Nuprl Lemma : projective-plane-axioms
∀g:ProjectivePlane
((∀p,q,r:Point. ∀s:p ≠ q. ∀s1:q ≠ r. (r ≠ p ∨ q ⇒ p ≠ q ∨ r))
∧ (∀p,q:Point. ∀l,m:Line. ∀s:p ≠ q. ∀s1:l ≠ m. (p ≠ l ⇒ q ≠ m ⇒ p I m ⇒ q I l ⇒ l ∧ m ≠ p ∨ q)))
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-meet: l ∧ m,
pgeo-join: p ∨ q,
pgeo-lsep: l ≠ m,
pgeo-psep: a ≠ b,
pgeo-incident: a I b,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
pgeo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
squash: ↓T,
sq_stable: SqStable(P),
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
projective-plane: ProjectivePlane,
implies: P ⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
all: ∀x:A. B[x],
triangle-axiom1: triangle-axiom1(g),
triangle-axiom2: triangle-axiom2(g)
Lemmas referenced :
pgeo-lsep_wf,
pgeo-meet_wf,
pgeo-point_wf,
pgeo-psep_wf,
pgeo-primitives_wf,
projective-plane-structure_wf,
basic-projective-plane_wf,
projective-plane_wf,
subtype_rel_transitivity,
projective-plane-subtype,
basic-projective-plane-subtype,
pgeo-plsep_wf,
pgeo-incident_wf,
projective-plane-structure_subtype,
pgeo-line_wf,
pgeo-join_wf,
sq_stable__pgeo-plsep
Rules used in proof :
independent_pairFormation,
independent_isectElimination,
instantiate,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
because_Cache,
productEquality,
sqequalRule,
isectElimination,
setEquality,
lambdaEquality,
productElimination,
applyEquality,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:ProjectivePlane
((\mforall{}p,q,r:Point. \mforall{}s:p \mneq{} q. \mforall{}s1:q \mneq{} r. (r \mneq{} p \mvee{} q {}\mRightarrow{} p \mneq{} q \mvee{} r))
\mwedge{} (\mforall{}p,q:Point. \mforall{}l,m:Line. \mforall{}s:p \mneq{} q. \mforall{}s1:l \mneq{} m.
(p \mneq{} l {}\mRightarrow{} q \mneq{} m {}\mRightarrow{} p I m {}\mRightarrow{} q I l {}\mRightarrow{} l \mwedge{} m \mneq{} p \mvee{} q)))
Date html generated:
2018_05_22-PM-00_41_24
Last ObjectModification:
2017_11_28-PM-03_56_02
Theory : euclidean!plane!geometry
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