### Nuprl Lemma : mk-nat-trans_wf

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

Proof

Definitions occuring in Statement :  mk-nat-trans: `x |→ T[x]` nat-trans: `nat-trans(C;D;F;G)` functor-arrow: `functor-arrow(F)` functor-ob: `functor-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]` so_apply: `x[s]` all: `∀x:A. B[x]` member: `t ∈ T` apply: `f a` function: `x:A ⟶ B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  prop: `ℙ` so_lambda: `λ2x.t[x]` all: `∀x:A. B[x]` so_apply: `x[s]` mk-nat-trans: `x |→ T[x]` nat-trans: `nat-trans(C;D;F;G)` uimplies: `b supposing a` member: `t ∈ T` uall: `∀[x:A]. B[x]`
Lemmas referenced :  small-category_wf cat-functor_wf functor-arrow_wf cat-comp_wf functor-ob_wf equal_wf all_wf cat-arrow_wf cat-ob_wf
Rules used in proof :  functionEquality isect_memberEquality equalitySymmetry equalityTransitivity axiomEquality because_Cache dependent_functionElimination sqequalRule lambdaFormation hypothesis thin isectElimination sqequalHypSubstitution extract_by_obid hypothesisEquality functionExtensionality applyEquality lambdaEquality dependent_set_memberEquality cut introduction isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

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

Date html generated: 2017_01_11-AM-09_18_05
Last ObjectModification: 2017_01_10-PM-04_23_33

Theory : small!categories

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