Nuprl Lemma : int_mod_ring_wf_field
∀[p:ℕ+]. int_mod_ring(p) ∈ Field{i} supposing prime(p)
Proof
Definitions occuring in Statement :
int_mod_ring: int_mod_ring(n),
prime: prime(a),
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
field: Field{i}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
field: Field{i},
subtype_rel: A ⊆r B,
crng: CRng,
rng: Rng,
prop: ℙ,
nat_plus: ℕ+,
int_mod_ring: int_mod_ring(n),
field_p: IsField(r),
rng_car: |r|,
pi1: fst(t),
rng_zero: 0,
pi2: snd(t),
rng_one: 1,
ring_divs: a | b in r,
rng_times: *,
infix_ap: x f y,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
int_mod: ℤ_n,
quotient: x,y:A//B[x; y],
false: False,
eqmod: a ≡ b mod m,
divides: b | a,
exists: ∃x:A. B[x],
prime: prime(a),
assoced: a ~ b,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
lelt: i ≤ j < k,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
int_mod_ring_wf,
cdrng_subtype_crng,
field_p_wf,
prime_wf,
nat_plus_wf,
nequal_wf,
int_mod_wf,
int-subtype-int_mod,
istype-int,
eqmod_wf,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
itermMinus_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
int_subtype_base,
set_subtype_base,
less_than_wf,
one_divs_any,
modulus-int_mod-nonzero,
gcd-reduce-prime,
modulus_wf_int_mod,
divisors_bound,
nat_plus_subtype_nat,
int_seg_subtype_nat_plus,
istype-false,
divides_wf,
int_seg_properties,
intformle_wf,
intformless_wf,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
le_wf,
quotient-member-eq,
eqmod_equiv_rel,
subtype_rel_self,
mod-eqmod,
eqmod_inversion,
squash_wf,
true_wf,
modulus_functionality_wrt_eqmod,
iff_weakening_equal,
multiply_wf_int_mod,
equal_wf,
istype-universe,
itermAdd_wf,
int_term_value_add_lemma,
subtract_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
universeIsType,
setElimination,
rename,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
because_Cache,
independent_pairFormation,
lambdaFormation_alt,
natural_numberEquality,
equalityIsType4,
baseClosed,
pertypeElimination,
productElimination,
productIsType,
inhabitedIsType,
independent_functionElimination,
dependent_pairFormation_alt,
minusEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
voidElimination,
baseApply,
closedConclusion,
intEquality,
applyLambdaEquality,
equalityIsType1,
pointwiseFunctionalityForEquality,
imageElimination,
imageMemberEquality,
instantiate,
universeEquality,
multiplyEquality,
equalityIsType3
Latex:
\mforall{}[p:\mBbbN{}\msupplus{}]. int\_mod\_ring(p) \mmember{} Field\{i\} supposing prime(p)
Date html generated:
2019_10_15-AM-11_38_23
Last ObjectModification:
2018_10_10-AM-11_03_17
Theory : general
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