Nuprl Lemma : omral_dom_inj
∀g:OCMon. ∀r:CDRng. ∀k:|g|. ∀v:|r|.
(dom(inj(k,v)) = if v =b 0 then 0{g↓oset} else mset_inj{g↓oset}(k) fi ∈ FiniteSet{g↓oset})
Proof
Definitions occuring in Statement :
omral_inj: inj(k,v),
omral_dom: dom(ps),
mset_inj: mset_inj{s}(x),
finite_set: FiniteSet{s},
null_mset: 0{s},
ifthenelse: if b then t else f fi ,
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
cdrng: CDRng,
rng_zero: 0,
rng_eq: =b,
rng_car: |r|,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
ocmon: OCMon,
omon: OMon,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
abmonoid: AbMon,
mon: Mon,
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,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
add_grp_of_rng: r↓+gp,
grp_car: |g|,
grp_eq: =b,
pi2: snd(t),
grp_id: e,
omral_inj: inj(k,v),
omral_dom: dom(ps)
Lemmas referenced :
oal_dom_inj,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
grp_car_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
equal_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
cdrng_is_abdmonoid,
cdrng_wf,
ocmon_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
hypothesis,
because_Cache,
lambdaEquality,
productEquality,
setElimination,
rename,
cumulativity,
universeEquality,
functionEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setEquality,
independent_pairFormation
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}k:|g|. \mforall{}v:|r|.
(dom(inj(k,v)) = if v =\msubb{} 0 then 0\{g\mdownarrow{}oset\} else mset\_inj\{g\mdownarrow{}oset\}(k) fi )
Date html generated:
2017_10_01-AM-10_05_21
Last ObjectModification:
2017_03_03-PM-01_11_22
Theory : polynom_3
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