Nuprl Lemma : module_act_minus_r
∀A:Rng. ∀m:A-Module. ∀a:|A|. ∀u:m.car. ((a m.act (m.minus u)) = (m.minus (a m.act u)) ∈ m.car)
Proof
Definitions occuring in Statement :
module: A-Module,
alg_act: a.act,
alg_minus: a.minus,
alg_car: a.car,
infix_ap: x f y,
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
rng: Rng,
module: A-Module,
grp_of_module: m↓grp,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
rng_of_alg: a↓rg,
rng_car: |r|,
grp_inv: ~,
pi2: snd(t),
rng_minus: -r,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
infix_ap: x f y,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
grp_op: *,
rng_plus: +r,
grp_id: e,
rng_zero: 0
Lemmas referenced :
alg_car_wf,
rng_car_wf,
module_wf,
rng_wf,
grp_eq_op_l,
grp_of_module_wf2,
grp_subtype_igrp,
abgrp_subtype_grp,
subtype_rel_transitivity,
abgrp_wf,
grp_wf,
igrp_wf,
alg_act_wf,
grp_inv_wf,
grp_of_module_wf,
grp_car_wf,
equal_wf,
squash_wf,
true_wf,
infix_ap_wf,
grp_op_wf,
grp_inverse,
iff_weakening_equal,
module_act_plus,
alg_minus_wf,
alg_zero_wf,
grp_id_wf,
module_act_zero_r
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
lambdaEquality,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}A:Rng. \mforall{}m:A-Module. \mforall{}a:|A|. \mforall{}u:m.car. ((a m.act (m.minus u)) = (m.minus (a m.act u)))
Date html generated:
2017_10_01-AM-09_51_51
Last ObjectModification:
2017_03_03-PM-00_46_44
Theory : algebras_1
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