Nuprl Lemma : cong-angle-out-all-iff
∀e:BasicGeometry. ∀a,b,c,x,y,z:Point.
((((b ≠ a ∧ b ≠ c) ∧ y ≠ x) ∧ y ≠ z)
⇒ (abc ≅a xyz
⇐⇒ ∀a',c',x',z':Point.
((((out(b a'a) ∧ out(b c'c)) ∧ out(y x'x)) ∧ out(y z'z) ∧ ba' ≅ yx' ∧ bc' ≅ yz') ⇒ a'c' ≅ x'z')))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
geo-strict-between: a-b-c,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
cand: A c∧ B
Lemmas referenced :
geo-out_wf,
geo-congruent_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-cong-angle_wf,
geo-sep_wf,
geo-point_wf,
out-preserves-angle-cong_1,
geo-out_inversion,
cong-angle-out-aux_1,
cong-angle-out-aux2_1,
geo-proper-extend-exists,
geo-between-out,
euclidean-plane-axioms,
geo-sep-sym,
geo-strict-between-sep1,
subtype_rel_self,
basic-geometry-_wf,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-add-length-between,
geo-congruent-symmetry,
geo-congruent-iff-length,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
geo-add-length-comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
sqequalRule,
productIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
inhabitedIsType,
functionIsType,
dependent_functionElimination,
independent_functionElimination,
rename,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,x,y,z:Point.
((((b \mneq{} a \mwedge{} b \mneq{} c) \mwedge{} y \mneq{} x) \mwedge{} y \mneq{} z)
{}\mRightarrow{} (abc \mcong{}\msuba{} xyz
\mLeftarrow{}{}\mRightarrow{} \mforall{}a',c',x',z':Point.
((((out(b a'a) \mwedge{} out(b c'c)) \mwedge{} out(y x'x)) \mwedge{} out(y z'z) \mwedge{} ba' \mcong{} yx' \mwedge{} bc' \mcong{} yz')
{}\mRightarrow{} a'c' \mcong{} x'z')))
Date html generated:
2019_10_16-PM-01_31_38
Last ObjectModification:
2018_10_09-PM-00_08_17
Theory : euclidean!plane!geometry
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