Nuprl Lemma : pgeo-minimum-order-proof2
∀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 :
pgeo-peq: a ≡ b,
quotient: x,y:A//B[x; y],
true: True,
subtract: n - m,
iff: P ⇐⇒ Q,
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uiff: uiff(P;Q),
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
cand: A c∧ B,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
and: P ∧ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
pgeo-order: order(pg) = n,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
length_of_nil_lemma,
length_of_cons_lemma,
pgeo-psep-sym,
cons_member,
member_wf,
l_member_wf,
member_singleton,
int_formula_prop_less_lemma,
intformless_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
int-seg-cardinality-le,
int_seg_wf,
cardinality-le_functionality_wrt_equipollence,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
decidable__lt,
cardinality-le_functionality,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
no_repeats_singleton,
no_repeats_cons,
nil_wf,
cons_wf,
le_wf,
false_wf,
quotient_wf,
cardinality-le-no_repeats-length,
pgeo-order-equiv_rel,
pgeo-peq_wf,
pgeo-incident_wf,
pgeo-primitives_wf,
projective-plane-structure_subtype,
pgeo-point_wf,
subtype_quotient,
projective-plane-structure_wf,
projective-plane-structure-complete_wf,
subtype_rel_transitivity,
projective-plane-subtype,
projective-plane-structure-complete_subtype,
pgeo-three-points-axiom,
decidable__le,
projective-plane_wf,
nat_wf,
pgeo-order_wf,
pgeo-non-trivial-dual
Rules used in proof :
productEquality,
pertypeElimination,
minusEquality,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
addEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
dependent_set_memberEquality,
lambdaEquality,
because_Cache,
setEquality,
productElimination,
sqequalRule,
independent_isectElimination,
isectElimination,
instantiate,
applyEquality,
unionElimination,
natural_numberEquality,
hypothesis,
rename,
setElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}pg:ProjectivePlane. \mforall{}n:\mBbbN{}. (order(pg) = n {}\mRightarrow{} (n \mgeq{} 2 ))
Date html generated:
2018_05_22-PM-00_58_50
Last ObjectModification:
2018_05_21-AM-01_31_45
Theory : euclidean!plane!geometry
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