Nuprl Lemma : equal-p-adics
∀[p:ℕ+]. ∀[x,y:p-adics(p)]. uiff(x = y ∈ p-adics(p);x = y ∈ (ℕ+ ⟶ ℤ))
Proof
Definitions occuring in Statement :
p-adics: p-adics(p),
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
p-adics: p-adics(p),
squash: ↓T,
so_lambda: λ2x.t[x],
nat: ℕ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
respects-equality: respects-equality(S;T)
Lemmas referenced :
p-adics-subtype,
subtype_rel_dep_function,
nat_plus_wf,
int_seg_wf,
exp_wf2,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
equal_functionality_wrt_subtype_rel2,
decidable__lt,
istype-less_than,
int_seg_properties,
exp_wf_nat_plus,
respects-equality-function,
subtype-base-respects-equality,
set_subtype_base,
lelt_wf,
int_subtype_base,
eqmod_wf,
itermAdd_wf,
int_term_value_add_lemma,
p-adics_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_pairFormation,
setElimination,
rename,
applyLambdaEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality_alt,
natural_numberEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
inhabitedIsType,
because_Cache,
intEquality,
lambdaFormation_alt,
functionEquality,
equalityIstype,
functionExtensionality_alt,
applyEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
productIsType,
functionIsType,
addEquality
Latex:
\mforall{}[p:\mBbbN{}\msupplus{}]. \mforall{}[x,y:p-adics(p)]. uiff(x = y;x = y)
Date html generated:
2019_10_15-AM-10_34_19
Last ObjectModification:
2018_12_08-AM-11_57_07
Theory : rings_1
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