Nuprl Lemma : cong-angle-out-exists-aux3
∀e:HeytingGeometry. ∀a,b,c,x,y,z:Point.
((∃a',c',x',z':Point
((Cong3(a'bc',x'yz') ∧ b # a'c' ∧ y # x'z') ∧ out(b a'a) ∧ out(b c'c) ∧ out(y x'x) ∧ out(y z'z)))
⇒ abc ≅a xyz)
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
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 :
so_apply: x[s],
heyting-geometry: Error :heyting-geometry,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
cand: A c∧ B,
geo-cong-tri: Cong3(abc,a'b'c')
Lemmas referenced :
geo-out_wf,
Error :geo-triangle_wf,
geo-cong-tri_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
Error :heyting-geometry_wf,
subtype_rel_transitivity,
heyting-geometry-subtype,
basic-geometry-subtype,
geo-point_wf,
exists_wf,
cong-angle-out-aux2,
geo-triangle-symmetry,
geo-out-triangle,
geo-length-flip,
geo-congruent-iff-length
Rules used in proof :
rename,
setElimination,
productEquality,
because_Cache,
lambdaEquality,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
independent_pairFormation,
equalitySymmetry,
equalityTransitivity,
dependent_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{} b \# a'c' \mwedge{} y \# x'z')
\mwedge{} out(b a'a)
\mwedge{} out(b c'c)
\mwedge{} out(y x'x)
\mwedge{} out(y z'z)))
{}\mRightarrow{} abc \00D0\msuba{} xyz)
Date html generated:
2017_10_02-PM-07_03_53
Last ObjectModification:
2017_08_06-PM-08_57_47
Theory : euclidean!plane!geometry
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