Nuprl Lemma : rng_of_alg_wf2
∀a:CRng. ∀m:algebra{i:l}(a). (m↓rg ∈ Rng)
Proof
Definitions occuring in Statement :
algebra: algebra{i:l}(A),
rng_of_alg: a↓rg,
all: ∀x:A. B[x],
member: t ∈ T,
crng: CRng,
rng: Rng
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
crng: CRng,
rng: Rng,
algebra: algebra{i:l}(A),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
module: A-Module,
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
ring_p: IsRing(T;plus;zero;neg;times;one),
rng_of_alg: a↓rg,
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
rng_minus: -r,
rng_times: *,
rng_one: 1,
cand: A c∧ B
Lemmas referenced :
algebra_wf,
crng_wf,
algebra_properties,
set_wf,
module_wf,
monoid_p_wf,
alg_car_wf,
rng_car_wf,
alg_times_wf,
alg_one_wf,
bilinear_wf,
alg_plus_wf,
all_wf,
dist_1op_2op_lr_wf,
alg_act_wf,
module_properties,
algebra_sig_wf,
group_p_wf,
alg_zero_wf,
alg_minus_wf,
comm_wf,
action_p_wf,
rng_times_wf,
rng_one_wf,
bilinear_p_wf,
rng_plus_wf,
rng_of_alg_wf,
ring_p_wf,
rng_zero_wf,
rng_minus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
applyEquality,
lambdaEquality,
productElimination,
instantiate,
isectElimination,
sqequalRule,
productEquality,
because_Cache,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation
Latex:
\mforall{}a:CRng. \mforall{}m:algebra\{i:l\}(a). (m\mdownarrow{}rg \mmember{} Rng)
Date html generated:
2016_05_16-AM-07_28_11
Last ObjectModification:
2015_12_28-PM-05_08_37
Theory : algebras_1
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