Nuprl Lemma : groupoid-cube-lemma2
∀[G:Groupoid]. ∀[x000,x100,x010,x110,x001,x101,x011,x111:cat-ob(cat(G))]. ∀[a:cat-arrow(cat(G)) x001 x011].
∀[b:cat-arrow(cat(G)) x011 x111]. ∀[c:cat-arrow(cat(G)) x001 x101]. ∀[d:cat-arrow(cat(G)) x101 x111].
∀[e:cat-arrow(cat(G)) x000 x010]. ∀[f:cat-arrow(cat(G)) x010 x110]. ∀[g:cat-arrow(cat(G)) x000 x100].
∀[h:cat-arrow(cat(G)) x100 x110].
uiff(a o b = c o d;e o f = g o h)
supposing (∃i:cat-arrow(cat(G)) x000 x001
∃j:cat-arrow(cat(G)) x010 x011
∃k:cat-arrow(cat(G)) x110 x111
∃l:cat-arrow(cat(G)) x100 x101. (e o j = i o a ∧ f o k = j o b ∧ l o d = h o k ∧ i o c = g o l))
∨ (∃i:cat-arrow(cat(G)) x001 x000
∃j:cat-arrow(cat(G)) x011 x010
∃k:cat-arrow(cat(G)) x111 x110
∃l:cat-arrow(cat(G)) x101 x100. (a o j = i o e ∧ b o k = j o f ∧ d o k = l o h ∧ i o g = c o l))
Proof
Definitions occuring in Statement :
groupoid-cat: cat(G),
groupoid: Groupoid,
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
apply: f a
Definitions unfolded in proof :
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
cand: A c∧ B
Lemmas referenced :
groupoid_wf,
cat-ob_wf,
cat-arrow_wf,
exists_wf,
or_wf,
groupoid-cat_wf,
cat-square-commutes_wf,
groupoid-cube-lemma,
groupoid-cube-lemma-rev
Rules used in proof :
productEquality,
because_Cache,
lambdaEquality,
sqequalRule,
applyEquality,
hypothesis,
hypothesisEquality,
isectElimination,
lemma_by_obid,
cut,
productElimination,
thin,
unionElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
isect_memberFormation,
introduction,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
independent_isectElimination
Latex:
\mforall{}[G:Groupoid]. \mforall{}[x000,x100,x010,x110,x001,x101,x011,x111:cat-ob(cat(G))].
\mforall{}[a:cat-arrow(cat(G)) x001 x011]. \mforall{}[b:cat-arrow(cat(G)) x011 x111].
\mforall{}[c:cat-arrow(cat(G)) x001 x101]. \mforall{}[d:cat-arrow(cat(G)) x101 x111].
\mforall{}[e:cat-arrow(cat(G)) x000 x010]. \mforall{}[f:cat-arrow(cat(G)) x010 x110].
\mforall{}[g:cat-arrow(cat(G)) x000 x100]. \mforall{}[h:cat-arrow(cat(G)) x100 x110].
uiff(a o b = c o d;e o f = g o h)
supposing (\mexists{}i:cat-arrow(cat(G)) x000 x001
\mexists{}j:cat-arrow(cat(G)) x010 x011
\mexists{}k:cat-arrow(cat(G)) x110 x111
\mexists{}l:cat-arrow(cat(G)) x100 x101
(e o j = i o a \mwedge{} f o k = j o b \mwedge{} l o d = h o k \mwedge{} i o c = g o l))
\mvee{} (\mexists{}i:cat-arrow(cat(G)) x001 x000
\mexists{}j:cat-arrow(cat(G)) x011 x010
\mexists{}k:cat-arrow(cat(G)) x111 x110
\mexists{}l:cat-arrow(cat(G)) x101 x100
(a o j = i o e \mwedge{} b o k = j o f \mwedge{} d o k = l o h \mwedge{} i o g = c o l))
Date html generated:
2016_05_18-AM-11_56_01
Last ObjectModification:
2015_12_28-PM-02_23_50
Theory : small!categories
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