Nuprl Lemma : cong-angle-out-exists-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)) ∧ a'bc' ≅a x'yz')))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃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,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
cand: A c∧ B,
geo-cong-angle: abc ≅a xyz,
geo-out: out(p ab)
Lemmas referenced :
geo-cong-angle_wf,
geo-out_wf,
geo-sep_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-point_wf,
cong-angle-out-exists2,
geo-sep-sym,
cong-angle-out-aux2_1,
geo-between-out,
geo-between-sep,
geo-out_transitivity,
geo-out_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
productIsType,
inhabitedIsType,
because_Cache,
applyEquality,
instantiate,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
rename
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{} \mexists{}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{} a'bc' \mcong{}\msuba{} x'yz')))
Date html generated:
2019_10_16-PM-01_31_27
Last ObjectModification:
2018_11_12-PM-03_55_35
Theory : euclidean!plane!geometry
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