-pt-exists,
geo-congruent-sep,
geo-congruent-symmetry,
geo-between-sep,
geo-add-length-between,
geo-eq_inversion,
geo-congruent-iff-length,
geo-between-exchange4,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry,
geo-between-congruent,
geo-length_functionality,
geo-eq_weakening,
iff_weakening_equal,
geo-length-flip,
true_wf,
squash_wf,
geo-mk-seg_wf,
geo-length_wf,
geo-length-type_wf,
equal_wf,
geo-add-length_wf,
geo-add-length-assoc,
geo-add-length-implies-eq2,
geo-add-length-zero2
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
universeEquality,
applyLambdaEquality,
hyp_replacement,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
lambdaEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point. (a \mneq{} b {}\mRightarrow{} ab\mleq{}cd {}\mRightarrow{} cd\mleq{}ab {}\mRightarrow{} ab \00D0 cd)
Date html generated:
2017_10_02-PM-06_46_37
Last ObjectModification:
2017_08_05-PM-04_51_28
Theory : euclidean!plane!geometry
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