Nuprl Lemma : groupoid-square-commutes-iff

[G:Groupoid]. ∀[x,y1,y2,z:cat-ob(cat(G))]. ∀[x_y1:cat-arrow(cat(G)) y1]. ∀[y1_z:cat-arrow(cat(G)) y1 z].
[x_y2:cat-arrow(cat(G)) y2]. ∀[y2_z:cat-arrow(cat(G)) y2 z].
  uiff(x_y1 y1_z x_y2 y2_z;y2_z
  (cat-comp(cat(G)) y2 groupoid-inv(G;x;y2;x_y2) (cat-comp(cat(G)) y1 x_y1 y1_z))
  ∈ (cat-arrow(cat(G)) y2 z))


Proof




Definitions occuring in Statement :  groupoid-inv: groupoid-inv(G;x;y;x_y) groupoid-cat: cat(G) groupoid: Groupoid cat-square-commutes: x_y1 y1_z x_y2 y2_z cat-comp: cat-comp(C) cat-arrow: cat-arrow(C) cat-ob: cat-ob(C) uiff: uiff(P;Q) uall: [x:A]. B[x] apply: a equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uiff: uiff(P;Q) and: P ∧ Q uimplies: supposing a cat-square-commutes: x_y1 y1_z x_y2 y2_z prop: true: True squash: T all: x:A. B[x] subtype_rel: A ⊆B guard: {T} iff: ⇐⇒ Q rev_implies:  Q implies:  Q
Lemmas referenced :  equal_wf cat-arrow_wf groupoid-cat_wf cat-comp_wf groupoid-inv_wf cat-square-commutes_wf cat-ob_wf groupoid_wf cat-comp-ident1 iff_weakening_equal squash_wf true_wf cat-comp-assoc groupoid_inv
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation sqequalHypSubstitution hypothesis thin hyp_replacement equalitySymmetry applyLambdaEquality extract_by_obid isectElimination applyEquality hypothesisEquality because_Cache sqequalRule axiomEquality productElimination independent_pairEquality isect_memberEquality equalityTransitivity natural_numberEquality lambdaEquality imageElimination dependent_functionElimination imageMemberEquality baseClosed independent_isectElimination independent_functionElimination universeEquality

Latex:
\mforall{}[G:Groupoid].  \mforall{}[x,y1,y2,z:cat-ob(cat(G))].  \mforall{}[x$_{y1}$:cat-arrow(cat(G))  x  y1].  \000C\mforall{}[y1$_{z}$:cat-arrow(cat(G)) 
                                                                                                                                                                  y1 
                                                                                                                                                                  z].
\mforall{}[x$_{y2}$:cat-arrow(cat(G))  x  y2].  \mforall{}[y2$_{z}$:cat-arrow(cat\000C(G))  y2  z].
    uiff(x$_{y1}$  o  y1$_{z}$  =  x$_{y2}$  o  \000Cy2$_{z}$;y2$_{z}$
    =  (cat-comp(cat(G))  y2  x  z  groupoid-inv(G;x;y2;x$_{y2}$)  (cat-comp(cat(G))  x  y\000C1  z  x$_{y1}$  y1$_{z}$)))



Date html generated: 2020_05_20-AM-07_55_55
Last ObjectModification: 2017_07_28-AM-09_20_16

Theory : small!categories


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