Nuprl Lemma : combination_functionality
∀[A,B:Type]. ∀n,m:ℤ. (A ~ B ⇒ Combination(n;A) ~ Combination(m;B) supposing n = m ∈ ℤ)
Proof
Definitions occuring in Statement :
combination: Combination(n;T),
equipollent: A ~ B,
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 :
prop: ℙ,
guard: {T},
sq_type: SQType(T),
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
equipollent: A ~ B,
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
biject: Bij(A;B;f),
and: P ∧ Q,
surject: Surj(A;B;f),
inject: Inj(A;B;f),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
compose: f o g,
top: Top,
combination: Combination(n;T)
Lemmas referenced :
equipollent_wf,
equal-wf-base,
biject_wf,
int_subtype_base,
subtype_base_sq,
subtype_rel_wf,
combination_wf,
subtype_rel_self,
map_wf_combination,
biject-inverse,
equal_wf,
list_wf,
iff_weakening_equal,
true_wf,
squash_wf,
map_wf,
map-map,
top_wf,
subtype_rel_list,
map-id,
length_wf,
equal-wf-T-base,
no_repeats_wf
Rules used in proof :
universeEquality,
independent_functionElimination,
dependent_functionElimination,
intEquality,
instantiate,
sqequalRule,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
independent_isectElimination,
because_Cache,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
isectElimination,
extract_by_obid,
lambdaEquality,
dependent_pairFormation,
productElimination,
sqequalHypSubstitution,
rename,
thin,
hypothesis,
axiomEquality,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
equalityTransitivity,
imageElimination,
voidEquality,
voidElimination,
isect_memberEquality,
setElimination,
productEquality,
dependent_set_memberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}n,m:\mBbbZ{}. (A \msim{} B {}\mRightarrow{} Combination(n;A) \msim{} Combination(m;B) supposing n = m)
Date html generated:
2018_05_21-PM-08_08_13
Last ObjectModification:
2017_12_07-PM-06_24_23
Theory : general
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