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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