### Nuprl Lemma : sp-join-meet-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-equal sp-join_wf sp-meet_wf sp-meet-is-top equal-wf-T-base Sierpinski_wf iff_wf equal-wf-base Sierpinski-equal2 Sierpinski-bottom_wf subtype-Sierpinski sp-join-is-bottom Sierpinski-unequal not-Sierpinski-top not-Sierpinski-bottom equal_wf iff_weakening_equal and_wf squash_wf true_wf sp-join-com sp-join-top
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  \mwedge{}  y  \mvee{}  z  =  x  \mvee{}  z  \mwedge{}  y  \mvee{}  z)

Date html generated: 2019_10_31-AM-07_18_34
Last ObjectModification: 2017_07_28-AM-09_12_18

Theory : synthetic!topology

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