Nuprl Lemma : unordered-combination_functionality
∀[A,B:Type]. ∀n,m:ℕ. (A ~ B ⇒ UnorderedCombination(n;A) ~ UnorderedCombination(m;B) supposing n = m ∈ ℤ)
Proof
Definitions occuring in Statement :
unordered-combination: UnorderedCombination(n;T),
equipollent: A ~ B,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
equipollent: A ~ B,
exists: ∃x:A. B[x],
prop: ℙ,
nat: ℕ,
unordered-combination: UnorderedCombination(n;T),
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
top: Top,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
biject: Bij(A;B;f),
inject: Inj(A;B;f),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
surject: Surj(A;B;f),
compose: f o g,
bag-map: bag-map(f;bs)
Lemmas referenced :
equal_wf,
equipollent_wf,
nat_wf,
bag-map_wf,
bag-map-no-repeats,
bag-size-map,
bag-no-repeats_wf,
bag-size_wf,
unordered-combination_wf,
inject_wf,
subtype_base_sq,
int_subtype_base,
biject-inverse,
squash_wf,
true_wf,
iff_weakening_equal,
biject_wf,
bag_wf,
bag-map-map,
bag-map-trivial,
map-id,
bag-subtype-list
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
sqequalHypSubstitution,
productElimination,
extract_by_obid,
isectElimination,
intEquality,
setElimination,
hypothesisEquality,
cumulativity,
universeEquality,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
independent_isectElimination,
independent_pairFormation,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
lambdaEquality,
functionEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
instantiate,
independent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[A,B:Type].
\mforall{}n,m:\mBbbN{}. (A \msim{} B {}\mRightarrow{} UnorderedCombination(n;A) \msim{} UnorderedCombination(m;B) supposing n = m)
Date html generated:
2017_10_01-AM-09_05_27
Last ObjectModification:
2017_07_26-PM-04_45_40
Theory : bags
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