Nuprl Lemma : p-sep-or
∀[p:ℕ+]. ∀[x:p-adics(p)]. ∀y:p-adics(p). (p-sep(x;y) ⇒ (∀z:p-adics(p). (p-sep(z;x) ∨ p-sep(z;y))))
Proof
Definitions occuring in Statement :
p-sep: p-sep(x;y),
p-adics: p-adics(p),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
p-sep: p-sep(x;y),
exists: ∃x:A. B[x],
member: t ∈ T,
p-adics: p-adics(p),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
not: ¬A,
false: False,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
and: P ∧ Q,
guard: {T}
Lemmas referenced :
decidable__equal_int,
int_seg_wf,
exp_wf2,
nat_plus_subtype_nat,
p-adics_wf,
p-sep_wf,
nat_plus_wf,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
equal_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
applyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
lambdaEquality,
isectElimination,
natural_numberEquality,
sqequalRule,
because_Cache,
unionElimination,
inlFormation,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
inrFormation
Latex:
\mforall{}[p:\mBbbN{}\msupplus{}]. \mforall{}[x:p-adics(p)]. \mforall{}y:p-adics(p). (p-sep(x;y) {}\mRightarrow{} (\mforall{}z:p-adics(p). (p-sep(z;x) \mvee{} p-sep(z;y))))
Date html generated:
2018_05_21-PM-03_23_15
Last ObjectModification:
2018_05_19-AM-08_21_31
Theory : rings_1
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