Nuprl Lemma : hp-angle-sum-eq-straight-angle2
∀e:EuclideanPlane. ∀a,b,c,x,y,z,a',b',c',x',y',z',i,j,k,i',j',k':Point.
(abc + xyz ≅ ijk ⇒ a'b'c' + x'y'z' ≅ i'j'k' ⇒ abc ≅a a'b'c' ⇒ xyz ≅a x'y'z' ⇒ i-j-k ⇒ i'-j'-k')
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry
Lemmas referenced :
hp-angle-sum-eq-straight-angle,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-cong-angle_wf,
hp-angle-sum_wf,
geo-point_wf,
hp-angle-sum-subst2,
geo-cong-angle-symm2,
hp-angle-sum-subst1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
independent_functionElimination,
hypothesis,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z,a',b',c',x',y',z',i,j,k,i',j',k':Point.
(abc + xyz \mcong{} ijk
{}\mRightarrow{} a'b'c' + x'y'z' \mcong{} i'j'k'
{}\mRightarrow{} abc \mcong{}\msuba{} a'b'c'
{}\mRightarrow{} xyz \mcong{}\msuba{} x'y'z'
{}\mRightarrow{} i-j-k
{}\mRightarrow{} i'-j'-k')
Date html generated:
2019_10_16-PM-02_25_48
Last ObjectModification:
2019_08_02-PM-00_34_51
Theory : euclidean!plane!geometry
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