### Nuprl Lemma : mk-groupoid_wf

`∀[C:SmallCategory]. ∀[inv:x:cat-ob(C) ⟶ y:cat-ob(C) ⟶ (cat-arrow(C) x y) ⟶ (cat-arrow(C) y x)].`
`  Groupoid(C;`
`           inv(x,y,f) = inv[x;y;f]) ∈ Groupoid `
`  supposing ∀x,y:cat-ob(C). ∀f:cat-arrow(C) x y.`
`              (((cat-comp(C) x y x f inv[x;y;f]) = (cat-id(C) x) ∈ (cat-arrow(C) x x))`
`              ∧ ((cat-comp(C) y x y inv[x;y;f] f) = (cat-id(C) y) ∈ (cat-arrow(C) y y)))`

Proof

Definitions occuring in Statement :  mk-groupoid: mk-groupoid groupoid: `Groupoid` cat-comp: `cat-comp(C)` cat-id: `cat-id(C)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2;s3]` all: `∀x:A. B[x]` and: `P ∧ Q` member: `t ∈ T` apply: `f a` function: `x:A ⟶ B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` mk-groupoid: mk-groupoid groupoid: `Groupoid` so_apply: `x[s1;s2;s3]` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` so_apply: `x[s]` all: `∀x:A. B[x]`
Lemmas referenced :  cat-ob_wf cat-arrow_wf all_wf equal_wf cat-comp_wf cat-id_wf small-category_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule dependent_pairEquality hypothesisEquality dependent_set_memberEquality lambdaEquality applyEquality functionExtensionality extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis because_Cache productEquality setEquality functionEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[inv:x:cat-ob(C)  {}\mrightarrow{}  y:cat-ob(C)  {}\mrightarrow{}  (cat-arrow(C)  x  y)  {}\mrightarrow{}  (cat-arrow(C)  y  x)].
Groupoid(C;
inv(x,y,f)  =  inv[x;y;f])  \mmember{}  Groupoid
supposing  \mforall{}x,y:cat-ob(C).  \mforall{}f:cat-arrow(C)  x  y.
(((cat-comp(C)  x  y  x  f  inv[x;y;f])  =  (cat-id(C)  x))
\mwedge{}  ((cat-comp(C)  y  x  y  inv[x;y;f]  f)  =  (cat-id(C)  y)))

Date html generated: 2017_10_05-AM-00_49_06
Last ObjectModification: 2017_07_28-AM-09_20_10

Theory : small!categories

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