Nuprl Lemma : geo-bet-sep-cong-tri-exists
∀e:BasicGeometry. ∀a,b,c,a',c':Point.
(a ≠ b ⇒ (∃b':Point. (a'_b'_c' ∧ Cong3(abc,a'b'c'))) supposing (a_b_c and ac ≅ a'c'))
Proof
Definitions occuring in Statement :
geo-cong-tri: Cong3(abc,a'b'c'),
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
cand: A c∧ B,
geo-cong-tri: Cong3(abc,a'b'c'),
and: P ∧ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
basic-geometry--subtype,
geo-sep_wf,
geo-congruent_wf,
geo-cong-tri_wf,
geo-between_wf,
geo-length-flip,
geo-congruent-iff-length,
geo-congruent-between-exists
Rules used in proof :
instantiate,
sqequalRule,
applyEquality,
productEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
isectElimination,
promote_hyp,
independent_pairFormation,
dependent_pairFormation,
productElimination,
independent_isectElimination,
isect_memberFormation,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,a',c':Point.
(a \mneq{} b {}\mRightarrow{} (\mexists{}b':Point. (a'\_b'\_c' \mwedge{} Cong3(abc,a'b'c'))) supposing (a\_b\_c and ac \00D0 a'c'))
Date html generated:
2017_10_02-PM-06_24_36
Last ObjectModification:
2017_08_05-PM-04_18_33
Theory : euclidean!plane!geometry
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