Nuprl Lemma : hp-angle-sum-subst3
∀g:EuclideanPlane. ∀a,b,c,d,e,f,x,y,z,i,j,k:Point. (abc + def ≅ xyz ⇒ xyz ≅a ijk ⇒ x # yz ⇒ 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-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,
geo-out: out(p ab),
cand: A c∧ B,
not: ¬A,
false: False,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
geo-tri: Triangle(a;b;c),
geo-cong-tri: Cong3(abc,a'b'c')
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-cong-angle_wf,
hp-angle-sum_wf,
geo-point_wf,
geo-sep-sym,
out-cong-angle,
geo-cong-angle-symm2,
geo-cong-angle-transitivity,
out-preserves-lsep,
lsep-all-sym,
geo-out-interior-point-exists,
geo-between-sep,
istype-void,
geo-congruent-strictbetween-exists,
geo-inner-five-segment,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-congruent-iff-length,
geo-length-flip,
p8geo,
geo-strict-between-sep2,
geo-congruent-symmetry,
geo-congruent-sep,
geo-strict-between-sep3,
geo-between-trivial,
geo-between_wf,
geo-out_wf,
geo-strict-between_wf,
out-preserves-angle-cong_1,
geo-out_transitivity,
geo-out_inversion,
geo-out_weakening,
geo-eq_weakening,
euclidean-plane-axioms
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,
dependent_functionElimination,
inhabitedIsType,
because_Cache,
independent_functionElimination,
independent_pairFormation,
voidElimination,
productIsType,
functionIsType,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt
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 \# yz {}\mRightarrow{} abc + def \mcong{} ijk)
Date html generated:
2019_10_16-PM-02_07_31
Last ObjectModification:
2019_06_11-PM-01_31_16
Theory : euclidean!plane!geometry
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