Nuprl Lemma : groupoid-cube
∀[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)) x000 x001]. ∀[j:cat-arrow(cat(G)) x010 x011].
∀[k:cat-arrow(cat(G)) x110 x111]. ∀[l:cat-arrow(cat(G)) x100 x101].
(a o b = c o d supposing e o j = i o a ∧ f o k = j o b ∧ l o d = h o k ∧ i o c = g o l ∧ e o f = g o h
∧ e o j = i o a supposing a o b = c o d ∧ f o k = j o b ∧ l o d = h o k ∧ i o c = g o l ∧ e o f = g o h
∧ f o k = j o b supposing e o j = i o a ∧ a o b = c o d ∧ l o d = h o k ∧ i o c = g o l ∧ e o f = g o h
∧ l o d = h o k supposing e o j = i o a ∧ f o k = j o b ∧ a o b = c o d ∧ i o c = g o l ∧ e o f = g o h
∧ i o c = g o l supposing e o j = i o a ∧ f o k = j o b ∧ l o d = h o k ∧ a o b = c o d ∧ e o f = g o h
∧ e o f = g o h supposing e o j = i o a ∧ f o k = j o b ∧ l o d = h o k ∧ i o c = g o l ∧ a o b = c o d)
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),
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,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
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 :
groupoid-square-commutes-iff,
cat-square-commutes-sym,
groupoid-cat_wf,
cat-square-commutes_wf,
cat-comp_wf,
groupoid-inv_wf,
groupoid-square-commutes-iff2,
cat-arrow_wf,
cat-ob_wf,
groupoid_wf,
equal_wf,
uiff_transitivity3,
squash_wf,
true_wf,
cat-comp-assoc,
groupoid-right-cancellation,
groupoid-left-cancellation,
subtype_rel_self,
iff_weakening_equal,
groupoid_inv,
cat-comp-ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
applyEquality,
sqequalRule,
axiomEquality,
productEquality,
independent_pairFormation,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
natural_numberEquality,
independent_functionElimination,
lambdaEquality,
imageElimination,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
instantiate
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)) x000 x001]. \mforall{}[j:cat-arrow(cat(G)) x010 x011].
\mforall{}[k:cat-arrow(cat(G)) x110 x111]. \mforall{}[l:cat-arrow(cat(G)) x100 x101].
(a o b = c o d
supposing 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 \mwedge{} e o f = g o h
\mwedge{} e o j = i o a
supposing a o b = c o d \mwedge{} f o k = j o b \mwedge{} l o d = h o k \mwedge{} i o c = g o l \mwedge{} e o f = g o h
\mwedge{} f o k = j o b
supposing e o j = i o a \mwedge{} a o b = c o d \mwedge{} l o d = h o k \mwedge{} i o c = g o l \mwedge{} e o f = g o h
\mwedge{} l o d = h o k
supposing e o j = i o a \mwedge{} f o k = j o b \mwedge{} a o b = c o d \mwedge{} i o c = g o l \mwedge{} e o f = g o h
\mwedge{} i o c = g o l
supposing e o j = i o a \mwedge{} f o k = j o b \mwedge{} l o d = h o k \mwedge{} a o b = c o d \mwedge{} e o f = g o h
\mwedge{} e o f = g o h
supposing 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 \mwedge{} a o b = c o d)
Date html generated:
2018_05_22-PM-09_56_47
Last ObjectModification:
2018_05_20-PM-10_10_30
Theory : small!categories
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