Nuprl Lemma : sp-meet-join-distrib
∀[x,y,z:Sierpinski]. (x ∨ y ∧ z = x ∧ z ∨ y ∧ z ∈ Sierpinski)
Proof
Definitions occuring in Statement :
sp-join: f ∨ g,
sp-meet: f ∧ g,
Sierpinski: Sierpinski,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
cand: A c∧ B,
squash: ↓T,
guard: {T},
true: True
Lemmas referenced :
Sierpinski-equal2,
sp-meet_wf,
sp-join_wf,
sp-join-is-bottom,
equal-wf-T-base,
Sierpinski_wf,
iff_wf,
equal-wf-base,
Sierpinski-equal,
Sierpinski-top_wf,
subtype-Sierpinski,
sp-meet-is-top,
Sierpinski-unequal,
not-Sierpinski-bottom,
not-Sierpinski-top,
equal_wf,
iff_weakening_equal,
and_wf,
squash_wf,
true_wf,
sp-meet-com,
sp-meet-bottom
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
addLevel,
independent_pairFormation,
impliesFunctionality,
independent_functionElimination,
productEquality,
baseClosed,
because_Cache,
andLevelFunctionality,
sqequalRule,
impliesLevelFunctionality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
isect_memberEquality,
axiomEquality,
lambdaFormation,
voidElimination,
promote_hyp,
lambdaEquality,
imageElimination,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
universeEquality
Latex:
\mforall{}[x,y,z:Sierpinski]. (x \mvee{} y \mwedge{} z = x \mwedge{} z \mvee{} y \mwedge{} z)
Date html generated:
2019_10_31-AM-07_18_37
Last ObjectModification:
2017_07_28-AM-09_12_19
Theory : synthetic!topology
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