Nuprl Lemma : pgeo-minimum-order
∀pg:ProjectivePlane. ∀n:ℕ. (order(pg) = n ⇒ (n ≥ 2 ))
Proof
Definitions occuring in Statement :
pgeo-order: order(pg) = n,
projective-plane: ProjectivePlane,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
pgeo-order: order(pg) = n,
member: t ∈ T,
sq_exists: ∃x:A [B[x]],
prop: ℙ,
equipollent: A ~ B,
exists: ∃x:A. B[x],
biject: Bij(A;B;f),
and: P ∧ Q,
not: ¬A,
quotient: x,y:A//B[x; y],
false: False,
cand: A c∧ B,
pgeo-peq: a ≡ b,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
respects-equality: respects-equality(S;T),
inject: Inj(A;B;f),
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
pgeo-non-trivial-dual,
pgeo-order_wf,
istype-nat,
projective-plane_wf,
pgeo-incident_wf,
projective-plane-structure_subtype,
projective-plane-structure-complete_subtype,
projective-plane-subtype,
subtype_rel_transitivity,
projective-plane-structure-complete_wf,
projective-plane-structure_wf,
pgeo-primitives_wf,
pgeo-peq_wf,
quotient_wf,
pgeo-point_wf,
pgeo-order-equiv_rel,
respects-equality-quotient1,
respects-equality-set-trivial,
pgeo-psep_wf,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
int_seg_properties,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-le,
istype-less_than,
set_subtype_base,
lelt_wf,
int_subtype_base,
pgeo-three-points-axiom,
subtype_quotient
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
universeIsType,
hypothesis,
productElimination,
sqequalRule,
pertypeElimination,
promote_hyp,
independent_functionElimination,
voidElimination,
productIsType,
equalityIstype,
setIsType,
because_Cache,
isectElimination,
applyEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
instantiate,
independent_isectElimination,
setEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
unionElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
independent_pairFormation,
addEquality,
intEquality,
sqequalBase,
functionIsType
Latex:
\mforall{}pg:ProjectivePlane. \mforall{}n:\mBbbN{}. (order(pg) = n {}\mRightarrow{} (n \mgeq{} 2 ))
Date html generated:
2019_10_16-PM-02_14_20
Last ObjectModification:
2018_12_13-PM-04_35_33
Theory : euclidean!plane!geometry
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