Nuprl Lemma : geo-vert-angle-SAS
∀e:BasicGeometry. ∀a,p,q,p',q':Point. (p-a-q ⇒ p'-a-q' ⇒ pa ≅ p'a ⇒ qa ≅ q'a ⇒ p'q ≅ pq')
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
basic-geometry-: BasicGeometry-,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
geo-midpoint: a=m=b
Lemmas referenced :
stable__geo-congruent,
false_wf,
or_wf,
geo-sep_wf,
not_wf,
geo-congruent_wf,
geo-congruent-iff-length,
geo-add-length-between,
geo-strict-between-implies-between,
subtype_rel_self,
basic-geometry-_wf,
geo-length-flip,
and_wf,
equal_wf,
geo-length-type_wf,
geo-add-length_wf,
geo-length_wf,
geo-mk-seg_wf,
minimal-double-negation-hyp-elim,
geo-congruent_functionality,
geo-eq_weakening,
geo-strict-between_functionality,
minimal-not-not-excluded-middle,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-strict-between_wf,
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
geo-strict-between-sep2,
geo-sep-sym,
geo-strict-between-sep3,
geo-proper-extend-exists,
geo-add-length-comm,
squash_wf,
true_wf,
geo-between-outer-trans,
geo-between-symmetry,
geo-between-exchange4,
geo-strict-between-sep1,
geo-midpoint-diagonals-congruent,
geo-midpoint-symmetry,
geo-inner-five-segment,
geo-out-unicity,
geo-out-iff-between1
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
functionEquality,
productElimination,
independent_isectElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
lambdaFormation,
independent_functionElimination,
unionElimination,
voidElimination,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,p,q,p',q':Point. (p-a-q {}\mRightarrow{} p'-a-q' {}\mRightarrow{} pa \mcong{} p'a {}\mRightarrow{} qa \mcong{} q'a {}\mRightarrow{} p'q \mcong{} pq')
Date html generated:
2018_05_22-PM-00_01_45
Last ObjectModification:
2018_05_11-PM-01_17_53
Theory : euclidean!plane!geometry
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