Nuprl Lemma : hp-angle-sum-subst2
∀g:EuclideanPlane. ∀a,b,c,d,e,f,x,y,z,i,j,k:Point. (abc + def ≅ xyz ⇒ abc ≅a ijk ⇒ ijk + def ≅ xyz)
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
hp-angle-sum: abc + xyz ≅ def,
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz
Lemmas referenced :
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
hp-angle-sum_wf,
geo-cong-angle_wf,
geo-sep_wf,
geo-between_wf,
geo-out_wf,
geo-cong-angle-symm2,
euclidean-plane-axioms,
geo-cong-angle-transitivity
Rules used in proof :
independent_isectElimination,
instantiate,
applyEquality,
inhabitedIsType,
dependent_functionElimination,
hypothesis,
hypothesisEquality,
sqequalRule,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
universeIsType,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
because_Cache,
productIsType,
independent_pairFormation,
dependent_pairFormation_alt,
productElimination,
independent_functionElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f,x,y,z,i,j,k:Point.
(abc + def \mcong{} xyz {}\mRightarrow{} abc \mcong{}\msuba{} ijk {}\mRightarrow{} ijk + def \mcong{} xyz)
Date html generated:
2019_10_16-PM-02_07_08
Last ObjectModification:
2019_06_07-PM-05_35_44
Theory : euclidean!plane!geometry
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