Nuprl Lemma : algebra_act_times_lr
∀A:Rng. ∀m:algebra{i:l}(A). ∀a,b:|A|. ∀u,v:m.car.
(((a * b) m.act (u m.times v)) = ((a m.act u) m.times (b m.act v)) ∈ m.car)
Proof
Definitions occuring in Statement :
algebra: algebra{i:l}(A),
alg_act: a.act,
alg_times: a.times,
alg_car: a.car,
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
rng: Rng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
rng: Rng,
algebra: algebra{i:l}(A),
module: A-Module,
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
infix_ap: x f y,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
alg_car_wf,
rng_car_wf,
algebra_wf,
rng_wf,
equal_wf,
squash_wf,
true_wf,
module_action_p,
alg_times_wf,
infix_ap_wf,
alg_act_wf,
iff_weakening_equal,
algebra_act_times_r
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
hypothesisEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}A:Rng. \mforall{}m:algebra\{i:l\}(A). \mforall{}a,b:|A|. \mforall{}u,v:m.car.
(((a * b) m.act (u m.times v)) = ((a m.act u) m.times (b m.act v)))
Date html generated:
2017_10_01-AM-09_51_55
Last ObjectModification:
2017_03_03-PM-00_46_34
Theory : algebras_1
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