Nuprl Lemma : cong-angle-out-exists-aux3-weak
∀e:HeytingGeometry. ∀a,b,c,x,y,z:Point.
((∃a',c',x',z':Point. (Cong3(a'bc',x'yz') ∧ out(b a'a) ∧ out(b c'c) ∧ out(y x'x) ∧ out(y z'z))) ⇒ abc ≅a xyz)
Proof
Definitions occuring in Statement :
heyting-geometry: HeytingGeometry,
geo-out: out(p ab),
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
geo-cong-tri: Cong3(abc,a'b'c'),
cand: A c∧ B,
uiff: uiff(P;Q)
Lemmas referenced :
geo-cong-tri_wf,
euclidean-plane-subtype-basic,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
basic-geometry_wf,
geo-out_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-congruent-iff-length,
geo-length-flip,
cong-angle-out-aux2-weak
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
productIsType,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
independent_functionElimination
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c,x,y,z:Point.
((\mexists{}a',c',x',z':Point. (Cong3(a'bc',x'yz') \mwedge{} out(b a'a) \mwedge{} out(b c'c) \mwedge{} out(y x'x) \mwedge{} out(y z'z)))
{}\mRightarrow{} abc \mcong{}\msuba{} xyz)
Date html generated:
2019_10_16-PM-02_08_47
Last ObjectModification:
2018_10_22-PM-02_48_39
Theory : euclidean!plane!geometry
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