### Nuprl Lemma : left-right-inverse-unique

`∀[C:SmallCategory]. ∀[x,y:cat-ob(C)]. ∀[f:cat-arrow(C) x y]. ∀[g2:cat-arrow(C) y x].`
`  ∀[g1:cat-arrow(C) y x]. g1 = g2 ∈ (cat-arrow(C) y x) supposing fg1=1 supposing g2f=1`

Proof

Definitions occuring in Statement :  cat-inverse: `fg=1` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` cat-inverse: `fg=1` true: `True` squash: `↓T` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` all: `∀x:A. B[x]`
Lemmas referenced :  cat-inverse_wf cat-arrow_wf cat-ob_wf small-category_wf cat-comp_wf equal_wf squash_wf true_wf iff_weakening_equal cat-comp-ident cat-comp-assoc
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry applyEquality natural_numberEquality lambdaEquality imageElimination universeEquality imageMemberEquality baseClosed independent_isectElimination productElimination independent_functionElimination dependent_functionElimination

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[x,y:cat-ob(C)].  \mforall{}[f:cat-arrow(C)  x  y].  \mforall{}[g2:cat-arrow(C)  y  x].
\mforall{}[g1:cat-arrow(C)  y  x].  g1  =  g2  supposing  fg1=1  supposing  g2f=1

Date html generated: 2017_10_05-AM-00_45_41
Last ObjectModification: 2017_07_28-AM-09_19_02

Theory : small!categories

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