Nuprl Lemma : groupoid-left-cancellation

[G:Groupoid]. ∀[x,y,z:cat-ob(cat(G))]. ∀[a,b:cat-arrow(cat(G)) z]. ∀[c:cat-arrow(cat(G)) y].
  uiff((cat-comp(cat(G)) a) (cat-comp(cat(G)) b) ∈ (cat-arrow(cat(G)) z);a
  b
  ∈ (cat-arrow(cat(G)) z))


Proof




Definitions occuring in Statement :  groupoid-cat: cat(G) groupoid: Groupoid 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 prop: true: True squash: T all: x:A. B[x] subtype_rel: A ⊆B guard: {T} iff: ⇐⇒ Q implies:  Q
Lemmas referenced :  equal_wf cat-arrow_wf groupoid-cat_wf cat-comp_wf and_wf cat-ob_wf groupoid_wf groupoid-inv_wf squash_wf true_wf cat-comp-assoc groupoid_inv cat-comp-ident iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin applyEquality hypothesisEquality equalitySymmetry dependent_set_memberEquality applyLambdaEquality setElimination rename productElimination sqequalRule independent_pairEquality isect_memberEquality axiomEquality because_Cache equalityTransitivity natural_numberEquality lambdaEquality imageElimination universeEquality dependent_functionElimination imageMemberEquality baseClosed independent_isectElimination independent_functionElimination

Latex:
\mforall{}[G:Groupoid].  \mforall{}[x,y,z:cat-ob(cat(G))].  \mforall{}[a,b:cat-arrow(cat(G))  y  z].  \mforall{}[c:cat-arrow(cat(G))  x  y].
    uiff((cat-comp(cat(G))  x  y  z  c  a)  =  (cat-comp(cat(G))  x  y  z  c  b);a  =  b)



Date html generated: 2020_05_20-AM-07_55_52
Last ObjectModification: 2017_07_28-AM-09_20_14

Theory : small!categories


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