Nuprl Lemma : sum-reindex2
∀[n:ℕ]. ∀[a:ℕn ⟶ ℤ]. ∀[m:ℕ]. ∀[b:ℕm ⟶ ℤ]. ∀[f:{i:ℕn| ¬(a[i] = 0 ∈ ℤ)} ⟶ {j:ℕm| ¬(b[j] = 0 ∈ ℤ)} ].
(Σ(a[i] | i < n) = Σ(b[j] | j < m) ∈ ℤ) supposing
((∀i:{i:ℕn| ¬(a[i] = 0 ∈ ℤ)} . (a[i] = b[f i] ∈ ℤ)) and
Bij({i:ℕn| ¬(a[i] = 0 ∈ ℤ)} ;{j:ℕm| ¬(b[j] = 0 ∈ ℤ)} ;f))
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
biject: Bij(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
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,
prop: ℙ,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
and: P ∧ Q
Lemmas referenced :
sum-reindex,
nat_wf,
biject_wf,
equal_wf,
equal-wf-T-base,
not_wf,
int_seg_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
intEquality,
applyEquality,
baseClosed,
sqequalRule,
lambdaEquality,
lambdaFormation,
dependent_set_memberEquality,
because_Cache,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
independent_isectElimination,
dependent_pairFormation,
independent_pairFormation,
productEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[m:\mBbbN{}]. \mforall{}[b:\mBbbN{}m {}\mrightarrow{} \mBbbZ{}]. \mforall{}[f:\{i:\mBbbN{}n| \mneg{}(a[i] = 0)\} {}\mrightarrow{} \{j:\mBbbN{}m| \mneg{}(b[j] = 0)\} ].
(\mSigma{}(a[i] | i < n) = \mSigma{}(b[j] | j < m)) supposing
((\mforall{}i:\{i:\mBbbN{}n| \mneg{}(a[i] = 0)\} . (a[i] = b[f i])) and
Bij(\{i:\mBbbN{}n| \mneg{}(a[i] = 0)\} ;\{j:\mBbbN{}m| \mneg{}(b[j] = 0)\} ;f))
Date html generated:
2016_05_15-PM-07_23_37
Last ObjectModification:
2016_01_16-AM-09_41_45
Theory : general
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