Nuprl Lemma : module_eqfun_p
∀[A:Rng]. ∀[m:A-DModule]. ∀[x,y:m.car]. uiff(↑(x m.eq y);x = y ∈ m.car)
Proof
Definitions occuring in Statement :
dmodule: A-DModule,
alg_eq: a.eq,
alg_car: a.car,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
dmodule: A-DModule,
module: A-Module,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
rng: Rng,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
eqfun_p: IsEqFun(T;eq)
Lemmas referenced :
rng_wf,
eqfun_p_wf,
rng_plus_wf,
bilinear_p_wf,
alg_act_wf,
rng_one_wf,
rng_times_wf,
action_p_wf,
comm_wf,
alg_minus_wf,
alg_zero_wf,
alg_plus_wf,
group_p_wf,
algebra_sig_wf,
set_wf,
alg_car_wf,
equal_wf,
assert_witness,
decidable__assert,
sq_stable_from_decidable,
rng_car_wf,
alg_eq_wf,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
setElimination,
thin,
rename,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
instantiate,
setEquality,
productEquality,
lambdaEquality,
lambdaFormation,
cumulativity,
universeEquality,
independent_isectElimination
Latex:
\mforall{}[A:Rng]. \mforall{}[m:A-DModule]. \mforall{}[x,y:m.car]. uiff(\muparrow{}(x m.eq y);x = y)
Date html generated:
2016_05_16-AM-07_26_41
Last ObjectModification:
2016_01_16-PM-09_59_58
Theory : algebras_1
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