Nuprl Lemma : quot_ring_wf
∀[r:CRng]. ∀[a:Ideal(r){i}]. ((∀x:|r|. SqStable(a x)) ⇒ (∀[d:detach_fun(|r|;a)]. (r / d ∈ CRng)))
Proof
Definitions occuring in Statement :
quot_ring: r / d,
ideal: Ideal(r){i},
crng: CRng,
rng_car: |r|,
detach_fun: detach_fun(T;A),
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
ideal: Ideal(r){i},
crng: CRng,
rng: Rng,
all: ∀x:A. B[x],
guard: {T},
detach_fun: detach_fun(T;A),
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
ring_p: IsRing(T;plus;zero;neg;times;one),
quot_ring: r / d,
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
rng_minus: -r,
rng_times: *,
rng_one: 1,
bilinear: BiLinear(T;pl;tm),
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
infix_ap: x f y,
ident: Ident(T;op;id),
assoc: Assoc(T;op),
inverse: Inverse(T;op;id;inv),
quot_ring_car: Carrier(r/d),
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
ideal_p: S Ideal of R,
subgrp_p: s SubGrp of g,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
grp_op: *,
grp_id: e,
comm: Comm(T;op)
Lemmas referenced :
detach_fun_properties,
rng_car_wf,
quot_ring_car_wf,
ideal_p_wf,
subtype_rel_self,
istype-assert,
quot_ring_sig,
detach_fun_wf,
sq_stable_wf,
ideal_wf,
crng_wf,
comm_wf,
rng_times_wf,
ring_p_wf,
rng_plus_wf,
rng_zero_wf,
rng_minus_wf,
rng_one_wf,
quotient-member-eq,
assert_wf,
ideal-detach-equiv,
rng_times_assoc,
iff_weakening_equal,
rng_times_one,
sq_stable_functionality,
rng_minus_over_plus,
rng_plus_assoc,
rng_plus_ac_1,
rng_plus_comm,
rng_plus_zero,
rng_minus_zero,
rng_plus_inv,
rng_times_over_plus,
rng_times_over_minus,
crng_times_ac_1,
crng_times_comm,
rng_plus_inv_assoc,
grp_car_wf,
add_grp_of_rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
because_Cache,
universeIsType,
sqequalRule,
functionIsType,
productIsType,
applyEquality,
instantiate,
universeEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
independent_pairFormation,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
productElimination,
independent_isectElimination,
equalityIstype,
sqequalBase,
independent_pairEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[a:Ideal(r)\{i\}]. ((\mforall{}x:|r|. SqStable(a x)) {}\mRightarrow{} (\mforall{}[d:detach\_fun(|r|;a)]. (r / d \mmember{} CRng)))
Date html generated:
2019_10_15-AM-10_33_46
Last ObjectModification:
2019_06_20-PM-06_43_59
Theory : rings_1
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