Nuprl Lemma : nsgrp_of_ideal_wf
∀[r:CRng]. ∀[a:Ideal(r){i}]. (a↓+nsgp ∈ NormSubGrp{i}(r↓+gp))
Proof
Definitions occuring in Statement :
nsgrp_of_ideal: a↓+nsgp,
ideal: Ideal(r){i},
add_grp_of_rng: r↓+gp,
crng: CRng,
norm_subgrp: NormSubGrp{i}(g),
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ideal: Ideal(r){i},
norm_subgrp: NormSubGrp{i}(g),
nsgrp_of_ideal: a↓+nsgp,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
and: P ∧ Q,
cand: A c∧ B,
crng: CRng,
rng: Rng,
prop: ℙ,
subgrp_p: s SubGrp of g,
grp_id: e,
pi2: snd(t),
grp_inv: ~,
grp_op: *,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
ideal_p: S Ideal of R,
norm_subset_p: norm_subset_p(g;s),
infix_ap: x f y,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
subgrp_p_wf,
add_grp_of_rng_wf,
norm_subset_p_wf,
ideal_wf,
crng_wf,
rng_car_wf,
rng_plus_wf,
rng_minus_wf,
rng_plus_comm,
rng_plus_ac_1,
rng_plus_inv_assoc,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
sqequalRule,
hypothesisEquality,
independent_pairFormation,
hypothesis,
productEquality,
lemma_by_obid,
isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
lambdaFormation,
applyEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality,
universeEquality,
independent_isectElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[a:Ideal(r)\{i\}]. (a\mdownarrow{}+nsgp \mmember{} NormSubGrp\{i\}(r\mdownarrow{}+gp))
Date html generated:
2016_05_15-PM-00_23_27
Last ObjectModification:
2015_12_27-AM-00_00_44
Theory : rings_1
Home
Index