Nuprl Lemma : geo-cong-angle_functionality
∀e:EuclideanPlane. ∀a1,a2,b1,b2,c1,c2,d1,d2,f1,f2,g1,g2:Point.
(a1 ≡ a2 ⇒ b1 ≡ b2 ⇒ c1 ≡ c2 ⇒ d1 ≡ d2 ⇒ f1 ≡ f2 ⇒ g1 ≡ g2 ⇒ (a1b1c1 ≅a d1f1g1 ⇐⇒ a2b2c2 ≅a d2f2g2))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅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,
geo-cong-angle: abc ≅a xyz,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x]
Lemmas referenced :
geo-cong-angle_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-between_wf,
geo-congruent_wf,
geo-sep_functionality,
geo-eq_weakening,
geo-between_functionality,
geo-congruent_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
inhabitedIsType,
productIsType,
productElimination,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a1,a2,b1,b2,c1,c2,d1,d2,f1,f2,g1,g2:Point.
(a1 \mequiv{} a2
{}\mRightarrow{} b1 \mequiv{} b2
{}\mRightarrow{} c1 \mequiv{} c2
{}\mRightarrow{} d1 \mequiv{} d2
{}\mRightarrow{} f1 \mequiv{} f2
{}\mRightarrow{} g1 \mequiv{} g2
{}\mRightarrow{} (a1b1c1 \mcong{}\msuba{} d1f1g1 \mLeftarrow{}{}\mRightarrow{} a2b2c2 \mcong{}\msuba{} d2f2g2))
Date html generated:
2019_10_16-PM-01_27_30
Last ObjectModification:
2018_11_07-PM-00_58_08
Theory : euclidean!plane!geometry
Home
Index