Nuprl Lemma : groupoid-cube-lemma-rev
∀[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]. ∀[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].
uiff(a o b = c o d;e o f = g o h) supposing 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],
and: P ∧ Q,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
prop: ℙ,
true: True,
implies: P ⇒ Q,
squash: ↓T,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
cat-square-commutes_wf,
groupoid-cat_wf,
cat-arrow_wf,
cat-ob_wf,
groupoid_wf,
groupoid-square-commutes-iff,
cat-square-commutes-sym,
equal_wf,
cat-comp_wf,
groupoid-inv_wf,
uiff_transitivity3,
squash_wf,
true_wf,
cat-comp-assoc,
groupoid-left-cancellation,
groupoid-right-cancellation,
iff_weakening_equal,
groupoid_inv,
cat-comp-ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
sqequalRule,
axiomEquality,
hypothesis,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productEquality,
applyEquality,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
independent_functionElimination,
lambdaEquality,
imageElimination,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed
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].
\mforall{}[i:cat-arrow(cat(G)) x001 x000]. \mforall{}[j:cat-arrow(cat(G)) x011 x010].
\mforall{}[k:cat-arrow(cat(G)) x111 x110]. \mforall{}[l:cat-arrow(cat(G)) x101 x100].
uiff(a o b = c o d;e o f = g o h)
supposing 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:
2020_05_20-AM-07_56_07
Last ObjectModification:
2017_07_28-AM-09_20_23
Theory : small!categories
Home
Index