Nuprl Lemma : rv-disjoint-rv-add
∀p:FinProbSpace. ∀n:ℕ. ∀X,Y,Z:RandomVariable(p;n).
(rv-disjoint(p;n;X;Y) ⇒ rv-disjoint(p;n;X;Z) ⇒ rv-disjoint(p;n;X;Y + Z))
Proof
Definitions occuring in Statement :
rv-disjoint: rv-disjoint(p;n;X;Y),
rv-add: X + Y,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-disjoint: rv-disjoint(p;n;X;Y),
member: t ∈ T,
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
so_apply: x[s],
subtype_rel: A ⊆r B,
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
guard: {T},
rv-add: X + Y,
squash: ↓T,
true: True,
nat: ℕ
Lemmas referenced :
all_wf,
p-outcome_wf,
not_wf,
equal_wf,
rationals_wf,
rv-add_wf,
qadd_wf,
int_seg_wf,
rv-disjoint_wf,
random-variable_wf,
nat_wf,
finite-prob-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
unionElimination,
inlFormation,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
sqequalRule,
lambdaEquality,
intEquality,
setElimination,
rename,
applyEquality,
functionExtensionality,
inrFormation,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}p:FinProbSpace. \mforall{}n:\mBbbN{}. \mforall{}X,Y,Z:RandomVariable(p;n).
(rv-disjoint(p;n;X;Y) {}\mRightarrow{} rv-disjoint(p;n;X;Z) {}\mRightarrow{} rv-disjoint(p;n;X;Y + Z))
Date html generated:
2018_05_22-AM-00_35_08
Last ObjectModification:
2017_07_26-PM-07_00_02
Theory : randomness
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