Nuprl Lemma : hp-angle-sum-subst4
∀g:EuclideanPlane. ∀a,b,c,d,e,f,x,y,z,i,j,k:Point.
(abc + def ≅ xyz ⇒ xyz ≅a ijk ⇒ x-y-z ⇒ a # bc ⇒ d # ef ⇒ abc + def ≅ ijk)
Proof
Definitions occuring in Statement :
hp-angle-sum: abc + xyz ≅ def,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
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,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
geo-cong-angle: abc ≅a xyz,
cand: A c∧ B,
geo-out: out(p ab)
Lemmas referenced :
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-strict-between_wf,
geo-cong-angle_wf,
hp-angle-sum_wf,
geo-point_wf,
Euclid-Prop23,
geo-sep-sym,
geo-between-trivial2,
geo-between_wf,
geo-out_wf,
geo-cong-angle-symmetry,
lsep-implies-sep,
geo-out_weakening,
geo-eq_weakening,
geo-out_inversion,
out-preserves-angle-cong_1,
supplementary-angles-preserve-congruence,
geo-cong-angle-transitivity,
euclidean-plane-axioms,
geo-cong-angle-symm2,
angle-cong-preserves-straight-angle,
geo-between-symmetry,
geo-strict-between-implies-between,
extended-out-preserves-between
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_functionElimination,
inhabitedIsType,
independent_functionElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f,x,y,z,i,j,k:Point.
(abc + def \mcong{} xyz {}\mRightarrow{} xyz \mcong{}\msuba{} ijk {}\mRightarrow{} x-y-z {}\mRightarrow{} a \# bc {}\mRightarrow{} d \# ef {}\mRightarrow{} abc + def \mcong{} ijk)
Date html generated:
2019_10_16-PM-02_31_50
Last ObjectModification:
2019_08_05-PM-03_23_27
Theory : euclidean!plane!geometry
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