Nuprl Lemma : is-nat-trans

`∀[C,D:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[trans:A:cat-ob(C) ⟶ (cat-arrow(D) (F A) (G A))].`
`  trans ∈ nat-trans(C;D;F;G) `
`  supposing ∀A,B:cat-ob(C). ∀g:cat-arrow(C) A B.`
`              ((cat-comp(D) (F A) (G A) (G B) (trans A) (G A B g))`
`              = (cat-comp(D) (F A) (F B) (G B) (F A B g) (trans B))`
`              ∈ (cat-arrow(D) (F A) (G B)))`

Proof

Definitions occuring in Statement :  nat-trans: `nat-trans(C;D;F;G)` functor-arrow: `arrow(F)` functor-ob: `ob(F)` cat-functor: `Functor(C1;C2)` cat-comp: `cat-comp(C)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` 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` nat-trans: `nat-trans(C;D;F;G)` all: `∀x:A. B[x]`
Lemmas referenced :  cat-ob_wf cat-arrow_wf functor-ob_wf cat-comp_wf functor-arrow_wf cat-functor_wf small-category_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut hypothesis dependent_set_memberEquality_alt hypothesisEquality sqequalRule functionIsType universeIsType extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache applyEquality equalityIstype axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality_alt isectIsTypeImplies inhabitedIsType

Latex:
\mforall{}[C,D:SmallCategory].  \mforall{}[F,G:Functor(C;D)].  \mforall{}[trans:A:cat-ob(C)  {}\mrightarrow{}  (cat-arrow(D)  (F  A)  (G  A))].
trans  \mmember{}  nat-trans(C;D;F;G)
supposing  \mforall{}A,B:cat-ob(C).  \mforall{}g:cat-arrow(C)  A  B.
((cat-comp(D)  (F  A)  (G  A)  (G  B)  (trans  A)  (G  A  B  g))
=  (cat-comp(D)  (F  A)  (F  B)  (G  B)  (F  A  B  g)  (trans  B)))

Date html generated: 2019_10_31-AM-07_24_02
Last ObjectModification: 2018_12_13-AM-09_45_27

Theory : small!categories

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