Nuprl Lemma : angle-sum-functionality
∀e:EuclideanPlane. ∀a,b,c,x,y,z,i,j,k,a',b',c',x',y',z',i',j',k':Point.
(a ≡ a'
⇒ b ≡ b'
⇒ c ≡ c'
⇒ x ≡ x'
⇒ y ≡ y'
⇒ z ≡ z'
⇒ i ≡ i'
⇒ j ≡ j'
⇒ k ≡ k'
⇒ (abc + xyz ≅a ijk ⇐⇒ a'b'c' + x'y'z' ≅a i'j'k'))
Proof
Definitions occuring in Statement :
angle-sum: abc + def ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
angle-sum: abc + def ≅a xyz,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
oriented-plane: OrientedPlane,
cand: A c∧ B,
basic-geometry: BasicGeometry
Lemmas referenced :
angle-sum_wf,
geo-eq_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-left_wf,
geo-cong-angle_wf,
geo-cong-angle_functionality,
geo-eq_weakening,
geo-left_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
inhabitedIsType,
dependent_pairFormation_alt,
productIsType,
independent_functionElimination,
promote_hyp
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z,i,j,k,a',b',c',x',y',z',i',j',k':Point.
(a \mequiv{} a'
{}\mRightarrow{} b \mequiv{} b'
{}\mRightarrow{} c \mequiv{} c'
{}\mRightarrow{} x \mequiv{} x'
{}\mRightarrow{} y \mequiv{} y'
{}\mRightarrow{} z \mequiv{} z'
{}\mRightarrow{} i \mequiv{} i'
{}\mRightarrow{} j \mequiv{} j'
{}\mRightarrow{} k \mequiv{} k'
{}\mRightarrow{} (abc + xyz \mcong{}\msuba{} ijk \mLeftarrow{}{}\mRightarrow{} a'b'c' + x'y'z' \mcong{}\msuba{} i'j'k'))
Date html generated:
2019_10_16-PM-02_03_59
Last ObjectModification:
2019_06_11-AM-09_42_55
Theory : euclidean!plane!geometry
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