Nuprl Lemma : zero-angles-congruent2
∀g:EuclideanPlane. ∀a,b,c,x,y,z:Point. (out(b ac) ⇒ out(y xz) ⇒ abc ≅a xyz)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
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,
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
member: t ∈ T,
geo-out: out(p ab),
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
cand: A c∧ B,
basic-geometry-: BasicGeometry-,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
uiff: uiff(P;Q),
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
geo-sep-sym,
geo-proper-extend-exists,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-congruent-iff-length,
geo-add-length-between,
geo-length-flip,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
geo-between_wf,
geo-congruent_wf,
geo-out_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-between-out,
geo-strict-between-sep1,
geo-out_transitivity,
geo-out_inversion,
geo-out-cong-cong
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
hypothesis,
because_Cache,
sqequalRule,
rename,
dependent_pairFormation_alt,
isectElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality_alt,
imageElimination,
universeIsType,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productIsType,
instantiate
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point. (out(b ac) {}\mRightarrow{} out(y xz) {}\mRightarrow{} abc \mcong{}\msuba{} xyz)
Date html generated:
2019_10_16-PM-01_48_48
Last ObjectModification:
2019_09_05-PM-02_46_34
Theory : euclidean!plane!geometry
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