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
Home
Index