Nuprl Lemma : pgeo-triangle-axiom1-dual
∀g:ProjectivePlane. ∀l,m,n:Line. ∀s:l ≠ m. ∀s1:m ≠ n. (l ∧ m ≠ n ⇒ m ∧ n ≠ l)
Proof
Definitions occuring in Statement :
projective-plane: ProjectivePlane,
pgeo-meet: l ∧ m,
pgeo-lsep: l ≠ m,
pgeo-plsep: a ≠ b,
pgeo-line: Line,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
and: P ∧ Q,
projective-plane: ProjectivePlane,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
false: False,
not: ¬A,
pgeo-incident: a I b
Lemmas referenced :
pgeo-line_wf,
pgeo-lsep_wf,
pgeo-incident_wf,
pgeo-point_wf,
pgeo-meet_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,
projective-plane-structure_subtype,
pgeo-plsep_wf,
PL-sep-or,
pgeo-meet-incident,
pgeo-meet-plsep-sym,
LP-sep-or2,
pgeo-psep-or,
Error :pgeo-psep-sym,
pgeo-meet-psep-sym,
pgeo-lsep-implies-plsep
Rules used in proof :
productEquality,
because_Cache,
setEquality,
lambdaEquality,
rename,
setElimination,
dependent_functionElimination,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination,
independent_functionElimination,
voidElimination,
productElimination
Latex:
\mforall{}g:ProjectivePlane. \mforall{}l,m,n:Line. \mforall{}s:l \mneq{} m. \mforall{}s1:m \mneq{} n. (l \mwedge{} m \mneq{} n {}\mRightarrow{} m \mwedge{} n \mneq{} l)
Date html generated:
2018_05_22-PM-00_47_19
Last ObjectModification:
2017_11_20-PM-03_39_04
Theory : euclidean!plane!geometry
Home
Index