Nuprl Lemma : hp-angle-sum-lt4
∀e:EuclideanPlane. ∀a,b,c,x,y,z,i,j,k,a',b',c',x',y',z',i',j',k':Point.
(x' # y'z'
⇒ abc + xyz ≅ ijk
⇒ a'b'c' + x'y'z' ≅ i'j'k'
⇒ ijk ≅a i'j'k'
⇒ a' # b'c'
⇒ x # yz
⇒ i # jk
⇒ x'y'z' < xyz
⇒ abc < a'b'c')
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
geo-lt-angle: abc < xyz,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
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,
basic-geometry: BasicGeometry,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
cong-angle-preserves-lsep_strong,
geo-cong-angle-symm2,
hp-angle-sum-symm,
geo-lt-angle_wf,
geo-lsep_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-lt3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z,i,j,k,a',b',c',x',y',z',i',j',k':Point.
(x' \# y'z'
{}\mRightarrow{} abc + xyz \mcong{} ijk
{}\mRightarrow{} a'b'c' + x'y'z' \mcong{} i'j'k'
{}\mRightarrow{} ijk \mcong{}\msuba{} i'j'k'
{}\mRightarrow{} a' \# b'c'
{}\mRightarrow{} x \# yz
{}\mRightarrow{} i \# jk
{}\mRightarrow{} x'y'z' < xyz
{}\mRightarrow{} abc < a'b'c')
Date html generated:
2019_10_16-PM-02_24_40
Last ObjectModification:
2019_08_26-PM-05_14_42
Theory : euclidean!plane!geometry
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