Nuprl Lemma : module_hom_is_grp_hom
∀A:Rng. ∀m,n:A-Module. ∀f:m.car ⟶ n.car. (module_hom_p(A; m; n; f) ⇒ IsMonHom{m↓grp,n↓grp}(f))
Proof
Definitions occuring in Statement :
module_hom_p: module_hom_p(a; m; n; f),
module: A-Module,
grp_of_module: m↓grp,
alg_car: a.car,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
rng: Rng,
monoid_hom_p: IsMonHom{M1,M2}(f)
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng: Rng,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
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|,
prop: ℙ,
module: A-Module,
module_hom_p: module_hom_p(a; m; n; f),
and: P ∧ Q,
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
grp_op: *,
pi2: snd(t),
rng_plus: +r
Lemmas referenced :
grp_hom_formation,
grp_of_module_wf2,
grp_subtype_igrp,
abgrp_subtype_grp,
subtype_rel_transitivity,
abgrp_wf,
grp_wf,
igrp_wf,
module_hom_p_wf,
alg_car_wf,
rng_car_wf,
module_wf,
rng_wf,
grp_car_wf,
grp_of_module_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
functionEquality,
productElimination
Latex:
\mforall{}A:Rng. \mforall{}m,n:A-Module. \mforall{}f:m.car {}\mrightarrow{} n.car. (module\_hom\_p(A; m; n; f) {}\mRightarrow{} IsMonHom\{m\mdownarrow{}grp,n\mdownarrow{}grp\}(f))
Date html generated:
2016_05_16-AM-07_27_09
Last ObjectModification:
2015_12_28-PM-05_07_48
Theory : algebras_1
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