Nuprl Lemma : vertical-angles-congruent
∀e:EuclideanPlane. ∀a,b,c,a',c':Point. (a-b-a' ⇒ c-b-c' ⇒ abc ≅a a'bc')
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
geo-midpoint: a=m=b,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
exists: ∃x:A. B[x]
Lemmas referenced :
geo-strict-between-sep2,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep-sym,
geo-strict-between-sep3,
geo-strict-between_wf,
geo-point_wf,
geo-midpoint-diagonals-congruent,
geo-strict-between-implies-between,
geo-between-outer-trans,
geo-between-symmetry,
geo-between-exchange4,
geo-strict-between-sep1,
geo-congruent-iff-length,
geo-length-flip,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
geo-proper-extend-exists,
geo-between_wf,
geo-congruent_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
because_Cache,
productElimination,
universeIsType,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
productIsType,
equalityIstype,
applyLambdaEquality,
setElimination,
rename,
dependent_pairFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,a',c':Point. (a-b-a' {}\mRightarrow{} c-b-c' {}\mRightarrow{} abc \mcong{}\msuba{} a'bc')
Date html generated:
2019_10_16-PM-01_32_00
Last ObjectModification:
2018_12_15-PM-09_32_43
Theory : euclidean!plane!geometry
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