Nuprl Lemma : groupoid-left-cancellation

`∀[G:Groupoid]. ∀[x,y,z:cat-ob(cat(G))]. ∀[a,b:cat-arrow(cat(G)) y z]. ∀[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) ∈ (cat-arrow(cat(G)) x z);a`
`  = b`
`  ∈ (cat-arrow(cat(G)) y 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: `f a` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` prop: `ℙ` true: `True` squash: `↓T` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` implies: `P `` 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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