### Nuprl Lemma : nat-trans-assoc-equation

`∀[C,D:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[T:nat-trans(C;D;F;G)]. ∀[A,B,B':cat-ob(C)]. ∀[g:cat-arrow(C) A B].`
`∀[h:cat-arrow(C) B B'].`
`  ((cat-comp(D) (ob(F) A) (ob(G) B) (ob(G) B') (cat-comp(D) (ob(F) A) (ob(F) B) (ob(G) B) (arrow(F) A B g) (T B)) `
`    (arrow(G) B B' h))`
`  = (cat-comp(D) (ob(F) A) (ob(F) B') (ob(G) B') `
`     (cat-comp(D) (ob(F) A) (ob(F) B) (ob(F) B') (arrow(F) A B g) (arrow(F) B B' h)) `
`     (T B'))`
`  ∈ (cat-arrow(D) (ob(F) A) (ob(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` uall: `∀[x:A]. B[x]` apply: `f a` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat-trans: `nat-trans(C;D;F;G)` true: `True` label: `...\$L... t` squash: `↓T` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` prop: `ℙ`
Lemmas referenced :  nat-trans-equation cat-comp_wf cat-arrow_wf cat-ob_wf nat-trans_wf cat-functor_wf small-category_wf functor-ob_wf functor-arrow_wf equal_wf cat-comp-assoc iff_weakening_equal squash_wf true_wf functor-arrow-comp
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality hypothesis because_Cache setElimination rename natural_numberEquality equalitySymmetry lambdaEquality imageElimination dependent_functionElimination sqequalRule imageMemberEquality baseClosed equalityTransitivity independent_isectElimination productElimination independent_functionElimination universeEquality

Latex:
\mforall{}[C,D:SmallCategory].  \mforall{}[F,G:Functor(C;D)].  \mforall{}[T:nat-trans(C;D;F;G)].  \mforall{}[A,B,B':cat-ob(C)].
\mforall{}[g:cat-arrow(C)  A  B].  \mforall{}[h:cat-arrow(C)  B  B'].
((cat-comp(D)  (ob(F)  A)  (ob(G)  B)  (ob(G)  B')
(cat-comp(D)  (ob(F)  A)  (ob(F)  B)  (ob(G)  B)  (arrow(F)  A  B  g)  (T  B))
(arrow(G)  B  B'  h))
=  (cat-comp(D)  (ob(F)  A)  (ob(F)  B')  (ob(G)  B')
(cat-comp(D)  (ob(F)  A)  (ob(F)  B)  (ob(F)  B')  (arrow(F)  A  B  g)  (arrow(F)  B  B'  h))
(T  B')))

Date html generated: 2017_10_05-AM-00_46_02
Last ObjectModification: 2017_07_28-AM-09_19_13

Theory : small!categories

Home Index