Nuprl Lemma : sub-bag-filter2
∀T:Type. ∀p1,p2:T ⟶ 𝔹. ∀b,c:bag(T).
(sub-bag(T;b;[x∈c|p1[x]]) ⇒ sub-bag(T;b;[x∈c|p2[x]]) ⇒ sub-bag(T;b;[x∈c|p1[x] ∧b p2[x]]))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
band: p ∧b q,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
sub-bag-filter,
band_wf,
assert_of_band,
bag-member_wf,
sub-bag_wf,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
bag_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
productElimination,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
setEquality,
setElimination,
rename,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}T:Type. \mforall{}p1,p2:T {}\mrightarrow{} \mBbbB{}. \mforall{}b,c:bag(T).
(sub-bag(T;b;[x\mmember{}c|p1[x]]) {}\mRightarrow{} sub-bag(T;b;[x\mmember{}c|p2[x]]) {}\mRightarrow{} sub-bag(T;b;[x\mmember{}c|p1[x] \mwedge{}\msubb{} p2[x]]))
Date html generated:
2016_05_15-PM-02_45_22
Last ObjectModification:
2015_12_27-AM-09_37_27
Theory : bags
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