Nuprl Lemma : hypotenuse-leg-congruence
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point.
(Rabc ⇒ Rxyz ⇒ ac ≅ xz ⇒ ab ≅ xy ⇒ a ≠ b ⇒ x ≠ y ⇒ b ≠ c ⇒ y ≠ z ⇒ bc ≅ yz)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
right-angle: Rabc,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
and: P ∧ Q,
exists: ∃x:A. B[x],
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
cand: A c∧ B
Lemmas referenced :
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-congruent_wf,
right-angle_wf,
geo-point_wf,
geo-proper-extend-exists,
geo-inner-five-segment,
geo-strict-between-implies-between,
geo-congruent-iff-length,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
adjacent-right-angles-supplementary,
geo-sep-sym,
right-angle-symmetry,
congruence-preserves-right-angle2,
geo-strict-between-sep3,
geo-length-flip,
geo-congruent-refl,
right-angle-SAS,
geo-congruent-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
productElimination,
dependent_functionElimination,
independent_functionElimination,
rename,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point.
(Rabc {}\mRightarrow{} Rxyz {}\mRightarrow{} ac \mcong{} xz {}\mRightarrow{} ab \mcong{} xy {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} x \mneq{} y {}\mRightarrow{} b \mneq{} c {}\mRightarrow{} y \mneq{} z {}\mRightarrow{} bc \mcong{} yz)
Date html generated:
2019_10_16-PM-01_53_16
Last ObjectModification:
2019_08_20-AM-10_59_50
Theory : euclidean!plane!geometry
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