Nuprl Lemma : bag-summation-equal2
∀[T:Type]. ∀[r:Rng]. ∀[f,g:T ⟶ |r|]. ∀[b,c:bag(T)].
Σ(x∈b). f[x] = Σ(x∈c). g[x] ∈ |r| supposing (∀x:T. (x ↓∈ b ⇒ (f[x] = g[x] ∈ |r|))) ∧ (b = c ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
comm: Comm(T;op),
cand: A c∧ B,
rng: Rng,
rng_sig: RngSig,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv)
Lemmas referenced :
rng_plus_comm,
rng_all_properties,
rng_properties,
bag-summation-equal,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
equal_wf,
bag-summation_wf,
all_wf,
bag-member_wf,
bag_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_pairFormation,
setElimination,
rename,
equalitySymmetry,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality,
equalityTransitivity,
productEquality,
functionEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[r:Rng]. \mforall{}[f,g:T {}\mrightarrow{} |r|]. \mforall{}[b,c:bag(T)].
\mSigma{}(x\mmember{}b). f[x] = \mSigma{}(x\mmember{}c). g[x] supposing (\mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (f[x] = g[x]))) \mwedge{} (b = c)
Date html generated:
2017_10_01-AM-09_01_37
Last ObjectModification:
2017_07_26-PM-04_42_59
Theory : bags
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