Nuprl Lemma : quot_ring_sig
∀[r:CRng]. ∀[a:Ideal(r){i}]. ((∀x:|r|. SqStable(a x)) ⇒ (∀[d:detach_fun(|r|;a)]. (r / d ∈ RngSig)))
Proof
Definitions occuring in Statement :
quot_ring: r / d,
ideal: Ideal(r){i},
crng: CRng,
rng_car: |r|,
rng_sig: RngSig,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
quot_ring: r / d,
rng_sig: RngSig,
quot_ring_car: Carrier(r/d),
quotient: x,y:A//B[x; y],
infix_ap: x f y,
uimplies: b supposing a,
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ideal_p: S Ideal of R,
subgrp_p: s SubGrp of g,
subtype_rel: A ⊆r B,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t)
Lemmas referenced :
detach_fun_properties,
rng_car_wf,
quot_ring_car_wf,
ideal_p_wf,
all_wf,
iff_wf,
assert_wf,
unit_wf2,
bool_wf,
detach_fun_wf,
sq_stable_wf,
ideal_wf,
crng_wf,
rng_plus_wf,
rng_minus_wf,
equal_wf,
equal-wf-base,
iff_imp_equal_bool,
ideal-detach-equiv,
btrue_wf,
quotient-member-eq,
rng_plus_assoc,
iff_weakening_equal,
infix_ap_wf,
rng_minus_over_plus,
rng_plus_comm,
rng_plus_ac_1,
rng_zero_wf,
quot_ring_car_subtype,
rng_one_wf,
rng_minus_minus,
rng_times_over_plus,
rng_times_over_minus,
rng_times_one,
rng_times_wf,
crng_times_comm,
rng_plus_inv_assoc,
it_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
dependent_set_memberEquality,
because_Cache,
functionExtensionality,
applyEquality,
sqequalRule,
lambdaEquality,
dependent_pairEquality,
functionEquality,
unionEquality,
productEquality,
cumulativity,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
allFunctionality,
promote_hyp,
universeEquality,
inrEquality
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{} RngSig)))
Date html generated:
2017_10_01-AM-08_17_54
Last ObjectModification:
2017_02_28-PM-02_03_58
Theory : rings_1
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