Nuprl Lemma : double_sum_functionality
∀[n,m:ℕ]. ∀[f,g:ℕn ⟶ ℕm ⟶ ℤ].
sum(f[x;y] | x < n; y < m) = sum(g[x;y] | x < n; y < m) ∈ ℤ supposing ∀x:ℕn. ∀y:ℕm. (f[x;y] = g[x;y] ∈ ℤ)
Proof
Definitions occuring in Statement :
double_sum: sum(f[x; y] | x < n; y < m)
,
int_seg: {i..j-}
,
nat: ℕ
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
so_apply: x[s1;s2]
,
all: ∀x:A. B[x]
,
function: x:A ⟶ B[x]
,
natural_number: $n
,
int: ℤ
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
double_sum: sum(f[x; y] | x < n; y < m)
,
so_lambda: λ2x.t[x]
,
so_apply: x[s1;s2]
,
nat: ℕ
,
so_apply: x[s]
,
all: ∀x:A. B[x]
,
squash: ↓T
,
prop: ℙ
,
true: True
,
subtype_rel: A ⊆r B
,
guard: {T}
,
iff: P
⇐⇒ Q
,
and: P ∧ Q
,
rev_implies: P
⇐ Q
,
implies: P
⇒ Q
Lemmas referenced :
sum_functionality,
sum_wf,
int_seg_wf,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
because_Cache,
independent_isectElimination,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
intEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
instantiate,
productElimination,
independent_functionElimination,
Error :functionIsType,
Error :universeIsType,
isect_memberEquality,
axiomEquality,
Error :inhabitedIsType
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[f,g:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m {}\mrightarrow{} \mBbbZ{}].
sum(f[x;y] | x < n; y < m) = sum(g[x;y] | x < n; y < m) supposing \mforall{}x:\mBbbN{}n. \mforall{}y:\mBbbN{}m. (f[x;y] = g[x;y])
Date html generated:
2019_06_20-PM-02_29_49
Last ObjectModification:
2018_09_26-PM-06_05_52
Theory : num_thy_1
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