Nuprl Lemma : prod_in_mset_prod
∀g:DMon. ∀a,b:MSet{g↓set}. ∀u,v:|g|. ((↑(u ∈b a)) ⇒ (↑(v ∈b b)) ⇒ (↑((u * v) ∈b a × b)))
Proof
Definitions occuring in Statement :
mset_prod: a × b,
mset_mem: mset_mem,
mset: MSet{s},
assert: ↑b,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
dset_of_mon: g↓set,
dmon: DMon,
grp_op: *,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
mset_prod: a × b,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
dmon: DMon,
mon: Mon,
tlambda: λx:T. b[x],
infix_ap: x f y,
subtype_rel: A ⊆r B,
grp_car: |g|,
so_lambda: λ2x.t[x],
mset_union_mon: <MSet{s},⋃,0>,
finite_set: FiniteSet{s},
so_apply: x[s],
mset: MSet{s},
quotient: x,y:A//B[x; y],
bool: 𝔹,
bor_mon: <𝔹,∨b>,
guard: {T},
uimplies: b supposing a,
monoid_hom: MonHom(M1,M2),
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
exists: ∃x:A. B[x],
cand: A c∧ B,
set_eq: =b,
pi2: snd(t)
Lemmas referenced :
assert_wf,
mset_mem_wf,
dset_of_mon_wf,
grp_car_wf,
mset_wf,
dmon_wf,
grp_op_wf,
subtype_rel_self,
set_car_wf,
dset_of_mon_wf0,
mset_for_wf,
mset_union_mon_wf,
abmonoid_subtype_iabmonoid,
mset_inj_wf_f,
finite_set_wf,
bor_mon_wf,
mset_inj_wf,
mon_subtype_grp_sig,
abmonoid_subtype_mon,
subtype_rel_transitivity,
abmonoid_wf,
mon_wf,
grp_sig_wf,
bool_wf,
mset_union_bor_mon_hom,
monoid_hom_p_wf,
assert_functionality_wrt_uiff,
dist_hom_over_mset_for,
mset_for_functionality,
bmsexists_char,
set_eq_wf,
mset_mem_char,
mset_for_mset_inj,
assert_of_mon_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
setElimination,
rename,
applyEquality,
lambdaEquality,
because_Cache,
instantiate,
independent_isectElimination,
dependent_set_memberEquality,
independent_functionElimination,
equalityTransitivity,
productElimination,
dependent_pairFormation,
independent_pairFormation,
productEquality
Latex:
\mforall{}g:DMon. \mforall{}a,b:MSet\{g\mdownarrow{}set\}. \mforall{}u,v:|g|. ((\muparrow{}(u \mmember{}\msubb{} a)) {}\mRightarrow{} (\muparrow{}(v \mmember{}\msubb{} b)) {}\mRightarrow{} (\muparrow{}((u * v) \mmember{}\msubb{} a \mtimes{} b)))
Date html generated:
2018_05_22-AM-07_45_56
Last ObjectModification:
2018_05_19-AM-08_31_14
Theory : mset
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