Nuprl Lemma : nat-trans-comp-equation
∀[C,D:SmallCategory]. ∀[F,G,H:Functor(C;D)]. ∀[t1:nat-trans(C;D;F;G)]. ∀[t2:nat-trans(C;D;G;H)]. ∀[A,B:cat-ob(C)].
∀[g:cat-arrow(C) A B].
((cat-comp(D) (ob(F) A) (ob(H) A) (ob(H) B) (cat-comp(D) (ob(F) A) (ob(G) A) (ob(H) A) (t1 A) (t2 A))
(arrow(H) A B g))
= (cat-comp(D) (ob(F) A) (ob(F) B) (ob(H) B) (arrow(F) A B g)
(cat-comp(D) (ob(F) B) (ob(G) B) (ob(H) B) (t1 B) (t2 B)))
∈ (cat-arrow(D) (ob(F) A) (ob(H) 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,
all: ∀x:A. B[x],
top: Top
Lemmas referenced :
nat-trans-equation,
trans-comp_wf,
trans_comp_ap_lemma,
cat-arrow_wf,
cat-ob_wf,
nat-trans_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
because_Cache
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F,G,H:Functor(C;D)]. \mforall{}[t1:nat-trans(C;D;F;G)]. \mforall{}[t2:nat-trans(C;D;G;H)].
\mforall{}[A,B:cat-ob(C)]. \mforall{}[g:cat-arrow(C) A B].
((cat-comp(D) (ob(F) A) (ob(H) A) (ob(H) B)
(cat-comp(D) (ob(F) A) (ob(G) A) (ob(H) A) (t1 A) (t2 A))
(arrow(H) A B g))
= (cat-comp(D) (ob(F) A) (ob(F) B) (ob(H) B) (arrow(F) A B g)
(cat-comp(D) (ob(F) B) (ob(G) B) (ob(H) B) (t1 B) (t2 B))))
Date html generated:
2017_01_19-PM-02_52_54
Last ObjectModification:
2017_01_11-PM-03_50_52
Theory : small!categories
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