Nuprl Lemma : hp-angle-sum-subst1
∀g:EuclideanPlane. ∀a,b,c,d,e,f,x,y,z,i,j,k:Point. (abc + def ≅ xyz ⇒ def ≅a ijk ⇒ abc + ijk ≅ 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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
hp-angle-sum: abc + xyz ≅ def,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
cand: A c∧ B,
basic-geometry: BasicGeometry,
geo-out: out(p ab),
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-cong-angle: abc ≅a xyz
Lemmas referenced :
geo-cong-angle-transitivity,
euclidean-plane-axioms,
geo-cong-angle-symm2,
geo-cong-angle_wf,
geo-out_wf,
geo-strict-between_wf,
geo-between_wf,
hp-angle-sum_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
hypothesisEquality,
cut,
hypothesis,
independent_pairFormation,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
independent_functionElimination,
because_Cache,
productIsType,
inhabitedIsType,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f,x,y,z,i,j,k:Point.
(abc + def \mcong{} xyz {}\mRightarrow{} def \mcong{}\msuba{} ijk {}\mRightarrow{} abc + ijk \mcong{} xyz)
Date html generated:
2019_10_16-PM-02_06_18
Last ObjectModification:
2019_06_05-AM-09_37_26
Theory : euclidean!plane!geometry
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