Nuprl Lemma : geo-conga-to-cong3
∀E:BasicGeometry. ∀a,b,c,d,e,f:Point.
(a ≠ b
⇒ c ≠ b
⇒ d ≠ e
⇒ f ≠ e
⇒ abc ≅a def
⇒ (∃a',c',d',f':Point. (out(b a'a) ∧ out(b cc') ∧ out(e d'd) ∧ out(e ff') ∧ Cong3(a'bc',d'ef'))))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-cong-tri: Cong3(abc,a'b'c'),
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],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
geo-out: out(p ab),
basic-geometry: BasicGeometry,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
geo-cong-tri: Cong3(abc,a'b'c'),
uiff: uiff(P;Q),
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
geo-between-sep,
geo-sep-sym,
geo-between_wf,
istype-void,
geo-congruent-iff-length,
geo-length-flip,
geo-out_wf,
geo-cong-tri_wf,
geo-cong-angle_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
hypothesis,
independent_pairFormation,
voidElimination,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
isectElimination,
applyEquality,
inhabitedIsType,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
instantiate
Latex:
\mforall{}E:BasicGeometry. \mforall{}a,b,c,d,e,f:Point.
(a \mneq{} b
{}\mRightarrow{} c \mneq{} b
{}\mRightarrow{} d \mneq{} e
{}\mRightarrow{} f \mneq{} e
{}\mRightarrow{} abc \mcong{}\msuba{} def
{}\mRightarrow{} (\mexists{}a',c',d',f':Point. (out(b a'a) \mwedge{} out(b cc') \mwedge{} out(e d'd) \mwedge{} out(e ff') \mwedge{} Cong3(a'bc',d'ef'))))
Date html generated:
2019_10_16-PM-01_28_29
Last ObjectModification:
2018_11_07-PM-00_57_51
Theory : euclidean!plane!geometry
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