Nuprl Lemma : quot_ring_car_subtype
∀[r:CRng]. ∀[a:Ideal(r){i}]. ((∀x:|r|. SqStable(a x)) ⇒ (∀[d:detach_fun(|r|;a)]. (|r| ⊆r Carrier(r/d))))
Proof
Definitions occuring in Statement :
quot_ring_car: Carrier(r/d),
ideal: Ideal(r){i},
crng: CRng,
rng_car: |r|,
detach_fun: detach_fun(T;A),
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
crng: CRng,
rng: Rng,
ideal: Ideal(r){i},
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
quot_ring_car: Carrier(r/d),
guard: {T},
detach_fun: detach_fun(T;A),
infix_ap: x f y,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
ideal_p: S Ideal of R,
subgrp_p: s SubGrp of g,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
pi1: fst(t)
Lemmas referenced :
rng_zero_wf,
iff_weakening_equal,
rng_plus_inv,
iff_wf,
ideal-detach-equiv,
quotient-member-eq,
rng_minus_wf,
rng_plus_wf,
assert_wf,
detach_fun_properties,
crng_wf,
ideal_wf,
sq_stable_wf,
all_wf,
detach_fun_wf,
rng_car_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lambdaEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
axiomEquality,
applyEquality,
dependent_functionElimination,
isect_memberEquality,
because_Cache,
independent_functionElimination,
independent_isectElimination,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
universeEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[a:Ideal(r)\{i\}].
((\mforall{}x:|r|. SqStable(a x)) {}\mRightarrow{} (\mforall{}[d:detach\_fun(|r|;a)]. (|r| \msubseteq{}r Carrier(r/d))))
Date html generated:
2016_05_15-PM-00_23_40
Last ObjectModification:
2016_01_15-AM-08_51_42
Theory : rings_1
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