Nuprl Lemma : geo-SCO_wf
∀[g:EuclideanPlaneStructure]
∀c,d,a:Point. ∀b:{b:Point| b ≠ a ∧ c_b_d} . (SCO(a;b;c;d) ∈ {u:Point| cu ≅ cd ∧ a_b_u ∧ (b ≠ d ⇒ b ≠ u)} )
Proof
Definitions occuring in Statement :
geo-SCO: SCO(a;b;c;d),
euclidean-plane-structure: EuclideanPlaneStructure,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
geo-SCO: SCO(a;b;c;d),
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
geo-SC_wf,
geo-sep_wf,
geo-between_wf,
set_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
setElimination,
thin,
rename,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
dependent_set_memberEquality,
hypothesis,
productEquality,
applyEquality,
because_Cache,
lambdaEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setEquality
Latex:
\mforall{}[g:EuclideanPlaneStructure]
\mforall{}c,d,a:Point. \mforall{}b:\{b:Point| b \mneq{} a \mwedge{} c\_b\_d\} .
(SCO(a;b;c;d) \mmember{} \{u:Point| cu \mcong{} cd \mwedge{} a\_b\_u \mwedge{} (b \mneq{} d {}\mRightarrow{} b \mneq{} u)\} )
Date html generated:
2018_05_22-AM-11_52_46
Last ObjectModification:
2018_03_30-PM-04_36_52
Theory : euclidean!plane!geometry
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