Nuprl Lemma : quot_ring_car_elim
∀[r:CRng]. ∀[a:Ideal(r){i}].
((∀x:|r|. SqStable(a x)) ⇒ (∀[d:detach_fun(|r|;a)]. ∀[u,v:|r|]. uiff(u = v ∈ Carrier(r/d);↑(d (u +r (-r v))))))
Proof
Definitions occuring in Statement :
quot_ring_car: Carrier(r/d),
ideal: Ideal(r){i},
crng: CRng,
rng_minus: -r,
rng_plus: +r,
rng_car: |r|,
detach_fun: detach_fun(T;A),
assert: ↑b,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
detach_fun: detach_fun(T;A),
infix_ap: x f y,
crng: CRng,
rng: Rng,
prop: ℙ,
subtype_rel: A ⊆r B,
ideal: Ideal(r){i},
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
quot_ring_car: Carrier(r/d),
quotient: x,y:A//B[x; y],
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
assert_witness,
rng_plus_wf,
rng_minus_wf,
equal_wf,
quot_ring_car_wf,
quot_ring_car_subtype,
assert_wf,
rng_car_wf,
detach_fun_wf,
all_wf,
sq_stable_wf,
ideal_wf,
crng_wf,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
and_wf,
member_wf,
quotient-member-eq,
det_ideal_defines_eqv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
lambdaEquality,
dependent_functionElimination,
pertypeElimination,
independent_isectElimination,
instantiate,
cumulativity,
natural_numberEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[a:Ideal(r)\{i\}].
((\mforall{}x:|r|. SqStable(a x)) {}\mRightarrow{} (\mforall{}[d:detach\_fun(|r|;a)]. \mforall{}[u,v:|r|]. uiff(u = v;\muparrow{}(d (u +r (-r v))))))
Date html generated:
2016_05_15-PM-00_24_25
Last ObjectModification:
2015_12_27-AM-00_00_32
Theory : rings_1
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