Nuprl Lemma : cong-angle-preserves-lsep_strong
∀g:EuclideanPlane. ∀a,b,c,x,y,z:Point. (x # yz ⇒ abc ≅a xyz ⇒ a # bc)
Proof
Definitions occuring in Statement :
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,
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
geo-cong-tri: Cong3(abc,a'b'c'),
euclidean-plane: EuclideanPlane,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
Euclid-Prop20_cycle,
geo-cong-angle_wf,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
cong-angle-out-exists-cong3,
geo-cong-preserves-lt,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
geo-congruent-iff-length,
geo-length-flip,
Prop22-inequality-implies-triangle,
geo-lt_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
subtype_rel_self,
iff_weakening_equal,
geo-add-length_functionality_wrt_cong,
out-preserves-lsep,
lsep-symmetry,
lsep-all-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
universeIsType,
isectElimination,
sqequalRule,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
because_Cache,
setElimination,
rename,
equalitySymmetry,
equalityTransitivity,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point. (x \# yz {}\mRightarrow{} abc \mcong{}\msuba{} xyz {}\mRightarrow{} a \# bc)
Date html generated:
2019_10_16-PM-02_23_46
Last ObjectModification:
2019_03_14-PM-10_55_49
Theory : euclidean!plane!geometry
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