### 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