Nuprl Lemma : algebra_hom_properties
∀A:RngSig. ∀M,N:algebra_sig{i:l}(|A|). ∀f:algebra_hom(A; M; N).
(FunThru2op(M.car;N.car;M.times;N.times;f) ∧ ((f M.one) = N.one ∈ N.car))
Proof
Definitions occuring in Statement :
algebra_hom: algebra_hom(A; M; N),
alg_one: a.one,
alg_times: a.times,
alg_car: a.car,
algebra_sig: algebra_sig{i:l}(A),
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T,
rng_car: |r|,
rng_sig: RngSig,
fun_thru_2op: FunThru2op(A;B;opa;opb;f)
Definitions unfolded in proof :
all: ∀x:A. B[x],
algebra_hom: algebra_hom(A; M; N),
uall: ∀[x:A]. B[x],
member: t ∈ T,
module_hom: module_hom(A; M; N),
prop: ℙ,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
rng_sig_wf,
algebra_sig_wf,
algebra_hom_wf,
squash_wf,
sq_stable__equal,
sq_stable__fun_thru_2op,
alg_one_wf,
equal_wf,
alg_times_wf,
rng_car_wf,
alg_car_wf,
fun_thru_2op_wf,
sq_stable__and
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
lemma_by_obid,
isectElimination,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
isect_memberEquality,
applyEquality,
independent_functionElimination,
introduction,
because_Cache,
sqequalRule,
lambdaEquality,
axiomEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}A:RngSig. \mforall{}M,N:algebra\_sig\{i:l\}(|A|). \mforall{}f:algebra\_hom(A; M; N).
(FunThru2op(M.car;N.car;M.times;N.times;f) \mwedge{} ((f M.one) = N.one))
Date html generated:
2016_05_16-AM-07_28_08
Last ObjectModification:
2016_01_16-PM-09_59_53
Theory : algebras_1
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