Nuprl Lemma : mset_prod_mem
∀g:DMon. ∀a,b:MSet{g↓set}. ∀u:|g|. u ∈b a × b = ∃b{g↓set} v ∈ a. ∃b{g↓set} w ∈ b. (u =b (v * w))
Proof
Definitions occuring in Statement :
mset_prod: a × b,
mset_for: mset_for,
mset_mem: mset_mem,
mset: MSet{s},
bool: 𝔹,
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
bor_mon: <𝔹,∨b>,
dset_of_mon: g↓set,
dmon: DMon,
grp_op: *,
grp_eq: =b,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
mset_prod: a × b,
member: t ∈ T,
uall: ∀[x:A]. B[x],
dmon: DMon,
mon: Mon,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
so_apply: x[s],
bor_mon: <𝔹,∨b>,
grp_car: |g|,
abmonoid: AbMon,
true: True,
infix_ap: x f y,
implies: P ⇒ Q,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
set_eq: =b,
pi2: snd(t)
Lemmas referenced :
grp_car_wf,
mset_wf,
dset_of_mon_wf,
dmon_wf,
equal_wf,
squash_wf,
true_wf,
bool_wf,
mset_mem_mon_for_union,
mset_for_wf,
mset_union_mon_wf,
mset_inj_wf_f,
infix_ap_wf,
set_car_wf,
dset_of_mon_wf0,
grp_op_wf,
bor_mon_wf,
abmonoid_subtype_iabmonoid,
grp_eq_wf,
abmonoid_wf,
mset_for_functionality,
mset_mem_wf,
mset_inj_wf,
assert_wf,
set_eq_wf,
mset_mem_char,
mset_for_mset_inj,
iff_weakening_equal,
grp_eq_sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
sqequalRule,
functionEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
independent_isectElimination,
productElimination
Latex:
\mforall{}g:DMon. \mforall{}a,b:MSet\{g\mdownarrow{}set\}. \mforall{}u:|g|. u \mmember{}\msubb{} a \mtimes{} b = \mexists{}\msubb{}\{g\mdownarrow{}set\} v \mmember{} a. \mexists{}\msubb{}\{g\mdownarrow{}set\} w \mmember{} b. (u =\msubb{} (v * w))
Date html generated:
2017_10_01-AM-10_00_54
Last ObjectModification:
2017_03_03-PM-01_03_18
Theory : mset
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