Nuprl Lemma : omral_alg_umap_tri_comm
∀g:OCMon. ∀a:CDRng. ∀n:algebra{i:l}(a). ∀f:|g| ⟶ n.car. ((alg_umap(n,f) o (λk.inj(k,1))) = f ∈ (|g| ⟶ n.car))
Proof
Definitions occuring in Statement :
omral_alg_umap: alg_umap(n,f),
omral_inj: inj(k,v),
algebra: algebra{i:l}(A),
alg_car: a.car,
compose: f o g,
all: ∀x:A. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
cdrng: CDRng,
rng_one: 1,
ocmon: OCMon,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
omral_alg_umap: alg_umap(n,f),
compose: f o g,
omralist: omral(g;r),
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
dset_list: s List,
set_prod: s × t,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
tlambda: λx:T. b[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
cdrng: CDRng,
crng: CRng,
rng: Rng,
algebra: algebra{i:l}(A),
module: A-Module,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
omon: OMon,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
infix_ap: x f y,
so_apply: x[s],
cand: A c∧ B,
rng_of_alg: a↓rg,
rng_car: |r|,
grp_car: |g|,
dset: DSet,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
finite_set: FiniteSet{s},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rng_mssum: rng_mssum,
top: Top,
rng_zero: 0,
abgrp: AbGrp,
grp: Group{i},
iabmonoid: IAbMonoid,
imon: IMonoid,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
grp_car_wf,
alg_car_wf,
rng_car_wf,
algebra_wf,
cdrng_wf,
ocmon_wf,
equal_wf,
squash_wf,
true_wf,
rng_mssum_functionality_wrt_equal,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
rng_of_alg_wf2,
set_car_wf,
dset_of_mon_wf0,
add_grp_of_rng_wf,
rng_of_alg_wf,
alg_act_wf,
lookup_wf,
oset_of_ocmon_wf0,
rng_zero_wf,
omral_inj_wf,
rng_one_wf,
omralist_wf,
dset_wf,
omral_dom_wf,
rng_eq_wf,
assert_of_rng_eq,
cdrng_subtype_drng,
null_mset_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
mset_inj_wf,
loset_wf,
omral_dom_inj,
finite_set_wf,
mset_mem_wf,
iff_weakening_equal,
uiff_transitivity,
equal-wf-T-base,
mset_for_null_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
module_over_triv_rng,
mset_for_mset_inj,
add_grp_of_rng_wf_b,
grp_sig_wf,
monoid_p_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
lookup_omral_inj,
mon_when_true,
assert_of_mon_eq,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
abdmonoid_wf,
dmon_wf,
module_action_p
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
functionExtensionality,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
functionEquality,
dependent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
because_Cache,
productEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
independent_functionElimination,
setEquality,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
cumulativity,
voidElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality,
voidEquality,
impliesFunctionality
Latex:
\mforall{}g:OCMon. \mforall{}a:CDRng. \mforall{}n:algebra\{i:l\}(a). \mforall{}f:|g| {}\mrightarrow{} n.car. ((alg\_umap(n,f) o (\mlambda{}k.inj(k,1))) = f)
Date html generated:
2017_10_01-AM-10_07_36
Last ObjectModification:
2017_03_03-PM-01_16_56
Theory : polynom_3
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