Nuprl Lemma : prod-ss-eq
∀[ss1,ss2:SeparationSpace]. ∀[x,y:Point(ss1 × ss2)]. uiff(x ≡ y;fst(x) ≡ fst(y) ∧ snd(x) ≡ snd(y))
Proof
Definitions occuring in Statement :
prod-ss: ss1 × ss2,
ss-eq: x ≡ y,
ss-point: Point(ss),
separation-space: SeparationSpace,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q
Definitions unfolded in proof :
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
or: P ∨ Q,
false: False,
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
top: Top,
member: t ∈ T,
ss-eq: x ≡ y,
uall: ∀[x:A]. B[x]
Lemmas referenced :
separation-space_wf,
prod-ss_wf,
or_wf,
not_wf,
pi1_wf_top,
ss-point_wf,
pi2_wf,
ss-sep_wf,
prod-ss-point,
prod-ss-sep
Rules used in proof :
productEquality,
unionElimination,
dependent_functionElimination,
inrFormation,
independent_pairEquality,
productElimination,
lambdaEquality,
hypothesisEquality,
inlFormation,
independent_functionElimination,
lambdaFormation,
independent_pairFormation,
because_Cache,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalRule,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[ss1,ss2:SeparationSpace]. \mforall{}[x,y:Point(ss1 \mtimes{} ss2)]. uiff(x \mequiv{} y;fst(x) \mequiv{} fst(y) \mwedge{} snd(x) \mequiv{} snd(y))
Date html generated:
2018_07_29-AM-10_11_11
Last ObjectModification:
2018_07_03-PM-05_28_56
Theory : constructive!algebra
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