Nuprl Lemma : grp_of_module_wf2
∀a:RngSig. ∀m:a-Module. (m↓grp ∈ AbGrp)
Proof
Definitions occuring in Statement :
module: A-Module,
grp_of_module: m↓grp,
all: ∀x:A. B[x],
member: t ∈ T,
rng_sig: RngSig,
abgrp: AbGrp
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
module: A-Module,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
group_p: IsGroup(T;op;id;inv),
monoid_p: IsMonoid(T;op;id),
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
grp_of_module: m↓grp,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
grp_id: e,
rng_of_alg: a↓rg,
rng_car: |r|,
rng_plus: +r,
rng_zero: 0,
grp_inv: ~,
rng_minus: -r
Lemmas referenced :
module_wf,
rng_sig_wf,
module_properties,
set_wf,
algebra_sig_wf,
rng_car_wf,
group_p_wf,
alg_car_wf,
alg_plus_wf,
alg_zero_wf,
alg_minus_wf,
comm_wf,
action_p_wf,
rng_times_wf,
rng_one_wf,
alg_act_wf,
bilinear_p_wf,
rng_plus_wf,
grp_of_module_wf,
monoid_p_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
productElimination,
instantiate,
isectElimination,
sqequalRule,
productEquality,
because_Cache,
cumulativity,
universeEquality,
dependent_set_memberEquality,
independent_pairFormation
Latex:
\mforall{}a:RngSig. \mforall{}m:a-Module. (m\mdownarrow{}grp \mmember{} AbGrp)
Date html generated:
2016_05_16-AM-07_26_43
Last ObjectModification:
2015_12_28-PM-05_08_31
Theory : algebras_1
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